@misc{curetonthornerQRH, title={A quasi-{R}iemann hypothesis for {D}edekind zeta functions}, key={hexagon:2610.00200v2}, howpublished={Hexagon preprint hexagon:2610.00200v2}, year={2026}, note={Contributors: Dean Cureton; Jesse Thorner}, url={https://hexagonmath.org/2610.00200v2} } @article {MR553223, AUTHOR = {Lagarias, J. C. and Montgomery, H. L. and Odlyzko, A. M.}, TITLE = {A bound for the least prime ideal in the {C}hebotarev density theorem}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {54}, YEAR = {1979}, NUMBER = {3}, PAGES = {271--296}, ISSN = {0020-9910,1432-1297}, MRCLASS = {12A75 (10H08)}, MRNUMBER = {553223}, MRREVIEWER = {Larry\ J.\ Goldstein}, DOI = {10.1007/BF01390234}, URL = {https://doi.org/10.1007/BF01390234}, } @misc{banks2026openaiprooflandausiegelzeros, title={On the {O}pen{AI} proof that {L}andau-{S}iegel zeros do not exist}, author={Banks, W.D.}, year={arXiv:2610.15483, 2026}, eprint={2610.15483}, archivePrefix={arXiv}, primaryClass={math.NT}, url={https://arxiv.org/abs/2610.15483}, } @misc{greensiegel, title={On {L}andau--{S}iegel zeros}, key={hexagon:2610.00311v2}, howpublished={Hexagon preprint hexagon:2610.00311v2}, year={2026}, note={Contributors: Ben Green}, url={https://hexagonmath.org/2610.00311v2} } @book {MR3822326, AUTHOR = {Marcus, D. A.}, TITLE = {Number fields}, SERIES = {Universitext}, EDITION = {Second}, NOTE = {With a foreword by Barry Mazur}, PUBLISHER = {Springer, Cham}, YEAR = {2018}, PAGES = {xviii+203}, ISBN = {978-3-319-90232-6; 978-3-319-90233-3}, MRCLASS = {11-01 (11Rxx 11Txx 12-01)}, MRNUMBER = {3822326}, DOI = {10.1007/978-3-319-90233-3}, URL = {https://doi.org/10.1007/978-3-319-90233-3}, } @article{zbMATH02163897, author = {Rosen, M.}, title = {A generalization of {Mertens}' theorem}, fjournal = {Journal of the Ramanujan Mathematical Society}, journal = {J. Ramanujan Math. Soc.}, issn = {0970-1249}, volume = {14}, number = {1}, pages = {1--19}, year = {1999}, language = {English}, keywords = {11R47,11R44,11R42}, zbMATH = {2163897}, Zbl = {1133.11317} } @misc{OAISiegel, author = {{OpenAI}}, title = {{Uniform exclusion of Landau--Siegel zeros}}, howpublished = {OpenAI Math Release preprint \href{https://github.com/openai/math/blob/main/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf}{OAI:Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026}}, year = {2026} } @article {MR608529, AUTHOR = {Masser, D. W.}, TITLE = {On polynomials and exponential polynomials in several complex variables}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {63}, YEAR = {1981}, NUMBER = {1}, PAGES = {81--95}, ISSN = {0020-9910,1432-1297}, MRCLASS = {10F35 (13A17)}, MRNUMBER = {608529}, MRREVIEWER = {Michel\ Waldschmidt}, DOI = {10.1007/BF01389194}, URL = {https://doi.org/10.1007/BF01389194}, } @misc{kerr, title={A simpler proof that there are no {L}andau--{S}iegel zeros}, author={Kerr, B.}, year={arXiv:2610.21099, 2026}, eprint={2610.21099}, archivePrefix={arXiv}, primaryClass={math.NT}, url={https://arxiv.org/abs/2610.21099}, } @article {MR684014, AUTHOR = {Weiss, Alfred}, TITLE = {The least prime ideal}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik. [Crelle's Journal]}, VOLUME = {338}, YEAR = {1983}, PAGES = {56--94}, } @misc{Leng, AUTHOR = {Leng, J.}, TITLE = {Efficient Equidistribution of Nilsequences}, YEAR = {arXiv:2312.10772}, } } @misc{FT, AUTHOR = {Frei, C. and Tinkova, M.}, TITLE = {Linear constellations in primes with arithmetic restrictions}, YEAR = {arXiv:2408.17441}, } @misc{KP, AUTHOR = {Koymans, P. and Pagano, C.}, TITLE = {Hilbert's tenth problem via additive combinatorics}, YEAR = {arXiv:2412.01768}, } @misc{Kai, AUTHOR = {Kai, W.}, TITLE = {Linear patterns of prime elements in number fields}, YEAR = {arXiv:2306.16983}, } @article{MR2180408, AUTHOR = {Green, B.}, TITLE = {Roth's theorem in the primes}, JOURNAL = {Ann. of Math. (2)}, VOLUME = {161}, NUMBER = {3}, YEAR = {2005}, PAGES = {1609--1636}, } @misc{LSS, AUTHOR = {Leng, J. and Sah, A. and Sawhney, M.}, TITLE = {Quasipolynomial bounds on the inverse theorem for the Gowers {$U^{s+1}[N]$}-norm}, YEAR = {arxiv:2402.17994}, } @article {MR3981313, AUTHOR = {Thorner, J. and Zaman, A.}, TITLE = {A unified and improved {C}hebotarev density theorem}, JOURNAL = {Algebra Number Theory}, FJOURNAL = {Algebra \& Number Theory}, VOLUME = {13}, YEAR = {2019}, NUMBER = {5}, PAGES = {1039--1068}, ISSN = {1937-0652,1944-7833}, } @incollection {MR447191, AUTHOR = {Lagarias, J. C. and Odlyzko, A. M.}, TITLE = {Effective versions of the {C}hebotarev density theorem}, BOOKTITLE = {Algebraic number fields: {$L$}-functions and {G}alois properties ({P}roc. {S}ympos., {U}niv. {D}urham, {D}urham, 1975)}, PAGES = {409--464}, PUBLISHER = {Academic Press, London-New York}, YEAR = {1977}, } @article {MR3518382, AUTHOR = {Baker, R.C. and Pollack, P.}, TITLE = {Bounded gaps between primes with a given primitive root, {II}}, JOURNAL = {Forum Math.}, FJOURNAL = {Forum Mathematicum}, VOLUME = {28}, YEAR = {2016}, NUMBER = {4}, PAGES = {675--687}, ISSN = {0933-7741,1435-5337}, } @article {MR3272281, AUTHOR = {Pollack, P.}, TITLE = {Bounded gaps between primes with a given primitive root}, JOURNAL = {Algebra Number Theory}, FJOURNAL = {Algebra \& Number Theory}, VOLUME = {8}, YEAR = {2014}, NUMBER = {7}, PAGES = {1769--1786}, ISSN = {1937-0652,1944-7833}, } @article {MR4517480, AUTHOR = {Tinkov\'a, M. and Waxman, E. and Zindulka, M.}, TITLE = {Artin twin primes}, JOURNAL = {J. Number Theory}, FJOURNAL = {Journal of Number Theory}, VOLUME = {245}, YEAR = {2023}, PAGES = {203--232}, ISSN = {0022-314X,1096-1658}, } @article {MR3272929, AUTHOR = {Maynard, J.}, TITLE = {Small gaps between primes}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {181}, YEAR = {2015}, NUMBER = {1}, PAGES = {383--413}, ISSN = {0003-486X,1939-8980}, } @article {MR3344173, AUTHOR = {Thorner, J.}, TITLE = {Bounded gaps between primes in {C}hebotarev sets}, JOURNAL = {Res. Math. Sci.}, FJOURNAL = {Research in the Mathematical Sciences}, VOLUME = {1}, YEAR = {2014}, PAGES = {Art. 4, 16}, ISSN = {2522-0144,2197-9847}, } @article {MR3667488, AUTHOR = {Vatwani, A. and Wong, P.-J.}, TITLE = {Patterns of primes in {C}hebotarev sets}, JOURNAL = {Int. J. Number Theory}, FJOURNAL = {International Journal of Number Theory}, VOLUME = {13}, YEAR = {2017}, NUMBER = {7}, PAGES = {1651--1677}, ISSN = {1793-0421,1793-7310}, } @article {MR4130085, AUTHOR = {Ji, C. and Kazdan, J. and McDonald, V.}, TITLE = {Primes with {B}eatty and {C}hebotarev conditions}, JOURNAL = {J. Number Theory}, FJOURNAL = {Journal of Number Theory}, VOLUME = {216}, YEAR = {2020}, PAGES = {307--334}, ISSN = {0022-314X,1096-1658}, } @article {MR3011564, AUTHOR = {Moree, P.}, TITLE = {Artin's primitive root conjecture---a survey}, JOURNAL = {Integers}, FJOURNAL = {Integers}, VOLUME = {12}, YEAR = {2012}, NUMBER = {6}, PAGES = {1305--1416}, ISSN = {1867-0652,1867-0660}, } @article {MR4736376, AUTHOR = {Arias-de-Reyna, S. and K\"onig, J.}, TITLE = {Locally cyclic extensions with {G}alois group {${\rm GL}_2(p)$}}, JOURNAL = {Int. J. Number Theory}, FJOURNAL = {International Journal of Number Theory}, VOLUME = {20}, YEAR = {2024}, NUMBER = {3}, PAGES = {781--796}, ISSN = {1793-0421,1793-7310}, } @article {MR4564765, AUTHOR = {Bienvenu, P.-Y. and Shao, X. and Ter\"av\"ainen, J.}, TITLE = {A transference principle for systems of linear equations, and applications to almost twin primes}, JOURNAL = {Algebra Number Theory}, FJOURNAL = {Algebra \& Number Theory}, VOLUME = {17}, YEAR = {2023}, NUMBER = {2}, PAGES = {497--539}, ISSN = {1937-0652,1944-7833}, } @article {MR369312, AUTHOR = {Szemer\'edi, E.}, TITLE = {On sets of integers containing no {$k$} elements in arithmetic progression}, JOURNAL = {Acta Arith.}, FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica}, VOLUME = {27}, YEAR = {1975}, PAGES = {199--245}, ISSN = {0065-1036}, } @misc {manners, AUTHOR = {Manners, F.}, TITLE = {Quantitative bounds in the inverse theorem for the {G}owers {$U^{s+1}$}-norms over cyclic groups}, YEAR = {arXiv:1811.00718}, } @article {taotera, AUTHOR = {Tao, T. and Ter\"av\"ainen, J.}, TITLE = {Quantitative bounds for {G}owers uniformity of the {M}\"obius and von {M}angoldt functions}, JOURNAL = {J. Eur. Math. Soc. (JEMS)}, FJOURNAL = {Journal of the European Mathematical Society (JEMS)}, VOLUME = {27}, YEAR = {2025}, NUMBER = {4}, PAGES = {1321--1384}, ISSN = {1435-9855,1435-9863}, } @article {MR3292295, AUTHOR = {Harpaz, Y. and Skorobogatov, A.N. and Wittenberg, O.}, TITLE = {The {H}ardy-{L}ittlewood conjecture and rational points}, JOURNAL = {Compos. Math.}, FJOURNAL = {Compositio Mathematica}, VOLUME = {150}, YEAR = {2014}, NUMBER = {12}, PAGES = {2095--2111}, ISSN = {0010-437X,1570-5846}, } @article {MR4094712, AUTHOR = {Bary-Soroker, L. and Schlank, T.~M.}, TITLE = {Sieves and the minimal ramification problem}, JOURNAL = {J. Inst. Math. Jussieu}, FJOURNAL = {Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institut de Math\'ematiques de Jussieu}, VOLUME = {19}, YEAR = {2020}, NUMBER = {3}, PAGES = {919--945}, ISSN = {1474-7480,1475-3030}, } @article {MR4177544, AUTHOR = {Kim, K.-S. and K\"onig, J.}, TITLE = {On {G}alois extensions with prescribed decomposition groups}, JOURNAL = {J. Number Theory}, FJOURNAL = {Journal of Number Theory}, VOLUME = {220}, YEAR = {2021}, PAGES = {266--294}, ISSN = {0022-314X,1096-1658}, } @article {MR4033249, AUTHOR = {Zoeteman, M.}, TITLE = {Uniformly counting primes with a given primitive root and in an arithmetic progression}, JOURNAL = {Int. J. Number Theory}, FJOURNAL = {International Journal of Number Theory}, VOLUME = {15}, YEAR = {2019}, NUMBER = {10}, PAGES = {2115--2134}, ISSN = {1793-0421,1793-7310}, } @article {MR3656911, AUTHOR = {Bienvenu, P.-Y.}, TITLE = {A higher-dimensional {S}iegel-{W}alfisz theorem}, JOURNAL = {Acta Arith.}, FJOURNAL = {Acta Arithmetica}, VOLUME = {179}, YEAR = {2017}, NUMBER = {1}, PAGES = {79--100}, ISSN = {0065-1036,1730-6264}, } @article {MR4436255, AUTHOR = {Dodos, P. and Kanellopoulos, V.}, TITLE = {Uniformity norms, their weaker versions, and applications}, JOURNAL = {Acta Arith.}, FJOURNAL = {Acta Arithmetica}, VOLUME = {203}, YEAR = {2022}, NUMBER = {3}, PAGES = {251--270}, ISSN = {0065-1036,1730-6264}, } @article {MR0644559, AUTHOR = {Serre, J.-P.}, TITLE = {Quelques applications du th\'{e}or\`eme de densit\'{e} de {C}hebotarev}, JOURNAL = {Inst. Hautes \'{E}tudes Sci. Publ. Math.}, FJOURNAL = {Institut des Hautes \'{E}tudes Scientifiques. Publications Math\'{e}matiques}, VOLUME = {54}, YEAR = {1981}, PAGES = {323--401}, } @book {MR2061214, AUTHOR = {Iwaniec, H. and Kowalski, E.}, TITLE = {Analytic number theory}, SERIES = {American Mathematical Society Colloquium Publications}, VOLUME = {53}, PUBLISHER = {American Mathematical Society, Providence, RI}, YEAR = {2004}, PAGES = {xii+615}, ISBN = {0-8218-3633-1}, MRCLASS = {11-02 (11Fxx 11Lxx 11Mxx 11Nxx)}, MRNUMBER = {2061214}, MRREVIEWER = {K.\ Soundararajan}, DOI = {10.1090/coll/053}, URL = {https://doi.org/10.1090/coll/053}, } @article {MR3060874, AUTHOR = {Kane, D.~M.}, TITLE = {An asymptotic for the number of solutions to linear equations in prime numbers from specified {C}hebotarev classes}, JOURNAL = {Int. J. Number Theory}, FJOURNAL = {International Journal of Number Theory}, VOLUME = 9, YEAR = 2013, NUMBER = 4, PAGES = {1073--1111}, ISSN = {1793-0421}, } @article {MR2490093, AUTHOR = {Moree, P.}, TITLE = {On primes in arithmetic progression having a prescribed primitive root. {II}}, JOURNAL = {Funct. Approx. Comment. Math.}, FJOURNAL = {Uniwersytet im. Adama Mickiewicza w Poznaniu. Wydzia\l Matematyki i Informatyki. Functiones et Approximatio Commentarii Mathematici}, VOLUME = {39}, YEAR = {2008}, NUMBER = {part 1}, PAGES = {133--144}, ISSN = {0208-6573}, } @article {MR207630, AUTHOR = {Hooley, C.}, TITLE = {On {A}rtin's conjecture}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle's Journal]}, VOLUME = {225}, YEAR = {1967}, PAGES = {209--220}, ISSN = {0075-4102}, } @article {MR2368279, AUTHOR = {Goldston, D. A. and Y\i ld\i r\i m, C. Y.}, TITLE = {Higher correlations of divisor sums related to primes. {III}. {S}mall gaps between primes}, JOURNAL = {Proc. Lond. Math. Soc. (3)}, FJOURNAL = {Proceedings of the London Mathematical Society. Third Series}, VOLUME = {95}, YEAR = {2007}, NUMBER = {3}, PAGES = {653--686}, ISSN = {0024-6115}, } @article {MR2552109, AUTHOR = {Goldston, D.~A. and Pintz, J. and Y\i ld\i r\i m, C.~Y.}, TITLE = {Primes in tuples. {I}}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {170}, YEAR = {2009}, NUMBER = {2}, PAGES = {819--862}, ISSN = {0003-486X}, MRCLASS = {11N05 (11N36)}, MRNUMBER = {2552109}, MRREVIEWER = {S. W. Graham}, DOI = {10.4007/annals.2009.170.819}, URL = {https://doi-org.manchester.idm.oclc.org/10.4007/annals.2009.170.819}, } @article{FLN2, author={Frei, C. and Loughran, D. and Newton, R.}, title={Number Fields with prescribed norms}, date={arXiv:1810.06024, 2018}, } @article {MR3990378, AUTHOR = {Elsholtz, C. and Frei, C.}, TITLE = {Arithmetic progressions in binary quadratic forms and norm forms}, JOURNAL = {Bull. Lond. Math. Soc.}, FJOURNAL = {Bulletin of the London Mathematical Society}, VOLUME = {51}, YEAR = {2019}, NUMBER = {4}, PAGES = {595--602}, ISSN = {0024-6093}, MRCLASS = {11E25 (11B25 11D57 11E12)}, MRNUMBER = {3990378}, DOI = {10.1112/blms.12256}, URL = {https://doi-org.manchester.idm.oclc.org/10.1112/blms.12256}, } @book{MR1697859, AUTHOR = {Neukirch, J.}, TITLE = {Algebraic number theory}, SERIES = {Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]}, VOLUME = {322}, NOTE = {Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder}, PUBLISHER = {Springer-Verlag, Berlin}, YEAR = {1999}, PAGES = {xviii+571}, ISBN = {3-540-65399-6}, MRCLASS = {11Rxx (11-02 11S15 11S31 14C40)}, MRNUMBER = {1697859}, MRREVIEWER = {Cornelius Greither}, DOI = {10.1007/978-3-662-03983-0}, URL = {https://doi-org.manchester.idm.oclc.org/10.1007/978-3-662-03983-0}, } @inproceedings {MR0236145, AUTHOR = {Fr\"{o}hlich, A.}, TITLE = {Local fields}, BOOKTITLE = {Algebraic {N}umber {T}heory ({P}roc. {I}nstructional {C}onf., {B}righton, 1965) }, PAGES = {1--41}, PUBLISHER = {Thompson, Washington, D.C.}, YEAR = {1967}, MRCLASS = {10.67}, MRNUMBER = {0236145}, MRREVIEWER = {L. R. McCulloh}, } @article {MR3171762, AUTHOR = {Green, B. and Tao, T.}, TITLE = {On the quantitative distribution of polynomial nilsequences---erratum}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {179}, YEAR = {2014}, NUMBER = {3}, PAGES = {1175--1183}, ISSN = {0003-486X}, MRCLASS = {37A15}, MRNUMBER = {3171762}, MRREVIEWER = {Volker Ziegler}, DOI = {10.4007/annals.2014.179.3.8}, URL = {http://dx.doi.org/10.4007/annals.2014.179.3.8}, } @article{yasudaMcKay1, AUTHOR = {Yasuda, T.}, TITLE = {The wild {M}c{K}ay correspondence and p-adic measures}, JOURNAL={J. Eur. Math. Soc}, YEAR = {to appear, 2016}, } @misc {Matthiesen16, AUTHOR = {Matthiesen, L.}, TITLE = {Linear correlations of multiplicative functions}, YEAR = {preprint, arXiv:1606.04482, 2016}, } @article {MR2805576, AUTHOR = {Widmer, M.}, TITLE = {On number fields with nontrivial subfields}, JOURNAL = {Int. J. Number Theory}, FJOURNAL = {International Journal of Number Theory}, VOLUME = {7}, YEAR = {2011}, NUMBER = {3}, PAGES = {695--720}, ISSN = {1793-0421}, MRCLASS = {11R32 (11G50 11R47 11Y40)}, MRNUMBER = {2805576}, MRREVIEWER = {J{\"u}rgen Kl{\"u}ners}, DOI = {10.1142/S1793042111004204}, URL = {http://dx.doi.org/10.1142/S1793042111004204}, } @article {destagnol, AUTHOR = {Destagnol, K.}, TITLE = {{L}a conjecture de {M}anin pour certaines surfaces de {C}h\^{a}telet}, JOURNAL = {Acta Arith.}, FJOURNAL = {Acta Arithmetica}, VOLUME = {174}, YEAR = {2016}, NUMBER = {1}, PAGES = {31--97}, } @article {MR3103132, AUTHOR = {de la Bret{\`e}che, R. and Tenenbaum, G.}, TITLE = {Sur la conjecture de {M}anin pour certaines surfaces de {C}h\^atelet}, JOURNAL = {J. Inst. Math. Jussieu}, FJOURNAL = {Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institut de Math\'ematiques de Jussieu}, VOLUME = {12}, YEAR = {2013}, NUMBER = {4}, PAGES = {759--819}, } @article {MR2927803, AUTHOR = {de la Bret{\`e}che, R. and Tenenbaum, G.}, TITLE = {Oscillations localis\'ees sur les diviseurs}, JOURNAL = {J. Lond. Math. Soc. (2)}, FJOURNAL = {Journal of the London Mathematical Society. Second Series}, VOLUME = {85}, YEAR = {2012}, NUMBER = {3}, PAGES = {669--693}, } @article {hooldel, AUTHOR = {Hooley, C.}, TITLE = {On a new technique and its applications to the theory of numbers}, JOURNAL = {Proc. London Math. Soc. (3)}, FJOURNAL = {Proceedings of the London Mathematical Society. Third Series}, VOLUME = {38}, YEAR = {1979}, NUMBER = {1}, PAGES = {115--151}, } @article {cubelinear, AUTHOR = {de la Bret{\`e}che, R. and Browning, T. D.}, TITLE = {Binary forms as sums of two squares and {C}h\^atelet surfaces}, JOURNAL = {Israel J. Math.}, FJOURNAL = {Israel Journal of Mathematics}, VOLUME = {191}, YEAR = {2012}, NUMBER = {2}, PAGES = {973--1012}, } @article {LLQ, AUTHOR = {de la Bret{\`e}che, R. and Browning, T. D.}, TITLE = {Le probl\`eme des diviseurs pour des formes binaires de degr\'e 4}, JOURNAL = {J. reine angew. Math.}, FJOURNAL = {Journal f\"ur die reine und angewandte Mathematik. [Crelle's Journal]}, VOLUME = {646}, YEAR = {2010}, PAGES = {1--44}, } @article {compoloi, AUTHOR = {de la Bret{\`e}che, R. and Browning, T. D.}, TITLE = {Binary linear forms as sums of two squares}, JOURNAL = {Compos. Math.}, FJOURNAL = {Compositio Mathematica}, VOLUME = {144}, YEAR = {2008}, NUMBER = {6}, PAGES = {1375--1402}, } @article {stephan, AUTHOR = {Daniel, S.}, TITLE = {On the divisor-sum problem for binary forms}, JOURNAL = {J. reine angew. Math.}, FJOURNAL = {Journal f\"ur die reine und angewandte Mathematik}, VOLUME = {507}, YEAR = {1999}, PAGES = {107--129} } @incollection {HB4L, AUTHOR = {Heath-Brown, D. R.}, TITLE = {Linear relations amongst sums of two squares}, BOOKTITLE = {Number theory and algebraic geometry}, SERIES = {London Math. Soc. Lecture Note Ser.}, VOLUME = {303}, PAGES = {133--176}, PUBLISHER = {Cambridge Univ. Press, Cambridge}, YEAR = {2003} } @incollection {bruedern, AUTHOR = {Br{\"u}dern, J.}, TITLE = {Daniel's twists of {H}ooley's delta function}, BOOKTITLE = {Contributions in analytic and algebraic number theory}, SERIES = {Springer Proc. Math.}, VOLUME = {9}, PAGES = {31--82}, PUBLISHER = {Springer, New York}, YEAR = {2012} } @misc{mella, AUTHOR = {Koll{\'a}r, J and Mella, M.}, TITLE = {Quadratic families of elliptic curves and unirationality of degree {$1$} conic bundles}, YEAR = {preprint, arXiv:1412.3673, 2014}, } @misc{fibpub, AUTHOR = {Frei, C. and Loughran, D. and Sofos, E.}, TITLE = {Rational points of bounded height on general conic bundle surfaces}, YEAR = {submitted, arXiv:1609.04330, 2016}, } @misc {MdC13, AUTHOR = {Mendes da Costa, D.}, TITLE = {Integral points on elliptic curves and the Bombieri-Pila bounds}, YEAR = {preprint, arXiv:1301.4116, 2013} } @article{salbergerunpub, AUTHOR = {Salberger, P.}, TITLE = {Counting rational points on projective varieties}, JOURNAL = {in preparation}, YEAR = {2013} } @article {rogerbest, AUTHOR = {Heath-Brown, D. R.}, TITLE = {The density of rational points on cubic surfaces}, JOURNAL = {Acta Arith.}, VOLUME = {79}, YEAR = {1997}, NUMBER = {1}, PAGES = {17--30} } @misc{divsumpub, AUTHOR = {Frei, C. and Sofos, E.}, TITLE = {Divisor sums for binary forms over number fields}, YEAR = {submitted, arXiv:1609.04002, 2016}, } @article {MR1200974, AUTHOR = {M{\"a}ki, S.}, TITLE = {The conductor density of abelian number fields}, JOURNAL = {J. London Math. Soc. (2)}, FJOURNAL = {Journal of the London Mathematical Society. Second Series}, VOLUME = {47}, YEAR = {1993}, NUMBER = {1}, PAGES = {18--30}, ISSN = {0024-6107}, CODEN = {JLMSAK}, MRCLASS = {11R45 (11R20 11R29)}, MRNUMBER = {1200974 (94b:11115)}, MRREVIEWER = {M. Ram Murty}, DOI = {10.1112/jlms/s2-47.1.18}, URL = {http://dx.doi.org/10.1112/jlms/s2-47.1.18}, } @incollection {MR2252828, AUTHOR = {Rudnick, Z.}, TITLE = {Eigenvalue statistics and lattice points}, BOOKTITLE = {Colloquium {D}e {G}iorgi 2006}, SERIES = {Colloquia}, VOLUME = {1}, PAGES = {45--50}, PUBLISHER = {Ed. Norm., Pisa}, YEAR = {2006}, MRCLASS = {11P21 (58J50)}, MRNUMBER = {2252828 (2008k:11106)}, MRREVIEWER = {Steven Joel Miller}, } @article {MR2630059, AUTHOR = {Vaserstein, L.}, TITLE = {Polynomial parametrization for the solutions of {D}iophantine equations and arithmetic groups}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {171}, YEAR = {2010}, NUMBER = {2}, PAGES = {979--1009}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {11D72 (20G20)}, MRNUMBER = {2630059 (2011e:11061)}, MRREVIEWER = {Timothy D. Browning}, DOI = {10.4007/annals.2010.171.979}, URL = {http://dx.doi.org/10.4007/annals.2010.171.979}, } @article {MR3319557, AUTHOR = {Derenthal, U. and Smeets, A. and Wei, D.}, TITLE = {Universal torsors and values of quadratic polynomials represented by norms}, JOURNAL = {Math. Ann.}, FJOURNAL = {Mathematische Annalen}, VOLUME = {361}, YEAR = {2015}, NUMBER = {3-4}, PAGES = {1021--1042}, ISSN = {0025-5831}, MRCLASS = {14G05 (11D57 14F22)}, MRNUMBER = {3319557}, DOI = {10.1007/s00208-014-1106-7}, URL = {http://dx.doi.org/10.1007/s00208-014-1106-7}, } @article {MR3293154, AUTHOR = {Wei, D.}, TITLE = {On the equation {$N_{K/k}(\Xi)=P(t)$}}, JOURNAL = {Proc. Lond. Math. Soc. (3)}, FJOURNAL = {Proceedings of the London Mathematical Society. Third Series}, VOLUME = {109}, YEAR = {2014}, NUMBER = {6}, PAGES = {1402--1434}, ISSN = {0024-6115}, MRCLASS = {11G35 (14G05)}, MRNUMBER = {3293154}, DOI = {10.1112/plms/pdu035}, URL = {http://dx.doi.org/10.1112/plms/pdu035}, } @misc {Szegedy12, AUTHOR = {Szegedy, B.}, TITLE = {On higher order fourier analysis}, YEAR = {preprint, arXiv:1203.2260, 2012}, } @misc {Szegedy10, AUTHOR = {Szegedy, B.}, TITLE = {Gowers norms, regularization and limits of functions on abelian groups}, YEAR = {preprint, arXiv:1010.6211, 2010}, } @article {MR3742557, AUTHOR = {Matthiesen, L.}, TITLE = {On the square-free representation function of a norm form and nilsequences}, JOURNAL = {J. Inst. Math. Jussieu}, FJOURNAL = {Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institut de Math\'ematiques de Jussieu}, VOLUME = {17}, YEAR = {2018}, NUMBER = {1}, PAGES = {107--135}, ISSN = {1474-7480,1475-3030}, MRCLASS = {11B30 (11D57 11N37 14G05)}, MRNUMBER = {3742557}, MRREVIEWER = {Sean\ M.\ Prendiville}, DOI = {10.1017/S1474748015000389}, URL = {https://doi.org/10.1017/S1474748015000389}, } @article {Matthiesen14, AUTHOR = {Matthiesen, L.}, TITLE = {On the square-free representation function of a norm form and nilsequences}, JOURNAL = {J. Inst. Math. Jussieu}, YEAR = {to appear, 2014}, } @article {HarpazWittenberg, AUTHOR = {Harpaz, Y. and Wittenberg, O.}, TITLE = {On the fibration method for zero-cycles and rational points}, JOURNAL = {Ann. of Math.}, VOLUME = {183}, NUMBER = {1}, PAGES = {229--295}, YEAR = {2016}, } @article {MR1961196, AUTHOR = {Heath-Brown, R. and Skorobogatov, A.}, TITLE = {Rational solutions of certain equations involving norms}, JOURNAL = {Acta Math.}, FJOURNAL = {Acta Mathematica}, VOLUME = {189}, YEAR = {2002}, NUMBER = {2}, PAGES = {161--177}, ISSN = {0001-5962}, MRCLASS = {14G05 (11D57 11G25 14F22 14G25)}, MRNUMBER = {1961196 (2004b:14033)}, MRREVIEWER = {Tam{\'a}s Szamuely}, DOI = {10.1007/BF02392841}, URL = {http://dx.doi.org/10.1007/BF02392841}, } @article {MR2989431, AUTHOR = {Browning, T. D. and Heath-Brown, D. R.}, TITLE = {Quadratic polynomials represented by norm forms}, JOURNAL = {Geom. Funct. Anal.}, FJOURNAL = {Geometric and Functional Analysis}, VOLUME = {22}, YEAR = {2012}, NUMBER = {5}, PAGES = {1124--1190}, ISSN = {1016-443X}, CODEN = {GFANFB}, MRCLASS = {14G05 (11D57 14G25)}, MRNUMBER = {2989431}, MRREVIEWER = {Liang-Chung Hsia}, DOI = {10.1007/s00039-012-0168-5}, URL = {http://dx.doi.org/10.1007/s00039-012-0168-5}, } @incollection {MR756082, AUTHOR = {Cohen, H. and Lenstra, Jr., H. W.}, TITLE = {Heuristics on class groups of number fields}, BOOKTITLE = {Number theory, {N}oordwijkerhout 1983}, SERIES = {Lecture Notes in Math.}, VOLUME = {1068}, PAGES = {33--62}, PUBLISHER = {Springer, Berlin}, YEAR = {1984}, MRCLASS = {11R29 (13C13)}, MRNUMBER = {756082 (85j:11144)}, MRREVIEWER = {F. J. van der Linden}, DOI = {10.1007/BFb0099440}, URL = {http://dx.doi.org/10.1007/BFb0099440}, } @incollection {MR2076244, AUTHOR = {Cohen, H.}, TITLE = {Counting {$A_4$} and {$S_4$} number fields with given resolvent cubic}, BOOKTITLE = {High primes and misdemeanours: lectures in honour of the 60th birthday of {H}ugh {C}owie {W}illiams}, SERIES = {Fields Inst. Commun.}, VOLUME = {41}, PAGES = {159--168}, PUBLISHER = {Amer. Math. Soc., Providence, RI}, YEAR = {2004}, MRCLASS = {11R16 (11R29 11R45 11Y40)}, MRNUMBER = {2076244 (2005e:11136)}, MRREVIEWER = {Daniel C. Mayer}, } @article {bhargavaHP, AUTHOR = {Bhargava, M.}, TITLE = {A positive proportion of plane cubics fail the Hasse principle}, YEAR = {preprint, arXiv:1402.1131, 2014}, } @article {MR3232765, AUTHOR = {Bret{\`e}che, R. de la and Browning, T.~D.}, TITLE = {Contre-exemples au principe de {H}asse pour certains tores coflasques}, JOURNAL = {J. Th\'eor. Nombres Bordeaux}, FJOURNAL = {Journal de Th\'eorie des Nombres de Bordeaux}, VOLUME = {26}, YEAR = {2014}, NUMBER = {1}, PAGES = {25--44}, ISSN = {1246-7405}, MRCLASS = {11G35 (11D99 14G05)}, MRNUMBER = {3232765}, MRREVIEWER = {Andrew Bremner}, URL = {http://jtnb.cedram.org/item?id=JTNB_2014__26_1_25_0}, } @article {MR3198755, AUTHOR = {Bret{\`e}che, R. de la and Browning, T. D.}, TITLE = {Density of {C}h\^atelet surfaces failing the {H}asse principle}, JOURNAL = {Proc. Lond. Math. Soc. (3)}, FJOURNAL = {Proceedings of the London Mathematical Society. Third Series}, VOLUME = {108}, YEAR = {2014}, NUMBER = {4}, PAGES = {1030--1078}, ISSN = {0024-6115}, MRCLASS = {11G25 (11G50 14G17)}, MRNUMBER = {3198755}, MRREVIEWER = {Tony Shaska}, DOI = {10.1112/plms/pdt060}, URL = {http://dx.doi.org/10.1112/plms/pdt060}, } @article {MR1670278, AUTHOR = {Daniel, S.}, TITLE = {On the divisor-sum problem for binary forms}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik}, VOLUME = {507}, YEAR = {1999}, PAGES = {107--129}, ISSN = {0075-4102}, CODEN = {JRMAA8}, MRCLASS = {11N37}, MRNUMBER = {1670278 (99k:11143)}, MRREVIEWER = {G. Greaves}, DOI = {10.1515/crll.1999.010}, URL = {http://dx.doi.org/10.1515/crll.1999.010}, } @article {sofosthesis, AUTHOR = {Sofos, E.}, TITLE = {Counting rational points on smooth cubic surfaces}, YEAR = {Ph.D. thesis, University of Bristol, 2015} } @article {MR0430336, AUTHOR = {Sato, M. and Kimura, T.}, TITLE = {A classification of irreducible prehomogeneous vector spaces and their relative invariants}, JOURNAL = {Nagoya Math. J.}, FJOURNAL = {Nagoya Mathematical Journal}, VOLUME = {65}, YEAR = {1977}, PAGES = {1--155}, ISSN = {0027-7630}, MRCLASS = {32M10 (20G05)}, MRNUMBER = {0430336 (55 \#3341)}, MRREVIEWER = {A. L. Oniscik}, } @incollection {MR2648698, AUTHOR = {Oh, H.}, TITLE = {Orbital counting via mixing and unipotent flows}, BOOKTITLE = {Homogeneous flows, moduli spaces and arithmetic}, SERIES = {Clay Math. Proc.}, VOLUME = {10}, PAGES = {339--375}, PUBLISHER = {Amer. Math. Soc., Providence, RI}, YEAR = {2010}, MRCLASS = {11G50 (11G35 14G05 22E55)}, MRNUMBER = {2648698 (2011f:11079)}, MRREVIEWER = {B. Sury}, } @misc {Yasuda14, AUTHOR = {Yasuda, T.}, TITLE = {Densities of rational points and number fields}, YEAR = {preprint, arXiv:1408.3912, 2014} } @article {MR1717617, AUTHOR = {Wong, S.}, TITLE = {Automorphic forms on {${\rm GL}(2)$} and the rank of class groups}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik}, VOLUME = {515}, YEAR = {1999}, PAGES = {125--153}, ISSN = {0075-4102}, CODEN = {JRMAA8}, MRCLASS = {11F70 (11F30 11R16 11R29)}, MRNUMBER = {1717617 (2000g:11042)}, MRREVIEWER = {K. Soundararajan}, DOI = {10.1515/crll.1999.075}, URL = {http://dx.doi.org/10.1515/crll.1999.075}, } @article {MR1918290, AUTHOR = {Cohen, H. and Diaz y Diaz, F. and Olivier, M.}, TITLE = {Enumerating quartic dihedral extensions of {$\Bbb Q$}}, JOURNAL = {Compositio Math.}, FJOURNAL = {Compositio Mathematica}, VOLUME = {133}, YEAR = {2002}, NUMBER = {1}, PAGES = {65--93}, ISSN = {0010-437X}, CODEN = {CMPMAF}, MRCLASS = {11R16 (11R29 11R45 11Y40)}, MRNUMBER = {1918290 (2003f:11167)}, MRREVIEWER = {T. Mets{\"a}nkyl{\"a}}, DOI = {10.1023/A:1016310902973}, URL = {http://dx.doi.org/10.1023/A:1016310902973}, } @article {MR1330934, AUTHOR = {Schmidt, W.~M.}, TITLE = {Number fields of given degree and bounded discriminant}, NOTE = {Columbia University Number Theory Seminar (New York, 1992)}, JOURNAL = {Ast\'erisque}, FJOURNAL = {Ast\'erisque}, NUMBER = {228}, YEAR = {1995}, PAGES = {4, 189--195}, ISSN = {0303-1179}, MRCLASS = {11R47 (11R29)}, MRNUMBER = {1330934 (96e:11153)}, MRREVIEWER = {Bart de Smit}, } @article {MR0466055, AUTHOR = {Billingsley, P.}, TITLE = {The probability theory of additive arithmetic functions}, JOURNAL = {Ann. Probability}, VOLUME = {2}, YEAR = {1974}, PAGES = {749--791}, MRCLASS = {10K20 (60F05)}, MRNUMBER = {0466055 (57 \#5938)}, MRREVIEWER = {Gutti J. Babu}, } @Article{zbMATH03809838, Author = {V.P. {Platonov}}, Title = {{The arithmetic theory of algebraic groups.}}, FJournal = {{Russian Mathematical Surveys}}, Journal = {{Russ. Math. Surv.}}, ISSN = {0036-0279; 1468-4829/e}, Volume = 37, Number = 3, Pages = {1--62}, Year = 1982, Publisher = {IOP Publishing, Bristol; London Mathematical Society, London; Turpion, London; Russian Academy of Sciences- RAS (Rossijskaya Akademiya Nauk - RAN), Moscow}, Language = {English}, DOI = {10.1070/RM1982v037n03ABEH003230}, MSC2010 = {20G30 20G10 14L35}, Zbl = {0513.20028}, } @misc {BhargavaShankarWang, AUTHOR = {Bhargava, M. and Shankar, A. and Wang, X.~J.}, TITLE = {Geometry-of-numbers methods and arithmetic density theorems over global fields}, YEAR = {preprint, 2015}, } @article {MR1185585, AUTHOR = {Wright, D.~J. and Yukie, A.}, TITLE = {Prehomogeneous vector spaces and field extensions}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {110}, YEAR = {1992}, NUMBER = {2}, PAGES = {283--314}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {12F10}, MRNUMBER = {1185585 (93j:12004)}, MRREVIEWER = {Martin Epkenhans}, DOI = {10.1007/BF01231334}, URL = {http://dx.doi.org/10.1007/BF01231334}, } @article {MR1932327, AUTHOR = {Gan, W.~T. and Gross, B. and Savin, G.}, TITLE = {Fourier coefficients of modular forms on {$G_2$}}, JOURNAL = {Duke Math. J.}, FJOURNAL = {Duke Mathematical Journal}, VOLUME = {115}, YEAR = {2002}, NUMBER = {1}, PAGES = {105--169}, ISSN = {0012-7094}, CODEN = {DUMJAO}, MRCLASS = {11F30 (11F55)}, MRNUMBER = {1932327 (2004a:11036)}, MRREVIEWER = {A. Raghuram}, DOI = {10.1215/S0012-7094-02-11514-2}, URL = {http://dx.doi.org/10.1215/S0012-7094-02-11514-2}, } @book {MR1633348, AUTHOR = {van den Dries, L.}, TITLE = {Tame topology and o-minimal structures}, SERIES = {London Mathematical Society Lecture Note Series}, VOLUME = {248}, PUBLISHER = {Cambridge University Press, Cambridge}, YEAR = {1998}, PAGES = {x+180}, ISBN = {0-521-59838-9}, MRCLASS = {03-02 (03C50 03C60 14P10 52-02 54-02 55-02 57-02)}, MRNUMBER = {1633348 (99j:03001)}, MRREVIEWER = {O. V. Belegradek}, DOI = {10.1017/CBO9780511525919}, URL = {http://dx.doi.org/10.1017/CBO9780511525919}, } @article {MR1404337, AUTHOR = {van den Dries, L. and Miller, C.}, TITLE = {Geometric categories and o-minimal structures}, JOURNAL = {Duke Math. J.}, FJOURNAL = {Duke Mathematical Journal}, VOLUME = {84}, YEAR = {1996}, NUMBER = {2}, PAGES = {497--540}, ISSN = {0012-7094}, CODEN = {DUMJAO}, MRCLASS = {32B20 (03C68)}, MRNUMBER = {1404337 (97i:32008)}, MRREVIEWER = {Anand Pillay}, DOI = {10.1215/S0012-7094-96-08416-1}, URL = {http://dx.doi.org/10.1215/S0012-7094-96-08416-1}, } @article {MR1289495, AUTHOR = {van den Dries, L. and Macintyre, A. and Marker, D.}, TITLE = {The elementary theory of restricted analytic fields with exponentiation}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {140}, YEAR = {1994}, NUMBER = {1}, PAGES = {183--205}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {12L12 (03C10 03C62)}, MRNUMBER = {1289495 (95k:12015)}, MRREVIEWER = {Thanases Pheidas}, DOI = {10.2307/2118545}, URL = {http://dx.doi.org/10.2307/2118545}, } @article {MR854552, AUTHOR = {van den Dries, L.}, TITLE = {A generalization of the {T}arski-{S}eidenberg theorem, and some nondefinability results}, JOURNAL = {Bull. Amer. Math. Soc. (N.S.)}, FJOURNAL = {American Mathematical Society. Bulletin. New Series}, VOLUME = {15}, YEAR = {1986}, NUMBER = {2}, PAGES = {189--193}, ISSN = {0273-0979}, CODEN = {BAMOAD}, MRCLASS = {03C40 (03E47)}, MRNUMBER = {854552 (88b:03048)}, MRREVIEWER = {Bienvenido F. Nebres}, DOI = {10.1090/S0273-0979-1986-15468-6}, URL = {http://dx.doi.org/10.1090/S0273-0979-1986-15468-6}, } @book {MR0028796, AUTHOR = {Tarski, A.}, TITLE = {A {D}ecision {M}ethod for {E}lementary {A}lgebra and {G}eometry}, PUBLISHER = {RAND Corporation, Santa Monica, Calif.}, YEAR = {1948}, PAGES = {iii+60}, MRCLASS = {02.0X}, MRNUMBER = {0028796 (10,499f)}, MRREVIEWER = {A. Heyting}, } @article {MR3354956, AUTHOR = {Beshenov, A. and Bilu, M. and Bilu, Yu. and Rath, P.}, TITLE = {Rational points on analytic varieties}, JOURNAL = {EMS Surv. Math. Sci.}, FJOURNAL = {EMS Surveys in Mathematical Sciences}, VOLUME = {2}, YEAR = {2015}, NUMBER = {1}, PAGES = {109--130}, ISSN = {2308-2151}, MRCLASS = {11G18 (03C64 11P21 32B20)}, MRNUMBER = {3354956}, DOI = {10.4171/EMSS/10}, URL = {http://dx.doi.org/10.4171/EMSS/10}, } @article{sem:Wilkie2007, AUTHOR = {Wilkie, A. J.}, TITLE={o-{M}inimal structures}, journal={S\'eminaire Bourbaki}, eprint={}, year={expos\'e 985, 2007}, } @article {MR1016893, AUTHOR = {Bombieri, E. and Pila, J.}, TITLE = {The number of integral points on arcs and ovals}, JOURNAL = {Duke Math. J.}, FJOURNAL = {Duke Mathematical Journal}, VOLUME = {59}, YEAR = {1989}, NUMBER = {2}, PAGES = {337--357}, ISSN = {0012-7094}, CODEN = {DUMJAO}, MRCLASS = {11P21 (11D99)}, MRNUMBER = {1016893 (90j:11099)}, MRREVIEWER = {Ulrich Rausch}, DOI = {10.1215/S0012-7094-89-05915-2}, URL = {http://dx.doi.org/10.1215/S0012-7094-89-05915-2}, } @article {MR0166155, AUTHOR = {Davenport, H.}, TITLE = {Corrigendum: ``{O}n a principle of {L}ipschitz``}, JOURNAL = {J. London Math. Soc.}, FJOURNAL = {Journal of the London Mathematical Society. Second Series}, VOLUME = {39}, YEAR = {1964}, PAGES = {580}, ISSN = {0024-6107}, MRCLASS = {10.30}, MRNUMBER = {0166155 (29 \#3433)}, } @article {MR2948473, AUTHOR = {Wood, M.~M.}, TITLE = {Parametrizing quartic algebras over an arbitrary base}, JOURNAL = {Algebra Number Theory}, FJOURNAL = {Algebra \& Number Theory}, VOLUME = {5}, YEAR = {2011}, NUMBER = {8}, PAGES = {1069--1094}, ISSN = {1937-0652}, MRCLASS = {14D20 (11E20 11R16 13A99)}, MRNUMBER = {2948473}, MRREVIEWER = {Detlev W. Hoffmann}, DOI = {10.2140/ant.2011.5.1069}, URL = {http://dx.doi.org/10.2140/ant.2011.5.1069}, } @incollection {MR1850611, AUTHOR = {Cohen, H. and Diaz y Diaz, F. and Olivier, M.}, TITLE = {Counting discriminants of number fields of degree up to four}, BOOKTITLE = {Algorithmic number theory ({L}eiden, 2000)}, SERIES = {Lecture Notes in Comput. Sci.}, VOLUME = {1838}, PAGES = {269--283}, PUBLISHER = {Springer, Berlin}, YEAR = {2000}, MRCLASS = {11R29 (11R21 11Y40)}, MRNUMBER = {1850611 (2002g:11148)}, MRREVIEWER = {Joe P. Buhler}, DOI = {10.1007/10722028_15}, URL = {http://dx.doi.org/10.1007/10722028_15}, } @book {MR0160744, AUTHOR = {Delone, B. N. and Faddeev, D. K.}, TITLE = {The theory of irrationalities of the third degree}, SERIES = {Translations of Mathematical Monographs, Vol. 10}, PUBLISHER = {American Mathematical Society, Providence, R.I.}, YEAR = {1964}, PAGES = {xvi+509}, MRCLASS = {10.25 (10.00)}, MRNUMBER = {0160744 (28 \#3955)}, } @article {MR0254010, AUTHOR = {Davenport, H. and Heilbronn, H.}, TITLE = {On the density of discriminants of cubic fields}, JOURNAL = {Bull. London Math. Soc.}, FJOURNAL = {The Bulletin of the London Mathematical Society}, VOLUME = {1}, YEAR = {1969}, PAGES = {345--348}, ISSN = {0024-6093}, MRCLASS = {10.65}, MRNUMBER = {0254010 (40 \#7223)}, MRREVIEWER = {J. V. Armitage}, } @article {MR0043823, AUTHOR = {Davenport, H.}, TITLE = {On the class-number of binary cubic forms. {II}}, JOURNAL = {J. London Math. Soc.}, FJOURNAL = {Journal of the London Mathematical Society. Second Series}, VOLUME = {26}, YEAR = {1951}, PAGES = {192--198}, ISSN = {0024-6107}, MRCLASS = {10.0X}, MRNUMBER = {0043823 (13,323f)}, MRREVIEWER = {W. H. Mills}, } @article {MR0043822, AUTHOR = {Davenport, H.}, TITLE = {On the class-number of binary cubic forms. {I}}, JOURNAL = {J. London Math. Soc.}, FJOURNAL = {Journal of the London Mathematical Society. Second Series}, VOLUME = {26}, YEAR = {1951}, PAGES = {183--192}, ISSN = {0024-6107}, MRCLASS = {10.0X}, MRNUMBER = {0043822 (13,323e)}, MRREVIEWER = {W. H. Mills}, } @misc {DerenthalWei, AUTHOR = {Derenthal, U. and Wei, D.}, TITLE = {Strong approximation and descent}, YEAR = {J. Reine Angew. Math., to appear, 2015}, } @incollection{BrowningCMSurvey, AUTHOR = {Browning, T.~D.}, TITLE = {A survey of applications of the circle method to rational points}, BOOKTITLE = {Arithmetic and geometry}, SERIES = {London Math. Soc. Lecture Note Ser.}, VOLUME = 420, PUBLISHER = {CUP}, YEAR = 2015, } @incollection {MR2011747, AUTHOR = {Colliot-Th{\'e}l{\`e}ne, J.-L.}, TITLE = {Points rationnels sur les fibrations}, BOOKTITLE = {Higher dimensional varieties and rational points ({B}udapest, 2001)}, SERIES = {Bolyai Soc. Math. Stud.}, VOLUME = {12}, PAGES = {171--221}, PUBLISHER = {Springer, Berlin}, YEAR = {2003}, MRCLASS = {14G25 (11G35 14G05)}, MRNUMBER = {2011747 (2005a:14027)}, MRREVIEWER = {Yuri Tschinkel}, DOI = {10.1007/978-3-662-05123-8_7}, URL = {http://dx.doi.org/10.1007/978-3-662-05123-8_7}, } @incollection {MR0427322, AUTHOR = {Manin, Y. I.}, TITLE = {Le groupe de {B}rauer-{G}rothendieck en g\'eom\'etrie diophantienne}, BOOKTITLE = {Actes du {C}ongr\`es {I}nternational des {M}ath\'ematiciens ({N}ice, 1970), {T}ome 1}, PAGES = {401--411}, PUBLISHER = {Gauthier-Villars, Paris}, YEAR = {1971}, MRCLASS = {14G25}, MRNUMBER = {0427322 (55 \#356)}, MRREVIEWER = {Loren D. Olson}, } @article {MR1666779, AUTHOR = {Skorobogatov, A. N.}, TITLE = {Beyond the {M}anin obstruction}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {135}, YEAR = {1999}, NUMBER = {2}, PAGES = {399--424}, ISSN = {0020-9910}, MRCLASS = {14F22 (11G35 14F42 14G25)}, MRNUMBER = {1666779 (2000c:14022)}, MRREVIEWER = {Tam{\'a}s Szamuely}, DOI = {10.1007/s002220050291}, URL = {http://dx.doi.org/10.1007/s002220050291}, } @article {MR3350224, AUTHOR = {Elsenhans, A.-S. and Jahnel, J.}, TITLE = {Cubic surfaces violating the {H}asse principle are {Z}ariski dense in the moduli scheme}, JOURNAL = {Adv. Math.}, FJOURNAL = {Advances in Mathematics}, VOLUME = {280}, YEAR = {2015}, PAGES = {360--378}, ISSN = {0001-8708}, MRCLASS = {11G35 (14G05 14G25 14J10 14J26)}, MRNUMBER = {3350224}, DOI = {10.1016/j.aim.2015.04.020}, URL = {http://dx.doi.org/10.1016/j.aim.2015.04.020}, } @article {MR0139989, AUTHOR = {Swinnerton-Dyer, H. P. F.}, TITLE = {Two special cubic surfaces}, JOURNAL = {Mathematika}, FJOURNAL = {Mathematika. A Journal of Pure and Applied Mathematics}, VOLUME = {9}, YEAR = {1962}, PAGES = {54--56}, ISSN = {0025-5793}, MRCLASS = {10.12 (14.40)}, MRNUMBER = {0139989 (25 \#3413)}, MRREVIEWER = {G. B. Huff}, } @misc {BrowningPrendiville, AUTHOR = {Browning, T.D. and Prendiville, S.}, TITLE = {Improvements in Birch's theorem on forms in many variables}, YEAR = {J. Reine Angew. Math., to appear, 2014}, } @misc {HanselmannThesis, AUTHOR = {Hanselmann, M.}, TITLE = {Rational points on quartic hypersurfaces}, YEAR = {Ph.D. thesis, Ludwig-Maximilians-Universit\"at M\"unchen, 2012}, } @article {MR2527413, AUTHOR = {Browning, T. D. and Heath-Brown, D. R.}, TITLE = {Rational points on quartic hypersurfaces}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik. [Crelle's Journal]}, VOLUME = {629}, YEAR = {2009}, PAGES = {37--88}, ISSN = {0075-4102}, CODEN = {JRMAA8}, MRCLASS = {14G25 (11G35 11P55 14G05)}, MRNUMBER = {2527413 (2010k:14032)}, MRREVIEWER = {Ulrich Derenthal}, DOI = {10.1515/CRELLE.2009.026}, URL = {http://dx.doi.org/10.1515/CRELLE.2009.026}, } @article {MR936992, AUTHOR = {Hooley, C.}, TITLE = {On nonary cubic forms}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik}, VOLUME = {386}, YEAR = {1988}, PAGES = {32--98}, ISSN = {0075-4102}, CODEN = {JRMAA8}, MRCLASS = {11D72 (11E76 11P55)}, MRNUMBER = {936992 (89h:11014)}, MRREVIEWER = {D. R. Heath-Brown}, DOI = {10.1515/crll.1988.386.32}, URL = {http://dx.doi.org/10.1515/crll.1988.386.32}, } @article {MR703978, AUTHOR = {Heath-Brown, D. R.}, TITLE = {Cubic forms in ten variables}, JOURNAL = {Proc. London Math. Soc. (3)}, FJOURNAL = {Proceedings of the London Mathematical Society. Third Series}, VOLUME = {47}, YEAR = {1983}, NUMBER = {2}, PAGES = {225--257}, ISSN = {0024-6115}, CODEN = {PLMTAL}, MRCLASS = {11D72 (11D88 11E76 11G25 11P55)}, MRNUMBER = {703978 (85b:11025)}, MRREVIEWER = {D. J. Lewis}, DOI = {10.1112/plms/s3-47.2.225}, URL = {http://dx.doi.org/10.1112/plms/s3-47.2.225}, } @article {MR2336082, AUTHOR = {Heath-Brown, D. R.}, TITLE = {Cubic forms in 14 variables}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {170}, YEAR = {2007}, NUMBER = {1}, PAGES = {199--230}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {11E76}, MRNUMBER = {2336082 (2010a:11063)}, MRREVIEWER = {Timothy D. Browning}, DOI = {10.1007/s00222-007-0062-1}, URL = {http://dx.doi.org/10.1007/s00222-007-0062-1}, } @article {MR0155800, AUTHOR = {Davenport, H.}, TITLE = {Cubic forms in sixteen variables}, JOURNAL = {Proc. Roy. Soc. Ser. A}, VOLUME = {272}, YEAR = {1963}, PAGES = {285--303}, MRCLASS = {10.17}, MRNUMBER = {0155800 (27 \#5734)}, MRREVIEWER = {B. J. Birch}, } @article {MR1421949, AUTHOR = {Heath-Brown, D. R.}, TITLE = {A new form of the circle method, and its application to quadratic forms}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik}, VOLUME = {481}, YEAR = {1996}, PAGES = {149--206}, ISSN = {0075-4102}, CODEN = {JRMAA8}, MRCLASS = {11P55 (11E20)}, MRNUMBER = {1421949 (97k:11139)}, MRREVIEWER = {R. C. Baker}, DOI = {10.1515/crll.1996.481.149}, URL = {http://dx.doi.org/10.1515/crll.1996.481.149}, } @article {MR2461509, AUTHOR = {Tao, Terence and Ziegler, Tamar}, TITLE = {The primes contain arbitrarily long polynomial progressions}, JOURNAL = {Acta Math.}, FJOURNAL = {Acta Mathematica}, VOLUME = {201}, YEAR = {2008}, NUMBER = {2}, PAGES = {213--305}, ISSN = {0001-5962}, CODEN = {ACMAA8}, MRCLASS = {11N13 (11B25 37A45)}, MRNUMBER = {2461509 (2009k:11144)}, MRREVIEWER = {Ben Joseph Green}, DOI = {10.1007/s11511-008-0032-5}, URL = {http://dx.doi.org/10.1007/s11511-008-0032-5}, } @article {MR2669681, AUTHOR = {Gowers, W.~T.}, TITLE = {Decompositions, approximate structure, transference, and the {H}ahn-{B}anach theorem}, JOURNAL = {Bull. Lond. Math. Soc.}, FJOURNAL = {Bulletin of the London Mathematical Society}, VOLUME = {42}, YEAR = {2010}, NUMBER = {4}, PAGES = {573--606}, ISSN = {0024-6093}, MRCLASS = {11B30 (05D99 11P32)}, MRNUMBER = {2669681 (2011k:11015)}, MRREVIEWER = {Julia Wolf}, DOI = {10.1112/blms/bdq018}, URL = {http://dx.doi.org/10.1112/blms/bdq018}, } @article {MR2817840, AUTHOR = {Green, B. and Tao, T. and Ziegler, T.}, TITLE = {An inverse theorem for the {G}owers {$U^{s+1}[N]$}-norm}, JOURNAL = {Electron. Res. Announc. Math. Sci.}, FJOURNAL = {Electronic Research Announcements in Mathematical Sciences}, VOLUME = {18}, YEAR = {2011}, PAGES = {69--90}, ISSN = {1935-9179}, MRCLASS = {11B30}, MRNUMBER = {2817840 (2012h:11013)}, MRREVIEWER = {Andreas O. Bender}, DOI = {10.3934/era.2011.18.69}, URL = {http://dx.doi.org/10.3934/era.2011.18.69}, } @article {MR2747135, AUTHOR = {Green, B. and Tao, T. and Ziegler, T.}, TITLE = {An inverse theorem for the {G}owers {$U^4$}-norm}, JOURNAL = {Glasg. Math. J.}, FJOURNAL = {Glasgow Mathematical Journal}, VOLUME = {53}, YEAR = {2011}, NUMBER = {1}, PAGES = {1--50}, ISSN = {0017-0895}, MRCLASS = {11B30 (11N13 11P70)}, MRNUMBER = {2747135 (2012b:11015)}, MRREVIEWER = {Julia Wolf}, DOI = {10.1017/S0017089510000546}, URL = {http://dx.doi.org/10.1017/S0017089510000546}, } @article {MR2138068, AUTHOR = {Bergelson, V. and Host, B. and Kra, B.}, TITLE = {Multiple recurrence and nilsequences}, NOTE = {With an appendix by Imre Ruzsa}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {160}, YEAR = {2005}, NUMBER = {2}, PAGES = {261--303}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {37A30 (05D10 28D05 37A05)}, MRNUMBER = {2138068 (2007i:37009)}, MRREVIEWER = {Randall McCutcheon}, DOI = {10.1007/s00222-004-0428-6}, URL = {http://dx.doi.org/10.1007/s00222-004-0428-6}, } @article {MR2150389, AUTHOR = {Host, B. and Kra, B.}, TITLE = {Nonconventional ergodic averages and nilmanifolds}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {161}, YEAR = {2005}, NUMBER = {1}, PAGES = {397--488}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {37A05 (28D05)}, MRNUMBER = {2150389 (2007b:37004)}, MRREVIEWER = {Randall McCutcheon}, DOI = {10.4007/annals.2005.161.397}, URL = {http://dx.doi.org/10.4007/annals.2005.161.397}, } @article {MR2391635, AUTHOR = {Green, B. and Tao, T.}, TITLE = {An inverse theorem for the {G}owers {$U^3(G)$} norm}, JOURNAL = {Proc. Edinb. Math. Soc. (2)}, FJOURNAL = {Proceedings of the Edinburgh Mathematical Society. Series II}, VOLUME = {51}, YEAR = {2008}, NUMBER = {1}, PAGES = {73--153}, ISSN = {0013-0915}, MRCLASS = {11B25 (11B75 11P55 11P70)}, MRNUMBER = {2391635 (2009g:11012)}, MRREVIEWER = {Serge{\u\i} V. Konyagin}, DOI = {10.1017/S0013091505000325}, URL = {http://dx.doi.org/10.1017/S0013091505000325}, } @article {MR1631259, AUTHOR = {Gowers, W.~T.}, TITLE = {A new proof of {S}zemer\'edi's theorem for arithmetic progressions of length four}, JOURNAL = {Geom. Funct. Anal.}, FJOURNAL = {Geometric and Functional Analysis}, VOLUME = {8}, YEAR = {1998}, NUMBER = {3}, PAGES = {529--551}, ISSN = {1016-443X}, CODEN = {GFANFB}, MRCLASS = {11B25 (11N13)}, MRNUMBER = {1631259 (2000d:11019)}, MRREVIEWER = {D. R. Heath-Brown}, DOI = {10.1007/s000390050065}, URL = {http://dx.doi.org/10.1007/s000390050065}, } @article {MR2473624, AUTHOR = {Green, B. and Tao, T.}, TITLE = {Quadratic uniformity of the {M}\"obius function}, JOURNAL = {Ann. Inst. Fourier (Grenoble)}, FJOURNAL = {Universit\'e de Grenoble. Annales de l'Institut Fourier}, VOLUME = {58}, YEAR = {2008}, NUMBER = {6}, PAGES = {1863--1935}, ISSN = {0373-0956}, CODEN = {AIFUA7}, MRCLASS = {11B30 (11A25)}, MRNUMBER = {2473624 (2010c:11012)}, MRREVIEWER = {Tamar Ziegler}, URL = {http://aif.cedram.org/item?id=AIF_2008__58_6_1863_0}, } @article {MR2415379, AUTHOR = {Green, B. and Tao, T.}, TITLE = {The primes contain arbitrarily long arithmetic progressions}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {167}, YEAR = {2008}, NUMBER = {2}, PAGES = {481--547}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {11N13 (11A41 11B25 37A45)}, MRNUMBER = {2415379 (2009e:11181)}, MRREVIEWER = {Tamar Ziegler}, DOI = {10.4007/annals.2008.167.481}, URL = {http://dx.doi.org/10.4007/annals.2008.167.481}, } @article {MR1844079, AUTHOR = {Gowers, W.~T.}, TITLE = {A new proof of {S}zemer\'edi's theorem}, JOURNAL = {Geom. Funct. Anal.}, FJOURNAL = {Geometric and Functional Analysis}, VOLUME = {11}, YEAR = {2001}, NUMBER = {3}, PAGES = {465--588}, ISSN = {1016-443X}, CODEN = {GFANFB}, MRCLASS = {11B25 (11K38 11K45)}, MRNUMBER = {1844079 (2002k:11014)}, MRREVIEWER = {Hillel Furstenberg}, DOI = {10.1007/s00039-001-0332-9}, URL = {http://dx.doi.org/10.1007/s00039-001-0332-9}, } @article {MR2877065, AUTHOR = {Green, B. and Tao, T.}, TITLE = {The quantitative behaviour of polynomial orbits on nilmanifolds}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {175}, YEAR = {2012}, NUMBER = {2}, PAGES = {465--540}, ISSN = {0003-486X}, MRCLASS = {37A15}, MRNUMBER = {2877065}, MRREVIEWER = {Tamar Ziegler}, DOI = {10.4007/annals.2012.175.2.2}, URL = {http://dx.doi.org/10.4007/annals.2012.175.2.2}, } @article {MR2877066, AUTHOR = {Green, B. and Tao, T.}, TITLE = {The {M}\"obius function is strongly orthogonal to nilsequences}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {175}, YEAR = {2012}, NUMBER = {2}, PAGES = {541--566}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {37A45 (11A25)}, MRNUMBER = {2877066}, MRREVIEWER = {Tamar Ziegler}, DOI = {10.4007/annals.2012.175.2.3}, URL = {http://dx.doi.org/10.4007/annals.2012.175.2.3}, } @article {MR2680398, AUTHOR = {Green, B. and Tao, T.}, TITLE = {Linear equations in primes}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {171}, YEAR = {2010}, NUMBER = {3}, PAGES = {1753--1850}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {11N13 (11B30 11P32)}, MRNUMBER = {2680398 (2011j:11177)}, MRREVIEWER = {Tamar Ziegler}, DOI = {10.4007/annals.2010.171.1753}, URL = {http://dx.doi.org/10.4007/annals.2010.171.1753}, } @article {MR2950773, AUTHOR = {Green, B. and Tao, T. and Ziegler, T.}, TITLE = {An inverse theorem for the {G}owers {$U^{s+1}[N]$}-norm}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {176}, YEAR = {2012}, NUMBER = {2}, PAGES = {1231--1372}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {11B30}, MRNUMBER = {2950773}, MRREVIEWER = {Julia Wolf}, DOI = {10.4007/annals.2012.176.2.11}, URL = {http://dx.doi.org/10.4007/annals.2012.176.2.11}, } @article {MR2908784, AUTHOR = {Matthiesen, L.}, TITLE = {Correlations of the divisor function}, JOURNAL = {Proc. Lond. Math. Soc. (3)}, FJOURNAL = {Proceedings of the London Mathematical Society. Third Series}, VOLUME = {104}, YEAR = {2012}, NUMBER = {4}, PAGES = {827--858}, ISSN = {0024-6115}, MRCLASS = {11N37 (11B30 11N64)}, MRNUMBER = {2908784}, MRREVIEWER = {Julia Wolf}, DOI = {10.1112/plms/pdr046}, URL = {http://dx.doi.org/10.1112/plms/pdr046}, } @article {MR2949879, AUTHOR = {Matthiesen, L.}, TITLE = {Linear correlations amongst numbers represented by positive definite binary quadratic forms}, JOURNAL = {Acta Arith.}, FJOURNAL = {Acta Arithmetica}, VOLUME = {154}, YEAR = {2012}, NUMBER = {3}, PAGES = {235--306}, ISSN = {0065-1036}, CODEN = {AARIA9}, MRCLASS = {11N37 (11B30 11E25)}, MRNUMBER = {2949879}, MRREVIEWER = {Tom Sanders}, DOI = {10.4064/aa154-3-2}, URL = {http://dx.doi.org/10.4064/aa154-3-2}, } @article {MR3040665, AUTHOR = {Matthiesen, L.}, TITLE = {Correlations of representation functions of binary quadratic forms}, JOURNAL = {Acta Arith.}, FJOURNAL = {Acta Arithmetica}, VOLUME = {158}, YEAR = {2013}, NUMBER = {3}, PAGES = {245--252}, ISSN = {0065-1036}, CODEN = {AARIA9}, MRCLASS = {11N37 (11E25)}, MRNUMBER = {3040665}, MRREVIEWER = {Gihan Marasingha}, DOI = {10.4064/aa158-3-4}, URL = {http://dx.doi.org/10.4064/aa158-3-4}, } @article {MR0025463, AUTHOR = {Delsarte, S.}, TITLE = {Fonctions de {M}\"obius sur les groupes abeliens finis}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {49}, YEAR = {1948}, PAGES = {600--609}, ISSN = {0003-486X}, MRCLASS = {20.0X}, MRNUMBER = {0025463 (10,9b)}, MRREVIEWER = {H. N. Shapiro}, } @article {MR2136219, AUTHOR = {Bekyel, E.}, TITLE = {Steinitz classes and discriminant counting}, JOURNAL = {Acta Arith.}, FJOURNAL = {Acta Arithmetica}, VOLUME = {118}, YEAR = {2005}, NUMBER = {1}, PAGES = {27--40}, ISSN = {0065-1036}, CODEN = {AARIA9}, MRCLASS = {11R47 (11R29 11R45)}, MRNUMBER = {2136219 (2006d:11139)}, MRREVIEWER = {F. Diaz y Diaz}, DOI = {10.4064/aa118-1-3}, URL = {http://dx.doi.org/10.4064/aa118-1-3}, } @article {MR2159764, AUTHOR = {Wong, S.}, TITLE = {Densities of quartic fields with even {G}alois groups}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {133}, YEAR = {2005}, NUMBER = {10}, PAGES = {2873--2881 (electronic)}, ISSN = {0002-9939}, CODEN = {PAMYAR}, MRCLASS = {11R45 (11G05 11R16 11R29)}, MRNUMBER = {2159764 (2006d:11138)}, MRREVIEWER = {D. R. Heath-Brown}, DOI = {10.1090/S0002-9939-05-07921-9}, URL = {http://dx.doi.org/10.1090/S0002-9939-05-07921-9}, } @article {MR2211303, AUTHOR = {Cohen, H. and Diaz y Diaz, F. and Olivier, M.}, TITLE = {Counting cyclic quartic extensions of a number field}, JOURNAL = {J. Th\'eor. Nombres Bordeaux}, FJOURNAL = {Journal de Th\'eorie des Nombres de Bordeaux}, VOLUME = {17}, YEAR = {2005}, NUMBER = {2}, PAGES = {475--510}, ISSN = {1246-7405}, MRCLASS = {11R45 (11R16 11R29 11R32)}, MRNUMBER = {2211303 (2006m:11164)}, MRREVIEWER = {T. Mets{\"a}nkyl{\"a}}, URL = {http://jtnb.cedram.org/item?id=JTNB_2005__17_2_475_0}, } @article {MR4201547, AUTHOR = {Frei, C. and Koymans, P. and Sofos, E.}, TITLE = {Vinogradov's three primes theorem with primes having given primitive roots}, JOURNAL = {Math. Proc. Cambridge Philos. Soc.}, FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society}, VOLUME = {170}, YEAR = {2021}, NUMBER = {1}, PAGES = {75--110}, ISSN = {0305-0041}, MRCLASS = {11P32 (11P55 11R45)}, MRNUMBER = {4201547}, MRREVIEWER = {Angel V. Kumchev}, DOI = {10.1017/S0305004119000331}, URL = {https://doi.org/10.1017/S0305004119000331}, } @article {MR2138067, AUTHOR = {Kable, A.~C. and Yukie, A.}, TITLE = {On the number of quintic fields}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {160}, YEAR = {2005}, NUMBER = {2}, PAGES = {217--259}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {11R21 (11R47)}, MRNUMBER = {2138067 (2006g:11217)}, MRREVIEWER = {Takeyoshi Kogiso}, DOI = {10.1007/s00222-004-0391-2}, URL = {http://dx.doi.org/10.1007/s00222-004-0391-2}, } @article {MR2196763, AUTHOR = {Kable, A.~C. and Wright, D.~J.}, TITLE = {Uniform distribution of the {S}teinitz invariants of quadratic and cubic extensions}, JOURNAL = {Compos. Math.}, FJOURNAL = {Compositio Mathematica}, VOLUME = {142}, YEAR = {2006}, NUMBER = {1}, PAGES = {84--100}, ISSN = {0010-437X}, MRCLASS = {11R45 (11M41 11R11 11R16 11S90)}, MRNUMBER = {2196763 (2006k:11225)}, MRREVIEWER = {Ming Guang Leu}, DOI = {10.1112/S0010437X05001740}, URL = {http://dx.doi.org/10.1112/S0010437X05001740}, } @article {MR2354798, AUTHOR = {Bhargava, M.}, TITLE = {Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants}, JOURNAL = {Int. Math. Res. Not. IMRN}, FJOURNAL = {International Mathematics Research Notices. IMRN}, YEAR = {2007}, NUMBER = {17}, PAGES = {Art. ID rnm052, 20}, ISSN = {1073-7928}, MRCLASS = {11S20 (11R29 11R45)}, MRNUMBER = {2354798 (2009e:11220)}, MRREVIEWER = {Laurent N. Berger}, DOI = {10.1093/imrn/rnm052}, URL = {http://dx.doi.org/10.1093/imrn/rnm052}, } @article {MR2362439, AUTHOR = {Datskovsky, B.~A. and Mammo, B.}, TITLE = {A mean value theorem for discriminants of abelian extensions of a number field}, JOURNAL = {J. Number Theory}, FJOURNAL = {Journal of Number Theory}, VOLUME = {127}, YEAR = {2007}, NUMBER = {2}, PAGES = {301--325}, ISSN = {0022-314X}, CODEN = {JNUTA9}, MRCLASS = {11R32 (11R20 11R29 11R47)}, MRNUMBER = {2362439 (2008j:11164)}, MRREVIEWER = {Lawrence Washington}, DOI = {10.1016/j.jnt.2007.06.002}, URL = {http://dx.doi.org/10.1016/j.jnt.2007.06.002}, } @article {MR2373587, AUTHOR = {Bhargava, M. and Wood, M.~M.}, TITLE = {The density of discriminants of {$S_3$}-sextic number fields}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {136}, YEAR = {2008}, NUMBER = {5}, PAGES = {1581--1587}, ISSN = {0002-9939}, CODEN = {PAMYAR}, MRCLASS = {11R45 (11R21 11R29)}, MRNUMBER = {2373587 (2009a:11234)}, MRREVIEWER = {J{\"u}rgen Kl{\"u}ners}, DOI = {10.1090/S0002-9939-07-09171-X}, URL = {http://dx.doi.org/10.1090/S0002-9939-07-09171-X}, } @article {MR2726582, AUTHOR = {Mammo, B.}, TITLE = {On the density of discriminants of abelian extensions of a number field}, JOURNAL = {Int. J. Number Theory}, FJOURNAL = {International Journal of Number Theory}, VOLUME = {6}, YEAR = {2010}, NUMBER = {6}, PAGES = {1273--1291}, ISSN = {1793-0421}, MRCLASS = {11R45 (11R20 11R29 11R47)}, MRNUMBER = {2726582 (2011i:11163)}, MRREVIEWER = {Roland Qu{\^e}me}, DOI = {10.1142/S1793042110003502}, URL = {http://dx.doi.org/10.1142/S1793042110003502}, } @article {MR2925852, AUTHOR = {Lagemann, T.}, TITLE = {Distribution of {A}rtin-{S}chreier extensions}, JOURNAL = {J. Number Theory}, FJOURNAL = {Journal of Number Theory}, VOLUME = {132}, YEAR = {2012}, NUMBER = {9}, PAGES = {1867--1887}, ISSN = {0022-314X}, CODEN = {JNUTA9}, MRCLASS = {11R58 (11M45 11R45)}, MRNUMBER = {2925852}, MRREVIEWER = {Cornelius Greither}, DOI = {10.1016/j.jnt.2012.03.011}, URL = {http://dx.doi.org/10.1016/j.jnt.2012.03.011}, } @article {MR2965759, AUTHOR = {Lee, Seok H. and Oh, G.}, TITLE = {On the distribution of cyclic number fields of prime degree}, JOURNAL = {Int. J. Number Theory}, FJOURNAL = {International Journal of Number Theory}, VOLUME = {8}, YEAR = {2012}, NUMBER = {6}, PAGES = {1463--1475}, ISSN = {1793-0421}, MRCLASS = {11R47 (11M45 11N45)}, MRNUMBER = {2965759}, MRREVIEWER = {{\c{S}}. A. Basarab}, DOI = {10.1142/S1793042112500868}, URL = {http://dx.doi.org/10.1142/S1793042112500868}, } @article {MR2999148, AUTHOR = {Wood, M.~M.}, TITLE = {The distribution of the number of points on trigonal curves over {$\Bbb F_q$}}, JOURNAL = {Int. Math. Res. Not. IMRN}, FJOURNAL = {International Mathematics Research Notices. IMRN}, YEAR = {2012}, NUMBER = {23}, PAGES = {5444--5456}, ISSN = {1073-7928}, MRCLASS = {11G20 (14G15)}, MRNUMBER = {2999148}, MRREVIEWER = {Benjamin Smith}, } @article {MR2996953, AUTHOR = {Larson, E. and Rolen, L.}, TITLE = {Upper bounds for the number of number fields with alternating {G}alois group}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {141}, YEAR = {2013}, NUMBER = {2}, PAGES = {499--503}, ISSN = {0002-9939}, CODEN = {PAMYAR}, MRCLASS = {11R32 (11G05)}, MRNUMBER = {2996953}, MRREVIEWER = {J{\"u}rgen Kl{\"u}ners}, DOI = {10.1090/S0002-9939-2012-11543-6}, URL = {http://dx.doi.org/10.1090/S0002-9939-2012-11543-6}, } @article {MR3165634, AUTHOR = {Cho, P.~J. and Kim, H.~H.}, TITLE = {Probabilistic properties of number fields}, JOURNAL = {J. Number Theory}, FJOURNAL = {Journal of Number Theory}, VOLUME = {133}, YEAR = {2013}, NUMBER = {12}, PAGES = {4175--4187}, ISSN = {0022-314X}, MRCLASS = {11R42 (11M41)}, MRNUMBER = {3165634}, MRREVIEWER = {Timothy Lee Gillespie}, DOI = {10.1016/j.jnt.2013.06.009}, URL = {http://dx.doi.org/10.1016/j.jnt.2013.06.009}, } @article {MR3215550, AUTHOR = {Cohen, H. and Thorne, F.}, TITLE = {Dirichlet series associated to cubic fields with given quadratic resolvent}, JOURNAL = {Michigan Math. J.}, FJOURNAL = {Michigan Mathematical Journal}, VOLUME = {63}, YEAR = {2014}, NUMBER = {2}, PAGES = {253--273}, ISSN = {0026-2285}, MRCLASS = {11R16 (11M41 11R29)}, MRNUMBER = {3215550}, MRREVIEWER = {St{\'e}phane R. Louboutin}, DOI = {10.1307/mmj/1401973050}, URL = {http://dx.doi.org/10.1307/mmj/1401973050}, } @article {MR3283181, AUTHOR = {Lagemann, T.}, TITLE = {Distribution of {A}rtin-{S}chreier-{W}itt extensions}, JOURNAL = {J. Number Theory}, FJOURNAL = {Journal of Number Theory}, VOLUME = {148}, YEAR = {2015}, PAGES = {288--310}, ISSN = {0022-314X}, MRCLASS = {11R58 (11M45 11R32 11R37 11R45)}, MRNUMBER = {3283181}, MRREVIEWER = {Martha Rzedowski-Calder{\'o}n}, DOI = {10.1016/j.jnt.2014.09.026}, URL = {http://dx.doi.org/10.1016/j.jnt.2014.09.026}, } @incollection {MR2159381, AUTHOR = {Ellenberg, J.~S. and Venkatesh, A.}, TITLE = {Counting extensions of function fields with bounded discriminant and specified {G}alois group}, BOOKTITLE = {Geometric methods in algebra and number theory}, SERIES = {Progr. Math.}, VOLUME = 235, PAGES = {151--168}, PUBLISHER = {Birkh\"auser Boston, Boston, MA}, YEAR = 2005, MRCLASS = {11R58 (12F12)}, MRNUMBER = {2159381 (2006f:11139)}, MRREVIEWER = {Xianke Zhang}, DOI = {10.1007/0-8176-4417-2_7}, URL = {http://dx.doi.org/10.1007/0-8176-4417-2_7}, } @article {MR3349443, AUTHOR = {T{\"u}rkelli, S.}, TITLE = {Connected components of {H}urwitz schemes and {M}alle's conjecture}, JOURNAL = {J. Number Theory}, FJOURNAL = {Journal of Number Theory}, VOLUME = {155}, YEAR = {2015}, PAGES = {163--201}, ISSN = {0022-314X}, MRCLASS = {11R32 (14H30)}, MRNUMBER = {3349443}, DOI = {10.1016/j.jnt.2015.03.005}, URL = {http://dx.doi.org/10.1016/j.jnt.2015.03.005}, } @article {MR744089, AUTHOR = {Taylor, M.~J.}, TITLE = {On the equidistribution of {F}robenius in cyclic extensions of a number field}, JOURNAL = {J. London Math. Soc. (2)}, FJOURNAL = {Journal of the London Mathematical Society. Second Series}, VOLUME = {29}, YEAR = {1984}, NUMBER = {2}, PAGES = {211--223}, ISSN = {0024-6107}, CODEN = {JLMSAK}, MRCLASS = {11R20 (11N37)}, MRNUMBER = {744089 (85f:11074)}, MRREVIEWER = {Kenz{\^o} Komatsu}, DOI = {10.1112/jlms/s2-29.2.211}, URL = {http://dx.doi.org/10.1112/jlms/s2-29.2.211}, } @article {MR791087, AUTHOR = {M{\"a}ki, S.}, TITLE = {On the density of abelian number fields}, JOURNAL = {Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes}, FJOURNAL = {Annales Academiae Scientiarum Fennicae. Series A I. Mathematica Dissertationes}, NUMBER = {54}, YEAR = {1985}, PAGES = {104}, ISSN = {0355-0087}, MRCLASS = {11N45 (11R21)}, MRNUMBER = {791087 (86h:11079)}, MRREVIEWER = {M. Ram Murty}, } @article {MR564533, AUTHOR = {Baily, A.~M.}, TITLE = {On the density of discriminants of quartic fields}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik}, VOLUME = {315}, YEAR = {1980}, PAGES = {190--210}, ISSN = {0075-4102}, CODEN = {JRMAA8}, MRCLASS = {12A30 (12A50)}, MRNUMBER = {564533 (81c:12006)}, MRREVIEWER = {Kenz{\^o} Komatsu}, DOI = {10.1515/crll.1980.315.190}, URL = {http://dx.doi.org/10.1515/crll.1980.315.190}, } @article {MR636193, AUTHOR = {Baily, A.~M.}, TITLE = {On octic fields of exponent {$2$}}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik}, VOLUME = {328}, YEAR = {1981}, PAGES = {33--38}, ISSN = {0075-4102}, CODEN = {JRMAA8}, MRCLASS = {12A40}, MRNUMBER = {636193 (83d:12002)}, MRREVIEWER = {L. Skula}, DOI = {10.1515/crll.1981.328.33}, URL = {http://dx.doi.org/10.1515/crll.1981.328.33}, } @article {MR0568295, AUTHOR = {Urazbaev, B. M.}, TITLE = {The distribution of cyclic fields}, JOURNAL = {Izv. Akad. Nauk Kazah. SSR Ser. Fiz.-Mat.}, FJOURNAL = {Izvestiya Akademii Nauk Kazakhsko\u\i\ SSR. Seriya Fiziko-Matematicheskaya}, YEAR = {1977}, NUMBER = {5}, PAGES = {66--70, 93}, ISSN = {0002-3191}, MRCLASS = {12A35 (10H25)}, MRNUMBER = {0568295 (58 \#27897)}, } @article {MR836624, AUTHOR = {Khushvaktov, M.}, TITLE = {Asymptotic expansions for the number of certain finite abelian extensions of the rational number field}, JOURNAL = {Dokl. Akad. Nauk UzSSR}, FJOURNAL = {Doklady Akademii Nauk UzSSR}, YEAR = {1986}, NUMBER = {2}, PAGES = {6--8}, ISSN = {0134-4307}, MRCLASS = {11N45 (11R20)}, MRNUMBER = {836624 (87f:11076)}, MRREVIEWER = {W. Narkiewicz}, } @article {MR0064076, AUTHOR = {Cohn, H.}, TITLE = {The density of abelian cubic fields}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {5}, YEAR = {1954}, PAGES = {476--477}, ISSN = {0002-9939}, MRCLASS = {10.0X}, MRNUMBER = {0064076 (16,222a)}, MRREVIEWER = {M. Krasner}, } @incollection {MR2647601, AUTHOR = {Chambert-Loir, Antoine}, TITLE = {Lectures on height zeta functions: at the confluence of algebraic geometry, algebraic number theory, and analysis}, BOOKTITLE = {Algebraic and analytic aspects of zeta functions and {$L$}-functions}, SERIES = {MSJ Mem.}, VOLUME = {21}, PAGES = {17--49}, PUBLISHER = {Math. Soc. Japan, Tokyo}, YEAR = {2010}, MRCLASS = {11M41 (11G50 14G10)}, MRNUMBER = {2647601 (2011g:11171)}, MRREVIEWER = {Robert Juricevic}, } @incollection {MR2498064, AUTHOR = {Tschinkel, Yu.}, TITLE = {Algebraic varieties with many rational points}, BOOKTITLE = {Arithmetic geometry}, SERIES = {Clay Math. Proc.}, VOLUME = {8}, PAGES = {243--334}, PUBLISHER = {Amer. Math. Soc., Providence, RI}, YEAR = {2009}, MRCLASS = {14G05 (11G35 11G50)}, MRNUMBER = {2498064 (2010j:14039)}, MRREVIEWER = {Timothy D. Browning}, } @incollection {MR0217026, AUTHOR = {Tate, J. T.}, TITLE = {Fourier analysis in number fields, and {H}ecke's zeta-functions}, BOOKTITLE = {Algebraic {N}umber {T}heory ({P}roc. {I}nstructional {C}onf., {B}righton, 1965)}, PAGES = {305--347}, PUBLISHER = {Thompson, Washington, D.C.}, YEAR = {1967}, MRCLASS = {10.41}, MRNUMBER = {0217026 (36 \#121)}, } @misc{TaklooTanimoto2015, AUTHOR={Takloo-Bighash, R. and Tanimoto, S.}, TITLE={Distribution of rational points of bounded height on equivariant compactifications of $\text{PGL}_2$}, YEAR={preprint, 2015}, } @misc{LehmannTanimotoTschinkel2014, AUTHOR={Lehmann, B. and Tanimoto, S. and Tschinkel, Yu.}, TITLE={Balanced line bundles on Fano varieties}, YEAR={preprint, 2014}, } @misc{ShalikaTschinkel2014, AUTHOR={Shalika, J. and Tschinkel, Yu.}, TITLE={Height zeta functions of equivariant compactifications of unipotent groups}, YEAR={to appear in Comm. Pure and Applied Math., 2014}, } @misc{HassetTanimotoTschinkelIMRN, AUTHOR={Hassett, B. and Tanimoto, S. and Tschinkel, Yu.}, TITLE={Balanced line bundles and equivariant compactifications of homogeneous spaces}, YEAR={IMRN, 2014}, } @misc{GorodnikTschinkeMixing, AUTHOR = {Gorodnik, A. and Takloo-Bighash, R. and Tschinkel, Yu.}, TITLE = {Multiple mixing for adele groups and rational points}, YEAR = {to appear in J. Eur. Math. Soc., 2015}, } @article {MR3117310, AUTHOR = {Takloo-Bighash, R. and Tschinkel, Yu.}, TITLE = {Integral points of bounded height on compactifications of semi-simple groups}, JOURNAL = {Amer. J. Math.}, FJOURNAL = {American Journal of Mathematics}, VOLUME = {135}, YEAR = {2013}, NUMBER = {5}, PAGES = {1433--1448}, ISSN = {0002-9327}, MRCLASS = {14G05 (11G50 14M27 20G30)}, MRNUMBER = {3117310}, MRREVIEWER = {Jacopo Gandini}, DOI = {10.1353/ajm.2013.0044}, URL = {http://dx.doi.org/10.1353/ajm.2013.0044}, } @misc {CLTschinkel2010, AUTHOR = {Chambert-Loir, A. and Tschinkel, Yu.}, TITLE = {Integral points of bounded height on toric varieties}, YEAR = {preprint, 2010}, } @article {MR2999313, AUTHOR = {Chambert-Loir, A. and Tschinkel, Yu.}, TITLE = {Integral points of bounded height on partial equivariant compactifications of vector groups}, JOURNAL = {Duke Math. J.}, FJOURNAL = {Duke Mathematical Journal}, VOLUME = {161}, YEAR = {2012}, NUMBER = {15}, PAGES = {2799--2836}, ISSN = {0012-7094}, CODEN = {DUMJAO}, MRCLASS = {11G50 (11G25 14G05 14G25)}, MRNUMBER = {2999313}, MRREVIEWER = {Yu Yasufuku}, DOI = {10.1215/00127094-1813638}, URL = {http://dx.doi.org/10.1215/00127094-1813638}, } @article {MR2740045, AUTHOR = {Chambert-Loir, A. and Tschinkel, Yu.}, TITLE = {Igusa integrals and volume asymptotics in analytic and adelic geometry}, JOURNAL = {Confluentes Math.}, FJOURNAL = {Confluentes Mathematici}, VOLUME = {2}, YEAR = {2010}, NUMBER = {3}, PAGES = {351--429}, ISSN = {1793-7442}, MRCLASS = {11G50 (11G35 14G05)}, MRNUMBER = {2740045 (2012d:11143)}, MRREVIEWER = {Boris {\`E}. Kunyavski{\u\i}}, DOI = {10.1142/S1793744210000223}, URL = {http://dx.doi.org/10.1142/S1793744210000223}, } @article {MR2328719, AUTHOR = {Shalika, J. and Takloo-Bighash, R. and Tschinkel, Yu.}, TITLE = {Rational points on compactifications of semi-simple groups}, JOURNAL = {J. Amer. Math. Soc.}, FJOURNAL = {Journal of the American Mathematical Society}, VOLUME = {20}, YEAR = {2007}, NUMBER = {4}, PAGES = {1135--1186 (electronic)}, ISSN = {0894-0347}, MRCLASS = {14G05 (11F70 11G50)}, MRNUMBER = {2328719 (2008g:14034)}, MRREVIEWER = {Tam{\'a}s Szamuely}, DOI = {10.1090/S0894-0347-07-00572-3}, URL = {http://dx.doi.org/10.1090/S0894-0347-07-00572-3}, } @incollection {MR2029871, AUTHOR = {Shalika, J. and Takloo-Bighash, R. and Tschinkel, Yu.}, TITLE = {Rational points on compactifications of semi-simple groups of rank 1}, BOOKTITLE = {Arithmetic of higher-dimensional algebraic varieties ({P}alo {A}lto, {CA}, 2002)}, SERIES = {Progr. Math.}, VOLUME = {226}, PAGES = {205--233}, PUBLISHER = {Birkh\"auser Boston, Boston, MA}, YEAR = {2004}, MRCLASS = {11G50 (11M41 14G05 14G25)}, MRNUMBER = {2029871 (2004m:11104)}, MRREVIEWER = {Robin de Jong}, } @incollection {MR2058627, AUTHOR = {Shalika, J.~A. and Tschinkel, Yu.}, TITLE = {Height zeta functions of equivariant compactifications of the {H}eisenberg group}, BOOKTITLE = {Contributions to automorphic forms, geometry, and number theory}, PAGES = {743--771}, PUBLISHER = {Johns Hopkins Univ. Press, Baltimore, MD}, YEAR = {2004}, MRCLASS = {11M41 (11G50)}, MRNUMBER = {2058627 (2005c:11110)}, MRREVIEWER = {Timothy D. Browning}, } @incollection {MR2058626, AUTHOR = {Shalika, J.~A. and Takloo-Bighash, R. and Tschinkel, Yu.}, TITLE = {Rational points and automorphic forms}, BOOKTITLE = {Contributions to automorphic forms, geometry, and number theory}, PAGES = {733--742}, PUBLISHER = {Johns Hopkins Univ. Press, Baltimore, MD}, YEAR = {2004}, MRCLASS = {11G50 (11F11 11G35 22E50)}, MRNUMBER = {2058626 (2005d:11100)}, MRREVIEWER = {Timothy D. Browning}, } @incollection {MR1875174, AUTHOR = {Hassett, B. and Tschinkel, Yu.}, TITLE = {Density of integral points on algebraic varieties}, BOOKTITLE = {Rational points on algebraic varieties}, SERIES = {Progr. Math.}, VOLUME = {199}, PAGES = {169--197}, PUBLISHER = {Birkh\"auser, Basel}, YEAR = {2001}, MRCLASS = {14G25 (11G35 14D10 14G05 14J20)}, MRNUMBER = {1875174 (2003a:14034)}, MRREVIEWER = {Tam{\'a}s Szamuely}, DOI = {10.1007/978-3-0348-8368-9_7}, URL = {http://dx.doi.org/10.1007/978-3-0348-8368-9_7}, } @incollection {MR1875177, AUTHOR = {Peyre, E. and Tschinkel, Yu.}, TITLE = {Tamagawa numbers of diagonal cubic surfaces of higher rank}, BOOKTITLE = {Rational points on algebraic varieties}, SERIES = {Progr. Math.}, VOLUME = {199}, PAGES = {275--305}, PUBLISHER = {Birkh\"auser, Basel}, YEAR = {2001}, MRCLASS = {11G35 (14G25)}, MRNUMBER = {1875177 (2003a:11076)}, MRREVIEWER = {Carlo Gasbarri}, } @article {MR1802710, AUTHOR = {Chambert-Loir, A. and Tschinkel, Yu.}, TITLE = {Points of bounded height on equivariant compactifications of vector groups. {II}}, JOURNAL = {J. Number Theory}, FJOURNAL = {Journal of Number Theory}, VOLUME = {85}, YEAR = {2000}, NUMBER = {2}, PAGES = {172--188}, ISSN = {0022-314X}, CODEN = {JNUTA9}, MRCLASS = {11G50 (11G35 14G05 14G40)}, MRNUMBER = {1802710 (2001k:11118)}, MRREVIEWER = {Matthias Strauch}, DOI = {10.1006/jnth.2000.2539}, URL = {http://dx.doi.org/10.1006/jnth.2000.2539}, } @article {MR1797654, AUTHOR = {Chambert-Loir, A. and Tschinkel, Yu.}, TITLE = {Points of bounded height on equivariant compactifications of vector groups. {I}}, JOURNAL = {Compositio Math.}, FJOURNAL = {Compositio Mathematica}, VOLUME = {124}, YEAR = {2000}, NUMBER = {1}, PAGES = {65--93}, ISSN = {0010-437X}, CODEN = {CMPMAF}, MRCLASS = {11G50 (11G35 14G05 14G40)}, MRNUMBER = {1797654 (2001j:11053)}, MRREVIEWER = {Matthias Strauch}, DOI = {10.1023/A:1002431622732}, URL = {http://dx.doi.org/10.1023/A:1002431622732}, } @incollection {MR1875171, AUTHOR = {Chambert-Loir, A. and Tschinkel, Yu.}, TITLE = {Fonctions z\^eta des hauteurs des espaces fibr\'es}, BOOKTITLE = {Rational points on algebraic varieties}, SERIES = {Progr. Math.}, VOLUME = {199}, PAGES = {71--115}, PUBLISHER = {Birkh\"auser, Basel}, YEAR = {2001}, MRCLASS = {11G50 (11G35 14G05 14G10 14G40)}, MRNUMBER = {1875171 (2003a:11079)}, MRREVIEWER = {Matthias Strauch}, } @incollection {MR1875170, AUTHOR = {Chambert-Loir, A. and Tschinkel, Yu.}, TITLE = {Torseurs arithm\'etiques et espaces fibr\'es}, BOOKTITLE = {Rational points on algebraic varieties}, SERIES = {Progr. Math.}, VOLUME = {199}, PAGES = {37--70}, PUBLISHER = {Birkh\"auser, Basel}, YEAR = {2001}, MRCLASS = {11G35 (11G50 14G05 14G40)}, MRNUMBER = {1875170 (2002m:11054)}, MRREVIEWER = {Matthias Strauch}, } @article {MR1723811, AUTHOR = {Strauch, M. and Tschinkel, Yu.}, TITLE = {Height zeta functions of toric bundles over flag varieties}, JOURNAL = {Selecta Math. (N.S.)}, FJOURNAL = {Selecta Mathematica. New Series}, VOLUME = {5}, YEAR = {1999}, NUMBER = {3}, PAGES = {325--396}, ISSN = {1022-1824}, CODEN = {SMATF6}, MRCLASS = {14G40 (11G35 11G50 14G05 14M15 14M25)}, MRNUMBER = {1723811 (2001h:14028)}, MRREVIEWER = {Emmanuel Peyre}, DOI = {10.1007/s000290050051}, URL = {http://dx.doi.org/10.1007/s000290050051}, } @article {MR1423638, AUTHOR = {Batyrev, V. and Tschinkel, Yu.}, TITLE = {Height zeta functions of toric varieties}, NOTE = {Algebraic geometry, 5}, JOURNAL = {J. Math. Sci.}, FJOURNAL = {Journal of Mathematical Sciences}, VOLUME = {82}, YEAR = {1996}, NUMBER = {1}, PAGES = {3220--3239}, ISSN = {1072-3374}, CODEN = {JMTSEW}, MRCLASS = {11G35 (14G05 14G10 14M25)}, MRNUMBER = {1423638 (98b:11067)}, MRREVIEWER = {Emmanuel Peyre}, DOI = {10.1007/BF02362469}, URL = {http://dx.doi.org/10.1007/BF02362469}, } @article {MR1453059, AUTHOR = {Strauch, M. and Tschinkel, Yu.}, TITLE = {Height zeta functions of twisted products}, JOURNAL = {Math. Res. Lett.}, FJOURNAL = {Mathematical Research Letters}, VOLUME = {4}, YEAR = {1997}, NUMBER = {2-3}, PAGES = {273--282}, ISSN = {1073-2780}, MRCLASS = {11G35 (11G40 14G05 14M15 14M25)}, MRNUMBER = {1453059 (98h:11084)}, MRREVIEWER = {Emmanuel Peyre}, DOI = {10.4310/MRL.1997.v4.n2.a8}, URL = {http://dx.doi.org/10.4310/MRL.1997.v4.n2.a8}, } @article {MR1214451, AUTHOR = {Manin, Yu. I. and Tschinkel, Yu.}, TITLE = {Points of bounded height on del {P}ezzo surfaces}, JOURNAL = {Compositio Math.}, FJOURNAL = {Compositio Mathematica}, VOLUME = {85}, YEAR = {1993}, NUMBER = {3}, PAGES = {315--332}, ISSN = {0010-437X}, CODEN = {CMPMAF}, MRCLASS = {11G35 (14G05 14J25)}, MRNUMBER = {1214451 (94k:11070)}, URL = {http://www.numdam.org/item?id=CM_1993__85_3_315_0}, } @article {MR2547614, AUTHOR = {Takloo-Bighash, R.}, TITLE = {Distribution of rational points: a survey}, JOURNAL = {Bull. Iranian Math. Soc.}, FJOURNAL = {Iranian Mathematical Society. Bulletin}, VOLUME = {35}, YEAR = {2009}, NUMBER = {1}, PAGES = {1--30, 273}, ISSN = {1017-060X}, MRCLASS = {14G05 (11D45 11G50)}, MRNUMBER = {2547614 (2011b:14052)}, } @misc {PoonenBourbaki, AUTHOR ={Poonen, B.}, TITLE ={Average rank of elliptic curves (after Manjul Bhargava and Arul Shankar)}, YEAR ={S\`eminaire Bourbaki, \'expos\'ee 1049, 2015}, } @misc {DanRachel, AUTHOR = {Frei, C. and Loughran, D. and Newton, R.}, TITLE = {The {H}asse norm principle for abelian extensions}, YEAR = {submitted, arXiv:1508.02518, 2015}, } @article {MR936994, AUTHOR = {Datskovsky, B. and Wright, D.~J.}, TITLE = {Density of discriminants of cubic extensions}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik}, VOLUME = {386}, YEAR = {1988}, PAGES = {116--138}, ISSN = {0075-4102}, CODEN = {JRMAA8}, MRCLASS = {11R29 (11R11 11R16)}, MRNUMBER = {936994 (90b:11112)}, MRREVIEWER = {Brian Peterson}, DOI = {10.1515/crll.1988.386.116}, URL = {http://dx.doi.org/10.1515/crll.1988.386.116}, } @incollection {MR2275598, AUTHOR = {Bhargava, M.}, TITLE = {Higher composition laws and applications}, BOOKTITLE = {International {C}ongress of {M}athematicians. {V}ol. {II}}, PAGES = {271--294}, PUBLISHER = {Eur. Math. Soc., Z\"urich}, YEAR = {2006}, MRCLASS = {11R29 (11E12 11E20 11E76 11R16 11R45)}, MRNUMBER = {2275598 (2008g:11185)}, MRREVIEWER = {A. G. Earnest}, } @article {MR2113024, AUTHOR = {Bhargava, M.}, TITLE = {Higher composition laws. {III}. {T}he parametrization of quartic rings}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {159}, YEAR = {2004}, NUMBER = {3}, PAGES = {1329--1360}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {11R16 (11E12 11E20)}, MRNUMBER = {2113024 (2005k:11214)}, MRREVIEWER = {A. G. Earnest}, DOI = {10.4007/annals.2004.159.1329}, URL = {http://dx.doi.org/10.4007/annals.2004.159.1329}, } @article {MR2373152, AUTHOR = {Bhargava, M.}, TITLE = {Higher composition laws. {IV}. {T}he parametrization of quintic rings}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {167}, YEAR = {2008}, NUMBER = {1}, PAGES = {53--94}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {11E76 (11R20 11R29)}, MRNUMBER = {2373152 (2009c:11057)}, MRREVIEWER = {A. G. Earnest}, DOI = {10.4007/annals.2008.167.53}, URL = {http://dx.doi.org/10.4007/annals.2008.167.53}, } @article{BrowningLoughran13, AUTHOR = {Browning, T.~D. and Loughran, D.}, TITLE = {Varieties with too many rational points}, JOURNAL = {Math. Z.}, YEAR = {to appear, 2015}, } @misc{LeRudulier, AUTHOR = {Le Rudulier, C.}, TITLE = {Points alg{\'e}briques de hauteur born{\'e}e sur une surface}, YEAR = {in preparation, available under http://cecile.lerudulier.fr/, 2015}, } @article {MR1401626, AUTHOR = {Batyrev, V.~V. and Tschinkel, Yu.}, TITLE = {Rational points on some {F}ano cubic bundles}, JOURNAL = {C. R. Acad. Sci. Paris S\'er. I Math.}, FJOURNAL = {Comptes Rendus de l'Acad\'emie des Sciences. S\'erie I. Math\'ematique}, VOLUME = {323}, YEAR = {1996}, NUMBER = {1}, PAGES = {41--46}, ISSN = {0764-4442}, CODEN = {CASMEI}, MRCLASS = {14G05 (11G35 14J20 14J45)}, MRNUMBER = {1401626 (97j:14023)}, MRREVIEWER = {Emmanuel Peyre}, } @article {MR2135320, AUTHOR = {Kl{\"u}ners, J.}, TITLE = {A counterexample to {M}alle's conjecture on the asymptotics of discriminants}, JOURNAL = {C. R. Math. Acad. Sci. Paris}, FJOURNAL = {Comptes Rendus Math\'ematique. Acad\'emie des Sciences. Paris}, VOLUME = {340}, YEAR = {2005}, NUMBER = {6}, PAGES = {411--414}, ISSN = {1631-073X}, MRCLASS = {11R45 (11R32 12F10 12F12)}, MRNUMBER = {2135320 (2005m:11214)}, MRREVIEWER = {T. Mets{\"a}nkyl{\"a}}, DOI = {10.1016/j.crma.2005.02.010}, URL = {http://dx.doi.org/10.1016/j.crma.2005.02.010}, } @article {MR2076117, AUTHOR = {Kl{\"u}ners, J. and Malle, G.}, TITLE = {Counting nilpotent {G}alois extensions}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik}, VOLUME = 572, YEAR = 2004, PAGES = {1--26}, ISSN = {0075-4102}, CODEN = {JRMAA8}, MRCLASS = {11R45 (11R32 12F10 12F12)}, MRNUMBER = {2076117 (2005f:11259)}, MRREVIEWER = {T. Mets{\"a}nkyl{\"a}}, DOI = {10.1515/crll.2004.050}, URL = {http://dx.doi.org/10.1515/crll.2004.050}, } @article {MR1884706, AUTHOR = {Malle, G.}, TITLE = {On the distribution of {G}alois groups}, JOURNAL = {J. Number Theory}, FJOURNAL = {Journal of Number Theory}, VOLUME = {92}, YEAR = {2002}, NUMBER = {2}, PAGES = {315--329}, ISSN = {0022-314X}, CODEN = {JNUTA9}, MRCLASS = {12F12 (11R32 11R47 12F10)}, MRNUMBER = {1884706 (2002k:12010)}, MRREVIEWER = {Helmut V{\"o}lklein}, DOI = {10.1006/jnth.2001.2713}, URL = {http://dx.doi.org/10.1006/jnth.2001.2713}, } @article {MR2068887, AUTHOR = {Malle, G.}, TITLE = {On the distribution of {G}alois groups. {II}}, JOURNAL = {Experiment. Math.}, FJOURNAL = {Experimental Mathematics}, VOLUME = {13}, YEAR = {2004}, NUMBER = {2}, PAGES = {129--135}, ISSN = {1058-6458}, MRCLASS = {11R32 (11R47 12F10)}, MRNUMBER = {2068887 (2005g:11216)}, MRREVIEWER = {F. Diaz y Diaz}, URL = {http://projecteuclid.org/euclid.em/1090350928}, } @article {MR1578996, AUTHOR = {Hermite, C.}, TITLE = {Extrait d'une lettre de {M}. {C}. {H}ermite \`a {M}. {B}orchardt sur le nombre limit\'e d'irrationalit\'es auxquelles se r\'eduisent les racines des \'equations \`a coefficients entiers complexes d'un degr\'e et d'un discriminant donn\'es}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik. [Crelle's Journal]}, VOLUME = {53}, YEAR = {1857}, PAGES = {182--192}, ISSN = {0075-4102}, MRCLASS = {Contributed Item}, MRNUMBER = {1578996}, DOI = {10.1515/crll.1857.53.182}, URL = {http://dx.doi.org/10.1515/crll.1857.53.182}, } @misc{Yasuda15, AUTHOR = {Yasuda, T.}, TITLE = {Manin's conjecture vs. {M}alle's conjecture}, YEAR = {preprint, arXiv:1505:04555, 2015}, } @misc{OliverThorne14, AUTHOR={Lemke Oliver, R.~J. and Thorne, F.}, TITLE ={The number of ramified primes in number fields of small degree}, YEAR ={preprint, arXiv:1408.1018, 2014}, } @article {MR2330428, AUTHOR = {Cohen, H. and Diaz y Diaz, F. and Olivier, M.}, TITLE = {Counting discriminants of number fields}, JOURNAL = {J. Th\'eor. Nombres Bordeaux}, FJOURNAL = {Journal de Th\'eorie des Nombres de Bordeaux}, VOLUME = {18}, YEAR = {2006}, NUMBER = {3}, PAGES = {573--593}, ISSN = {1246-7405}, MRCLASS = {11R45 (11R29 11R32)}, MRNUMBER = {2330428 (2008d:11127)}, MRREVIEWER = {T. Mets{\"a}nkyl{\"a}}, URL = {http://jtnb.cedram.org/item?id=JTNB_2006__18_3_573_0}, } @article {MR2199231, AUTHOR = {Ellenberg, J.~S. and Venkatesh, A.}, TITLE = {The number of extensions of a number field with fixed degree and bounded discriminant}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {163}, YEAR = {2006}, NUMBER = {2}, PAGES = {723--741}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {11R45 (11R29)}, MRNUMBER = {2199231 (2006j:11159)}, MRREVIEWER = {T. Mets{\"a}nkyl{\"a}}, DOI = {10.4007/annals.2006.163.723}, URL = {http://dx.doi.org/10.4007/annals.2006.163.723}, } @article {MR1925912, AUTHOR = {Cohen, H. and Diaz y Diaz, F. and Olivier, M.}, TITLE = {On the density of discriminants of cyclic extensions of prime degree}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik}, VOLUME = 550, YEAR = 2002, PAGES = {169--209}, ISSN = {0075-4102}, CODEN = {JRMAA8}, MRCLASS = {11R20 (11N45 11R29 11R45)}, MRNUMBER = {1925912 (2004a:11115)}, MRREVIEWER = {T. Mets{\"a}nkyl{\"a}}, DOI = {10.1515/crll.2002.071}, URL = {http://dx.doi.org/10.1515/crll.2002.071}, } @article {MR1104780, AUTHOR = {Ekedahl, T.}, TITLE = {An infinite version of the {C}hinese remainder theorem}, JOURNAL = {Comment. Math. Univ. St. Paul.}, FJOURNAL = {Commentarii Mathematici Universitatis Sancti Pauli}, VOLUME = {40}, YEAR = {1991}, NUMBER = {1}, PAGES = {53--59}, ISSN = {0010-258X}, CODEN = {COMAAC}, MRCLASS = {11D79 (11A07)}, MRNUMBER = {1104780 (92h:11027)}, MRREVIEWER = {M. Ram Murty}, } @article {MR1740984, AUTHOR = {Poonen, B. and Stoll, M.}, TITLE = {The {C}assels-{T}ate pairing on polarized abelian varieties}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {150}, YEAR = {1999}, NUMBER = {3}, PAGES = {1109--1149}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {11G10 (11G30 14H40 14K15)}, MRNUMBER = {1740984 (2000m:11048)}, MRREVIEWER = {Tam{\'a}s Szamuely}, DOI = {10.2307/121064}, URL = {http://dx.doi.org/10.2307/121064}, } @article {MR1980998, AUTHOR = {Poonen, B.}, TITLE = {Squarefree values of multivariable polynomials}, JOURNAL = {Duke Math. J.}, FJOURNAL = {Duke Mathematical Journal}, VOLUME = {118}, YEAR = {2003}, NUMBER = {2}, PAGES = {353--373}, ISSN = {0012-7094}, CODEN = {DUMJAO}, MRCLASS = {11N32 (11D75)}, MRNUMBER = {1980998 (2004d:11094)}, MRREVIEWER = {G. Greaves}, DOI = {10.1215/S0012-7094-03-11826-8}, URL = {http://dx.doi.org/10.1215/S0012-7094-03-11826-8}, } @article {MR0491593, AUTHOR = {Davenport, H. and Heilbronn, H.}, TITLE = {On the density of discriminants of cubic fields. {II}}, JOURNAL = {Proc. Roy. Soc. London Ser. A}, FJOURNAL = {Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences}, VOLUME = {322}, YEAR = {1971}, NUMBER = {1551}, PAGES = {405--420}, ISSN = {0962-8444}, MRCLASS = {12A30 (12A45 12A75)}, MRNUMBER = {0491593 (58 \#10816)}, MRREVIEWER = {J. V. Armitage}, } @article {MR2183288, AUTHOR = {Bhargava, M.}, TITLE = {The density of discriminants of quartic rings and fields}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {162}, YEAR = {2005}, NUMBER = {2}, PAGES = {1031--1063}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {11R45 (11R16 11R29)}, MRNUMBER = {2183288 (2006m:11163)}, MRREVIEWER = {F. Diaz y Diaz}, DOI = {10.4007/annals.2005.162.1031}, URL = {http://dx.doi.org/10.4007/annals.2005.162.1031}, } @article {MR3272925, AUTHOR = {Bhargava, M. and Shankar, A.}, TITLE = {Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {181}, YEAR = {2015}, NUMBER = {1}, PAGES = {191--242}, ISSN = {0003-486X}, MRCLASS = {11E20 (14G25)}, MRNUMBER = {3272925}, MRREVIEWER = {John M. Voight}, DOI = {10.4007/annals.2015.181.1.3}, URL = {http://dx.doi.org/10.4007/annals.2015.181.1.3}, } @article {MR3275847, AUTHOR = {Bhargava, M. and Shankar, A.}, TITLE = {Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {181}, YEAR = {2015}, NUMBER = {2}, PAGES = {587--621}, ISSN = {0003-486X}, MRCLASS = {11G05 (11E20 14Gxx)}, MRNUMBER = {3275847}, DOI = {10.4007/annals.2015.181.2.4}, URL = {http://dx.doi.org/10.4007/annals.2015.181.2.4}, } @article {MR2641942, AUTHOR = {Belabas, K. and Bhargava, M. and Pomerance, C.}, TITLE = {Error estimates for the {D}avenport-{H}eilbronn theorems}, JOURNAL = {Duke Math. J.}, FJOURNAL = {Duke Mathematical Journal}, VOLUME = {153}, YEAR = {2010}, NUMBER = {1}, PAGES = {173--210}, ISSN = {0012-7094}, CODEN = {DUMJAO}, MRCLASS = {11R29 (11R11 11R16 11R45)}, MRNUMBER = {2641942 (2011d:11260)}, MRREVIEWER = {Cornelius Greither}, DOI = {10.1215/00127094-2010-007}, URL = {http://dx.doi.org/10.1215/00127094-2010-007}, } @article {MR2745272, AUTHOR = {Bhargava, M.}, TITLE = {The density of discriminants of quintic rings and fields}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {172}, YEAR = {2010}, NUMBER = {3}, PAGES = {1559--1591}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {11R29 (11R21 11R45)}, MRNUMBER = {2745272 (2011k:11152)}, MRREVIEWER = {Jerzy Browkin}, DOI = {10.4007/annals.2010.172.1559}, URL = {http://dx.doi.org/10.4007/annals.2010.172.1559}, } @article {MR3264252, AUTHOR = {Shankar, A. and Tsimerman, J.}, TITLE = {Counting {$S_5$}-fields with a power saving error term}, JOURNAL = {Forum Math. Sigma}, FJOURNAL = {Forum of Mathematics. Sigma}, VOLUME = {2}, YEAR = {2014}, PAGES = {e13, 8}, ISSN = {2050-5094}, MRCLASS = {11N36 (11R21 11R29 11R45 12F10)}, MRNUMBER = {3264252}, MRREVIEWER = {Mitsuko Horie}, DOI = {10.1017/fms.2014.10}, URL = {http://dx.doi.org/10.1017/fms.2014.10}, } @misc{Bhargava14, AUTHOR = {M. Bhargava}, TITLE = {The geometric sieve and the density of squarefree values of invariant polynomials}, YEAR = {preprint, arXiv:1402.0031, 2014}, } @misc{BrightBrowningLoughran15, AUTHOR = {M. Bright and T.~D. Browning and D. Loughran}, TITLE = {Failure of weak approximation in families}, YEAR = {preprint, arXiv:1506.0181, 2015}, } @misc{Loughran13, AUTHOR = {D. Loughran}, TITLE = {The number of varieties in a family which contain a rational point}, YEAR = {preprint, arXiv:1310.6219, 2013}, } @misc{BrowningMatthiesen14, AUTHOR = {T.~D. Browning and L. Matthiesen}, TITLE = {Norm forms for arbitrary number fields as products of linear polynomials}, YEAR = {preprint, 2014}, } @article {MR3194818, AUTHOR = {Browning, T. D. and Matthiesen, L. and Skorobogatov, A. N.}, TITLE = {Rational points on pencils of conics and quadrics with many degenerate fibers}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {180}, YEAR = {2014}, NUMBER = {1}, PAGES = {381--402}, ISSN = {0003-486X}, MRCLASS = {14G05 (14C21)}, MRNUMBER = {3194818}, DOI = {10.4007/annals.2014.180.1.8}, URL = {http://dx.doi.org/10.4007/annals.2014.180.1.8}, } @article {MR2881334, AUTHOR = {Frei, C.}, TITLE = {On rings of integers generated by their units}, JOURNAL = {Bull. Lond. Math. Soc.}, FJOURNAL = {Bulletin of the London Mathematical Society}, VOLUME = {44}, YEAR = {2012}, NUMBER = {1}, PAGES = {167--182}, ISSN = {0024-6093}, MRCLASS = {11R04 (11R27)}, MRNUMBER = {2881334 (2012m:11147)}, MRREVIEWER = {W. Narkiewicz}, DOI = {10.1112/blms/bdr089}, URL = {http://dx.doi.org/10.1112/blms/bdr089}, } @article {MR3742196, AUTHOR = {Browning, T. D. and Matthiesen, L.}, TITLE = {Norm forms for arbitrary number fields as products of linear polynomials}, JOURNAL = {Ann. Sci. \'Ec. Norm. Sup\'er. (4)}, FJOURNAL = {Annales Scientifiques de l'\'Ecole Normale Sup\'erieure. Quatri\`eme S\'erie}, VOLUME = {50}, YEAR = {2017}, NUMBER = {6}, PAGES = {1383--1446}, ISSN = {0012-9593,1873-2151}, } @article {Damaris, AUTHOR = {Schindler, D.}, TITLE = {Manin's conjecture for certain biprojective hypersurfaces}, JOURNAL = {J. reine angew. Math}, YEAR = {to appear, 2014}, } @article {MR3229043, AUTHOR = {Browning, T.~D. and Vishe, P.}, TITLE = {Cubic hypersurfaces and a version of the circle method for number fields}, JOURNAL = {Duke Math. J.}, FJOURNAL = {Duke Mathematical Journal}, VOLUME = {163}, YEAR = {2014}, NUMBER = {10}, PAGES = {1825--1883}, ISSN = {0012-7094}, MRCLASS = {11D72 (11D25 11D45 11P55 11R47 14G05)}, MRNUMBER = {3229043}, MRREVIEWER = {D. R. Heath-Brown}, DOI = {10.1215/00127094-2738530}, URL = {http://dx.doi.org/10.1215/00127094-2738530}, } @article {MR3188633, AUTHOR = {Schindler, D. and Skorobogatov, A.}, TITLE = {Norms as products of linear polynomials}, JOURNAL = {J. Lond. Math. Soc. (2)}, FJOURNAL = {Journal of the London Mathematical Society. Second Series}, VOLUME = {89}, YEAR = {2014}, NUMBER = {2}, PAGES = {559--580}, ISSN = {0024-6107}, MRCLASS = {11D57 (11G35 11P55 14G05)}, MRNUMBER = {3188633}, DOI = {10.1112/jlms/jdt080}, URL = {http://dx.doi.org/10.1112/jlms/jdt080}, } @article {MR969545, AUTHOR = {Wright, D.~J.}, TITLE = {Distribution of discriminants of abelian extensions}, JOURNAL = {Proc. London Math. Soc. (3)}, FJOURNAL = {Proceedings of the London Mathematical Society. Third Series}, VOLUME = {58}, YEAR = {1989}, NUMBER = {1}, PAGES = {17--50}, ISSN = {0024-6115}, CODEN = {PLMTAL}, MRCLASS = {11R29 (11R37 11R45)}, MRNUMBER = {969545 (90b:11115)}, MRREVIEWER = {Boris Datskovsky}, DOI = {10.1112/plms/s3-58.1.17}, URL = {http://dx.doi.org/10.1112/plms/s3-58.1.17}, } @article {MR2581243, AUTHOR = {Wood, M.~M.}, TITLE = {On the probabilities of local behaviors in abelian field extensions}, JOURNAL = {Compos. Math.}, FJOURNAL = {Compositio Mathematica}, VOLUME = {146}, YEAR = {2010}, NUMBER = {1}, PAGES = {102--128}, ISSN = {0010-437X}, MRCLASS = {11R20 (11R45)}, MRNUMBER = {2581243 (2011a:11195)}, MRREVIEWER = {Roland Qu{\^e}me}, DOI = {10.1112/S0010437X0900431X}, URL = {http://dx.doi.org/10.1112/S0010437X0900431X}, } @article {Manfred, AUTHOR = {Frei, C. and Madritsch, M.}, TITLE = {Forms of differing degrees over number fields}, JOURNAL = {Mathematika}, YEAR = {to appear, 2016}, } @misc {salberger, AUTHOR = {Salberger, P.}, TITLE = {Uniform bounds for rational points on cubic hypersurfaces}, YEAR = {in preparation, 2014}, } @article{BHB, AUTHOR = {Browning, T.~D. and Heath-Brown, R.}, TITLE = {Forms in many variables and differing degrees}, JOURNAL = {J. Eur. Math. Soc.}, YEAR = {to appear, 2015}, } @article {MR2881307, AUTHOR = {Habegger, P. and Pila, J.}, TITLE = {Some unlikely intersections beyond {A}ndr\'e-{O}ort}, JOURNAL = {Compos. Math.}, FJOURNAL = {Compositio Mathematica}, VOLUME = {148}, YEAR = {2012}, NUMBER = {1}, PAGES = {1--27}, ISSN = {0010-437X}, MRCLASS = {11G18 (03C64 11G15 11G50)}, MRNUMBER = {2881307}, MRREVIEWER = {Alexandra Shlapentokh}, DOI = {10.1112/S0010437X11005604}, URL = {http://dx.doi.org/10.1112/S0010437X11005604}, } @article {MR2800724, AUTHOR = {Pila, J.}, TITLE = {O-minimality and the {A}ndr\'e-{O}ort conjecture for {$\Bbb C^n$}}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {173}, YEAR = {2011}, NUMBER = {3}, PAGES = {1779--1840}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {11G18 (03C64 11U09 14G35)}, MRNUMBER = {2800724 (2012j:11129)}, MRREVIEWER = {Alexandra Shlapentokh}, DOI = {10.4007/annals.2011.173.3.11}, URL = {http://dx.doi.org/10.4007/annals.2011.173.3.11}, } @article {MR2411018, AUTHOR = {Pila, J. and Zannier, U.}, TITLE = {Rational points in periodic analytic sets and the {M}anin-{M}umford conjecture}, JOURNAL = {Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl.}, FJOURNAL = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni}, VOLUME = {19}, YEAR = {2008}, NUMBER = {2}, PAGES = {149--162}, ISSN = {1120-6330}, MRCLASS = {11J95 (11G10 14G05)}, MRNUMBER = {2411018 (2009d:11110)}, MRREVIEWER = {Pavlos Tzermias}, DOI = {10.4171/RLM/514}, URL = {http://dx.doi.org/10.4171/RLM/514}, } @article {MR2228464, AUTHOR = {Pila, J. and Wilkie, A. J.}, TITLE = {The rational points of a definable set}, JOURNAL = {Duke Math. J.}, FJOURNAL = {Duke Mathematical Journal}, VOLUME = {133}, YEAR = {2006}, NUMBER = {3}, PAGES = {591--616}, ISSN = {0012-7094}, CODEN = {DUMJAO}, MRCLASS = {03C64 (11G99 11U09)}, MRNUMBER = {2228464 (2007f:03048)}, MRREVIEWER = {Alexandra Shlapentokh}, DOI = {10.1215/S0012-7094-06-13336-7}, URL = {http://dx.doi.org/10.1215/S0012-7094-06-13336-7}, } @article{ Breteche2007, AUTHOR = "Bret{\`e}che, R. de la and Swinnerton-Dyer, P.", TITLE = "Fonction z{\^e}ta des hauteurs associ{\'e}e {\`a} une certaine surface cubique", JOURNAL = "Bull. Soc. Math. France", VOLUME = 135, YEAR = 2007, NUMBER = 1, PAGES = "65--92", ISSN = "0037-9484", CODEN = "BSMFAA", FJOURNAL = "Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France", MRCLASS = "14G10 (11G35 11G50 14G05)", MRNUMBER = "2430199 (2009f:14041)", MRREVIEWER = "C{\'i}cero Carvalho" } @Article{ Fouvry1998, AUTHOR = "{\'E}. Fouvry", TITLE = "Sur la hauteur des points d'une certaine surface cubique singuli{\`e}re", NOTE = "Nombre et r{\'e}partition de points de hauteur born{\'e}e (Paris, 1996)", JOURNAL = "Ast{\'e}risque", NUMBER = "251", YEAR = "1998", PAGES = "31--49", ISSN = "0303-1179", FJOURNAL = "Ast{\'e}risque", MRCLASS = "11G50 (11G35 14G25)", MRNUMBER = "1679838 (2000b:11075)", MRREVIEWER = "Boris {\`E}. Kunyavski{\u\i}" } @Article{ Breteche1998, AUTHOR = "Bret{\`e}che, R. de la", TITLE = "Sur le nombre de points de hauteur born{\'e}e d'une certaine surface cubique singuli{\`e}re", NOTE = "Nombre et r{\'e}partition de points de hauteur born{\'e}e (Paris, 1996)", JOURNAL = "Ast{\'e}risque", NUMBER = "251", YEAR = "1998", PAGES = "51--77", ISSN = "0303-1179", FJOURNAL = "Ast{\'e}risque", MRCLASS = "11G50 (11G35 14G25)", MRNUMBER = "1679839 (2000b:11074)", MRREVIEWER = "Boris {\`E}. Kunyavski{\u\i}" } @article {MR3227528, AUTHOR = {Ange, T.}, TITLE = {Le th\'eor\`eme de {S}chanuel dans les fibr\'es ad\'eliques hermitiens}, JOURNAL = {Manuscripta Math.}, FJOURNAL = {Manuscripta Mathematica}, VOLUME = {144}, YEAR = {2014}, NUMBER = {3-4}, PAGES = {565--608}, ISSN = {0025-2611}, MRCLASS = {11G50 (11H06 11R04)}, MRNUMBER = {3227528}, DOI = {10.1007/s00229-014-0665-y}, URL = {http://dx.doi.org/10.1007/s00229-014-0665-y}, } @article {MR2039627, AUTHOR = {Loher, T. and Masser, D.}, TITLE = {Uniformly counting points of bounded height}, JOURNAL = {Acta Arith.}, FJOURNAL = {Acta Arithmetica}, VOLUME = {111}, YEAR = {2004}, NUMBER = {3}, PAGES = {277--297}, ISSN = {0065-1036}, CODEN = {AARIA9}, MRCLASS = {11R04 (11G50)}, MRNUMBER = {2039627 (2005a:11162)}, MRREVIEWER = {Yuri Tschinkel}, DOI = {10.4064/aa111-3-5}, URL = {http://dx.doi.org/10.4064/aa111-3-5}, } @article {Sierpinski1906, AUTHOR = {Sierpi{\'n}ski, W.}, TITLE = {O pewnem zagadnieniu z rachunku funkcyj asymptotycznych ({O}n a problem of the theory of asymptotic functions)}, JOURNAL = {Prace Mat.-Fiz.}, VOLUME = 17, YEAR = 1906, PAGES = {77-118}, } @article {MR3180592, AUTHOR = {Derenthal, U.}, TITLE = {Singular del {P}ezzo surfaces whose universal torsors are hypersurfaces}, JOURNAL = {Proc. Lond. Math. Soc. (3)}, FJOURNAL = {Proceedings of the London Mathematical Society. Third Series}, VOLUME = {108}, YEAR = {2014}, NUMBER = {3}, PAGES = {638--681}, ISSN = {0024-6115}, MRCLASS = {14J26 (14C20 14G05)}, MRNUMBER = {3180592}, DOI = {10.1112/plms/pdt041}, URL = {http://dx.doi.org/10.1112/plms/pdt041}, } @article {mikew, AUTHOR = {Browning, T.~D. and Swarbrick Jones, M.}, TITLE = {Counting rational points on del {P}ezzo surfaces with a conic bundle structure}, JOURNAL = {Acta Arith.}, FJOURNAL = {Acta Arithmetica}, VOLUME = 163, YEAR = 2014, PAGES = {271--298} } @article {FourThirds, AUTHOR = {Heath-Brown, D. R.}, TITLE = {The density of rational points on cubic surfaces}, JOURNAL = {Acta Arith.}, FJOURNAL = {Acta Arithmetica}, VOLUME = {79}, YEAR = {1997}, NUMBER = {1}, PAGES = {17--30} } @article {MR0033094, AUTHOR = {Northcott, D. G.}, TITLE = {An inequality in the theory of arithmetic on algebraic varieties}, JOURNAL = {Proc. Cambridge Philos. Soc.}, VOLUME = {45}, YEAR = {1949}, PAGES = {502--509}, MRCLASS = {14.0X}, MRNUMBER = {0033094 (11,390a)}, MRREVIEWER = {B. Segre}, } @article {MR0414556, AUTHOR = {Colliot-Th{\'e}l{\`e}ne, J.-L. and Sansuc, J.-J.}, TITLE = {Torseurs sous des groupes de type multiplicatif; applications \`a l'\'etude des points rationnels de certaines vari\'et\'es alg\'ebriques}, JOURNAL = {C. R. Acad. Sci. Paris S\'er. A-B}, VOLUME = {282}, YEAR = {1976}, NUMBER = {18}, PAGES = {Aii, A1113--A1116}, MRCLASS = {14G05}, MRNUMBER = {0414556 (54 \#2657)}, MRREVIEWER = {Loren D. Olson}, } @article {MR1679836, AUTHOR = {Slater, J.~B. and Swinnerton-Dyer, P.}, TITLE = {Counting points on cubic surfaces. {I}}, NOTE = {Nombre et r{\'e}partition de points de hauteur born{\'e}e (Paris, 1996)}, JOURNAL = {Ast\'erisque}, FJOURNAL = {Ast\'erisque}, NUMBER = {251}, YEAR = {1998}, PAGES = {1--12}, ISSN = {0303-1179}, MRCLASS = {11G35 (14G05)}, MRNUMBER = {1679836 (2000d:11087)}, MRREVIEWER = {Teresa de Diego}, } @article{IMRN, AUTHOR = {Derenthal, U. and Frei, C.}, TITLE={On {M}anin's conjecture for a certain singular cubic surface over imaginary quadratic fields}, journal = {Int. Math. Res. Not. IMRN}, volume = {2015}, number = {10}, pages = {2728-2750}, year={2015}, } @article{Marta, AUTHOR = {Frei, C. and Pieropan, M.}, TITLE = {O-minimality on twisted universal torsors and {M}anin's conjecture over number fields}, JOURNAL = {Ann. Sci. {\'E}c. Norm. Sup{\'e}r.}, YEAR = {to appear, 2015}, } @misc{Sofos14, AUTHOR = {Sofos, E.}, TITLE = {Rational points on the {F}ermat cubic surface}, YEAR = {preprint, arXiv:1402.0303v1, 2014}, } @article{Efthymios, AUTHOR = {Frei, C. and Sofos, E.}, TITLE = {Counting rational points on smooth cubic surfaces}, JOURNAL = {Math. Res. Lett.}, VOLUME={23}, NUMBER={1}, PAGES={127--143}, YEAR = {2016}, } @article{Annali, author = {Frei, C. and Widmer, M.}, title = {Schanuel's theorem for heights defined via extension fields}, journal = {Ann. Sc. Norm. Super. Pisa Cl. Sci.}, year = {to appear, 2014} } @article {MR2099200, AUTHOR = {Bret{\`e}che, R. de la and Fouvry, {\'E}.}, TITLE = {L'\'eclat\'e du plan projectif en quatre points dont deux conjugu\'es}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik}, VOLUME = {576}, YEAR = {2004}, PAGES = {63--122}, ISSN = {0075-4102}, CODEN = {JRMAA8}, MRCLASS = {11G50 (14G05)}, MRNUMBER = {2099200 (2005f:11131)}, MRREVIEWER = {Timothy D. Browning}, DOI = {10.1515/crll.2004.091}, URL = {http://dx.doi.org/10.1515/crll.2004.091}, } @incollection {MR2362193, AUTHOR = {Browning, T.~D.}, TITLE = {An overview of {M}anin's conjecture for del {P}ezzo surfaces}, BOOKTITLE = {Analytic number theory}, SERIES = {Clay Math. Proc.}, VOLUME = {7}, PAGES = {39--55}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {2007}, MRCLASS = {14G05 (11G35 14G25 14J26)}, MRNUMBER = {2362193 (2008j:14041)}, MRREVIEWER = {Hizuru Yamagishi}, } @article {arXiv:1210.1792, AUTHOR = {Loughran, D.}, TITLE = {Rational points of bounded height and the {W}eil restriction}, journal = {Israel J. Math.}, year={to appear, 2014} } @article {MR1398816, AUTHOR = {Wilkie, A.~J.}, TITLE = {Model completeness results for expansions of the ordered field of real numbers by restricted {P}faffian functions and the exponential function}, JOURNAL = {J. Amer. Math. Soc.}, FJOURNAL = {Journal of the American Mathematical Society}, VOLUME = {9}, YEAR = {1996}, NUMBER = {4}, PAGES = {1051--1094}, ISSN = {0894-0347}, MRCLASS = {03C62 (03C60 03C65 14P15)}, MRNUMBER = {1398816 (98j:03052)}, MRREVIEWER = {Luc B{\'e}lair}, DOI = {10.1090/S0894-0347-96-00216-0}, URL = {http://dx.doi.org/10.1090/S0894-0347-96-00216-0}, } @article {MR0047697, AUTHOR = {Hua, L.-K.}, TITLE = {On exponential sums over an algebraic number field}, JOURNAL = {Canadian J. Math.}, FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de Math\'ematiques}, VOLUME = {3}, YEAR = {1951}, PAGES = {44--51}, ISSN = {0008-414X}, MRCLASS = {10.0X}, MRNUMBER = {0047697 (13,916b)}, MRREVIEWER = {G. Pall}, } @Article{arXiv:1210.5943, title = "{Counting lattice points and o-minimal structures}", author = "F. Barroero and M. Widmer", JOURNAL={Int. Math. Res. Not. IMRN}, VOLUME = {2014}, NUMBER = {18}, PAGES = {4932--4957}, year = {2014}, } @book {MR1434478, AUTHOR = {Cassels, J.~W.~S.}, TITLE = {An introduction to the geometry of numbers}, SERIES = {Classics in Mathematics}, NOTE = {Corrected reprint of the 1971 edition}, PUBLISHER = {Springer-Verlag}, ADDRESS = {Berlin}, YEAR = {1997}, PAGES = {viii+344}, ISBN = {3-540-61788-4}, MRCLASS = {11Hxx}, MRNUMBER = {1434478 (97i:11074)}, } @book{coste2000introduction, title={An Introduction to Semialgebraic Geometry}, author={Coste, M.}, isbn={9788881472253}, series={Dottorato di ricerca in matematica / Universit{\`a} di Pisa, Dipartimento di Matematica}, url={http://books.google.de/books?id=X4jXIAAACAAJ}, year={2000}, publisher={Istituti Editoriali e Poligrafici Internazionali} } @article {MR2846337, AUTHOR = {Widmer, M.}, TITLE = {Lipschitz class, narrow class, and counting lattice points}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {140}, YEAR = {2012}, NUMBER = {2}, PAGES = {677--689}, ISSN = {0002-9939}, CODEN = {PAMYAR}, MRCLASS = {11P21 (11H06 52C07)}, MRNUMBER = {2846337 (2012h:11140)}, MRREVIEWER = {Lenny Fukshansky}, DOI = {10.1090/S0002-9939-2011-10926-2}, URL = {http://dx.doi.org/10.1090/S0002-9939-2011-10926-2}, } @article {MR0043821, AUTHOR = {Davenport, H.}, TITLE = {On a principle of {L}ipschitz}, JOURNAL = {J. London Math. Soc.}, FJOURNAL = {Journal of the London Mathematical Society. Second Series}, VOLUME = {26}, YEAR = {1951}, PAGES = {179--183}, ISSN = {0024-6107}, MRCLASS = {10.0X}, MRNUMBER = {0043821 (13,323d)}, MRREVIEWER = {W. H. Mills}, } @InCollection{ MR2290499, AUTHOR = "U. Derenthal and Yu. Tschinkel", TITLE = "Universal torsors over del {P}ezzo surfaces and rational points", BOOKTITLE = "Equidistribution in number theory, an introduction", SERIES = "NATO Sci. Ser. II Math. Phys. Chem.", VOLUME = "237", PAGES = "169--196", PUBLISHER = "Springer", ADDRESS = "Dordrecht", YEAR = "2007", MRCLASS = "14G05 (11G35 11G50)", MRNUMBER = "MR2290499 (2007j:14024)", MRREVIEWER = "Timothy D. Browning" } @article {MR2769338, AUTHOR = {Loughran, D.}, TITLE = {Manin's conjecture for a singular sextic del {P}ezzo surface}, JOURNAL = {J. Th\'eor. Nombres Bordeaux}, FJOURNAL = {Journal de Th\'eorie des Nombres de Bordeaux}, VOLUME = {22}, YEAR = {2010}, NUMBER = {3}, PAGES = {675--701}, ISSN = {1246-7405}, MRCLASS = {14G05 (11D45 11G35 11G50 14G10 14J26)}, MRNUMBER = {2769338 (2012a:14048)}, MRREVIEWER = {Ulrich Derenthal}, URL = {http://jtnb.cedram.org/item?id=JTNB_2010__22_3_675_0}, } @Article{ MR2377367, AUTHOR = "U. Derenthal and M. Joyce and Z. Teitler", TITLE = "The nef cone volume of generalized del {P}ezzo surfaces", JOURNAL = "Algebra Number Theory", FJOURNAL = "Algebra \& Number Theory", VOLUME = 2, YEAR = 2008, NUMBER = 2, PAGES = "157--182", ISSN = "1937-0652", MRCLASS = "14J26 (14G05)", MRNUMBER = "MR2377367" } @Article{ MR2520770, AUTHOR = "U. Derenthal", TITLE = "Counting integral points on universal torsors", JOURNAL = "Int. Math. Res. Not.", FJOURNAL = "International Mathematics Research Notices. IMRN", YEAR = "2009", NUMBER = "14", PAGES = "2648--2699", ISSN = "1073-7928", MRCLASS = "14G05 (11G35 11G50 14G25)", MRNUMBER = "MR2520770 (2010f:14019)", MRREVIEWER = "Yuri Tschinkel" } @Article{ MR2318651, AUTHOR = "U. Derenthal", TITLE = "On a constant arising in {M}anin's conjecture for del {P}ezzo surfaces", JOURNAL = "Math. Res. Lett.", FJOURNAL = "Mathematical Research Letters", VOLUME = "14", YEAR = "2007", NUMBER = "3", PAGES = "481--489", ISSN = "1073-2780", MRCLASS = "14J26 (14G05)", MRNUMBER = "MR2318651" } @Article{ MR2543667, AUTHOR = {Browning, T.~D. and Derenthal, U.}, TITLE = "Manin's conjecture for a quartic del {P}ezzo surface with {$A_4$} singularity", JOURNAL = "Ann. Inst. Fourier (Grenoble)", FJOURNAL = "Universit{\'e} de Grenoble. Annales de l'Institut Fourier", VOLUME = "59", YEAR = "2009", NUMBER = "3", PAGES = "1231--1265", ISSN = "0373-0956", CODEN = "AIFUA7", MRCLASS = "11G35 (11G50 14G05 14G40)", MRNUMBER = "MR2543667", MRREVIEWER = "Oscar G. Villareal", URL = "http://aif.cedram.org/item?id=AIF_2009__59_3_1231_0" } @Article{ MR2247898, AUTHOR = "D. Masser and J.~D. Vaaler", TITLE = "Counting algebraic numbers with large height. {II}", JOURNAL = "Trans. Amer. Math. Soc.", VOLUME = "359", YEAR = "2007", NUMBER = "1", PAGES = "427--445 (electronic)", ISSN = "0002-9947", DOI = "10.1090/S0002-9947-06-04115-8", URL = "http://dx.doi.org/10.1090/S0002-9947-06-04115-8", CODEN = "TAMTAM", FJOURNAL = "Transactions of the American Mathematical Society", MRCLASS = "11R04 (11G50 11Y40)", MRNUMBER = "2247898 (2008m:11208)", MRREVIEWER = "Vichian Laohakosol" } @Article{ MR1330740, AUTHOR = "W.~M. Schmidt", TITLE = "Northcott's theorem on heights. {II}. {T}he quadratic case", JOURNAL = "Acta Arith.", FJOURNAL = "Acta Arithmetica", VOLUME = "70", YEAR = "1995", NUMBER = "4", PAGES = "343--375", ISSN = "0065-1036", CODEN = "AARIA9", MRCLASS = "11G35 (11D75 14G25)", MRNUMBER = "1330740 (96a:11059)", MRREVIEWER = "Joseph H. Silverman" } @Misc{ arXiv:1205.0373, title = "{Quadratic congruences on average and rational points on cubic surfaces }", author = "S. Baier and U. Derenthal", eprint = "arXiv:1205.0373", YEAR = "arXiv:1205.0373, 2012" } @Article{ MR1340296, AUTHOR = "E. Peyre", TITLE = "Hauteurs et mesures de {T}amagawa sur les vari{\'e}t{\'e}s de {F}ano", JOURNAL = "Duke Math. J.", FJOURNAL = "Duke Mathematical Journal", VOLUME = "79", YEAR = "1995", NUMBER = "1", PAGES = "101--218", ISSN = "0012-7094", CODEN = "DUMJAO", MRCLASS = "11G35 (14G05 14J45)", MRNUMBER = "MR1340296 (96h:11062)", MRREVIEWER = "Shouwu Zhang" } @Article{ MR1620682, AUTHOR = "V.~V. Batyrev and Yu. Tschinkel", TITLE = "Manin's conjecture for toric varieties", JOURNAL = "J. Algebraic Geom.", FJOURNAL = "Journal of Algebraic Geometry", VOLUME = "7", YEAR = "1998", NUMBER = "1", PAGES = "15--53", ISSN = "1056-3911", MRCLASS = "11G50 (11G35 14G05 14G40 14M25)", MRNUMBER = "MR1620682 (2000c:11107)" } @Article{ MR1032922, AUTHOR = "V.~V. Batyrev and Yu. I. Manin", TITLE = "Sur le nombre des points rationnels de hauteur born{\'e} des vari{\'e}t{\'e}s alg{\'e}briques", JOURNAL = "Math. Ann.", FJOURNAL = "Mathematische Annalen", VOLUME = "286", YEAR = "1990", NUMBER = "1-3", PAGES = "27--43", ISSN = "0025-5831", CODEN = "MAANA", MRCLASS = "11G35 (14G10 14G40)", MRNUMBER = "MR1032922 (91g:11069)", MRREVIEWER = "Marc Hindry" } @Article{ MR1679843, AUTHOR = "V.~V. Batyrev and Yu. Tschinkel", TITLE = "Tamagawa numbers of polarized algebraic varieties", NOTE = "Nombre et r{\'e}partition de points de hauteur born{\'e}e (Paris, 1996)", JOURNAL = "Ast{\'e}risque", FJOURNAL = "Ast{\'e}risque", NUMBER = "251", YEAR = "1998", PAGES = "299--340", ISSN = "0303-1179", MRCLASS = "11G50 (11G35 14G05 14G10)", MRNUMBER = "MR1679843 (2000d:11090)", MRREVIEWER = "Carlo Gasbarri" } @Article{ MR89m:11060, AUTHOR = "J. Franke and Yu.~I. Manin and Yu. Tschinkel", TITLE = "Rational points of bounded height on {F}ano varieties", JOURNAL = "Invent. Math.", FJOURNAL = "Inventiones Mathematicae", VOLUME = "95", YEAR = "1989", NUMBER = "2", PAGES = "421--435", ISSN = "0020-9910", CODEN = "INVMBH", MRCLASS = "11G35 (14G25 14J20)", MRNUMBER = "89m:11060", MRREVIEWER = "Joseph H. Silverman" } @Article{ arXiv:1105.2807, title = "{Gaussian rational points on a singular cubic surface}", author = "U. Derenthal and F. Janda", JOURNAL = "In Torsors, {\'e}tale homotopy and applications to rational points - Proceedings of the ICMS workshop in Edinburgh, 10-14 January 2011 in London Mathematical Society Lecture Note Series", volume = {405}, pages="210-231", year = "2011" } @article{ANT, title = "{Counting rational points over number fields on a singular cubic surface}", author = "C. Frei", journal = {Algebra Number Theory}, volume = {7}, number = {6}, pages = {1451-1479}, YEAR = {2013}, } @Book{ MR1282723, AUTHOR = "S. Lang", TITLE = "Algebraic number theory", SERIES = "Graduate Texts in Mathematics", VOLUME = "110", EDITION = "Second", PUBLISHER = "Springer-Verlag", ADDRESS = "New York", YEAR = "1994", PAGES = "xiv+357", ISBN = "0-387-94225-4", MRCLASS = "11Rxx (11-01 11-02)", MRNUMBER = "1282723 (95f:11085)", MRREVIEWER = "M. Ram Murty" } @article {MR0150129, AUTHOR = {Birch, B.~J.}, TITLE = {Forms in many variables}, JOURNAL = {Proc. Roy. Soc. Ser. A}, VOLUME = {265}, YEAR = {1961/1962}, PAGES = {245--263}, MRCLASS = {10.15}, MRNUMBER = {MR0150129 (27 \#132)}, MRREVIEWER = {J. V. Armitage}, } @article {MR1290198, AUTHOR = {Skinner, C.~M.}, TITLE = {Rational points on nonsingular cubic hypersurfaces}, JOURNAL = {Duke Math. J.}, FJOURNAL = {Duke Mathematical Journal}, VOLUME = {75}, YEAR = {1994}, NUMBER = {2}, PAGES = {409--466}, ISSN = {0012-7094}, CODEN = {DUMJAO}, MRCLASS = {11P55 (11D72 11R47)}, MRNUMBER = {1290198 (95d:11138)}, MRREVIEWER = {D. R. Heath-Brown}, DOI = {10.1215/S0012-7094-94-07512-1}, URL = {http://dx.doi.org/10.1215/S0012-7094-94-07512-1}, } @article {MR1446148, AUTHOR = {Skinner, C.~M.}, TITLE = {Forms over number fields and weak approximation}, JOURNAL = {Compositio Math.}, FJOURNAL = {Compositio Mathematica}, VOLUME = {106}, YEAR = {1997}, NUMBER = {1}, PAGES = {11--29}, ISSN = {0010-437X}, CODEN = {CMPMAF}, MRCLASS = {14G05 (11G35 11P55)}, MRNUMBER = {1446148 (98b:14021)}, MRREVIEWER = {Dan Abramovich}, DOI = {10.1023/A:1000129818730}, URL = {http://dx.doi.org/10.1023/A:1000129818730}, } @article {MR1681100, AUTHOR = {Peyre, E. and Tschinkel, Yu.}, TITLE = {Tamagawa numbers of diagonal cubic surfaces, numerical evidence}, JOURNAL = {Math. Comp.}, FJOURNAL = {Mathematics of Computation}, VOLUME = {70}, YEAR = {2001}, NUMBER = {233}, PAGES = {367--387}, ISSN = {0025-5718}, CODEN = {MCMPAF}, MRCLASS = {11G35 (11G50 14G05)}, MRNUMBER = {MR1681100 (2001f:11100)}, MRREVIEWER = {Tam{\'a}s Szamuely}, } @article {MR89f:11082, AUTHOR = {Colliot-Th{\'e}l{\`e}ne, J.-L. and Sansuc, J.-J.}, TITLE = {La descente sur les vari\'et\'es rationnelles. {II}}, JOURNAL = {Duke Math. J.}, FJOURNAL = {Duke Mathematical Journal}, VOLUME = {54}, YEAR = {1987}, NUMBER = {2}, PAGES = {375--492}, ISSN = {0012-7094}, CODEN = {DUMJAO}, MRCLASS = {11G35 (14G25)}, MRNUMBER = {89f:11082}, MRREVIEWER = {Daniel Coray}, } @article {MR1679841, AUTHOR = {Salberger, P.}, TITLE = {Tamagawa measures on universal torsors and points of bounded height on {F}ano varieties}, NOTE = {Nombre et r\'epartition de points de hauteur born\'ee (Paris, 1996)}, JOURNAL = {Ast\'erisque}, FJOURNAL = {Ast\'erisque}, NUMBER = {251}, YEAR = {1998}, PAGES = {91--258}, ISSN = {0303-1179}, MRCLASS = {11G50 (11G35 14G05 14G25 14J45)}, MRNUMBER = {MR1679841 (2000d:11091)}, MRREVIEWER = {David Harari}, } @article {MR1679842, AUTHOR = {Peyre, E.}, TITLE = {Terme principal de la fonction z\^eta des hauteurs et torseurs universels}, NOTE = {Nombre et r\'epartition de points de hauteur born\'ee (Paris, 1996)}, JOURNAL = {Ast\'erisque}, FJOURNAL = {Ast\'erisque}, NUMBER = {251}, YEAR = {1998}, PAGES = {259--298}, ISSN = {0303-1179}, MRCLASS = {11G50 (14G05 14G10)}, MRNUMBER = {MR1679842 (2000f:11081)}, MRREVIEWER = {Carlo Gasbarri}, } @article {MR1906155, AUTHOR = {Chambert-Loir, A. and Tschinkel, Yu.}, TITLE = {On the distribution of points of bounded height on equivariant compactifications of vector groups}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {148}, YEAR = {2002}, NUMBER = {2}, PAGES = {421--452}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {11G50 (11G35 14G05 14G10)}, MRNUMBER = {MR1906155 (2003d:11094)}, MRREVIEWER = {Tam{\'a}s Szamuely}, } @article {MR1909606, AUTHOR = {Bret{\`e}che, R. de la}, TITLE = {Nombre de points de hauteur born\'ee sur les surfaces de del {P}ezzo de degr\'e 5}, JOURNAL = {Duke Math. J.}, FJOURNAL = {Duke Mathematical Journal}, VOLUME = {113}, YEAR = {2002}, NUMBER = {3}, PAGES = {421--464}, ISSN = {0012-7094}, CODEN = {DUMJAO}, MRCLASS = {14G05 (11G35 11G50 14G25 14J45)}, MRNUMBER = {MR1909606 (2003m:14033)}, } @article {MR2373960, AUTHOR = {Bret{\`e}che, R. de la and Browning, T.~D.}, TITLE = {On {M}anin's conjecture for singular del {P}ezzo surfaces of degree four. {II}}, JOURNAL = {Math. Proc. Cambridge Philos. Soc.}, FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society}, VOLUME = 143, YEAR = 2007, NUMBER = 3, PAGES = {579--605}, ISSN = {0305-0041}, CODEN = {MPCPCO}, MRCLASS = {14G05 (14J26)}, MRNUMBER = {MR2373960}, MRREVIEWER = {Yuri Tschinkel}, } @article {MR2332351, AUTHOR = {Bret{\`e}che, R. de la and Browning, T.~D. and Derenthal, U.}, TITLE = {On {M}anin's conjecture for a certain singular cubic surface}, JOURNAL = {Ann. Sci. \'Ecole Norm. Sup. (4)}, FJOURNAL = {Annales Scientifiques de l'\'Ecole Normale Sup\'erieure. Quatri\`eme S\'erie}, VOLUME = 40, YEAR = 2007, NUMBER = 1, PAGES = {1--50}, ISSN = {0012-9593}, CODEN = {ASENAH}, MRCLASS = {14G25 (11G35 11G50 14Jxx)}, MRNUMBER = {MR2332351}, } @article{arXiv:1011.3434, title = {Inhomogeneous cubic congruences and rational points on del {P}ezzo surfaces}, author = {Baier, S. and Browning, T.~D.}, JOURNAL = {J. reine angew. Math.}, YEAR = {2013}, VOLUME={680}, PAGES={69-151} } @article {MR2329549, AUTHOR = {Bret{\`e}che, R. de la}, TITLE = {R\'epartition des points rationnels sur la cubique de {S}egre}, JOURNAL = {Proc. Lond. Math. Soc. (3)}, FJOURNAL = {Proceedings of the London Mathematical Society. Third Series}, VOLUME = {95}, YEAR = {2007}, NUMBER = {1}, PAGES = {69--155}, ISSN = {0024-6115}, MRCLASS = {11D72 (11G35 14G05)}, MRNUMBER = {MR2329549 (2008f:11041)}, MRREVIEWER = {Timothy D. Browning}, } @misc{arXiv:1205.0190, title = {{On a senary cubic form}}, author = {Blomer, V. and Br\"udern, J. and Salberger, P.}, eprint = {arXiv:1205.0190}, YEAR = {arXiv:1205.0190, 2012} } @article {MR557080, AUTHOR = {Schanuel, S.}, TITLE = {Heights in number fields}, JOURNAL = {Bull. Soc. Math. France}, FJOURNAL = {Bulletin de la Soci\'et\'e Math\'ematique de France}, VOLUME = {107}, YEAR = {1979}, NUMBER = {4}, PAGES = {433--449}, ISSN = {0037-9484}, CODEN = {BSMFAA}, MRCLASS = {12A99 (10B45 14G25)}, MRNUMBER = {557080 (81c:12025)}, MRREVIEWER = {Michel Waldschmidt}, URL = {http://www.numdam.org/item?id=BSMF_1979__107__433_0}, } @article {MR2320172, AUTHOR = {Bret{\`e}che, R. de la and Browning, T.~D.}, TITLE = {On {M}anin's conjecture for singular del {P}ezzo surfaces of degree 4. {I}}, JOURNAL = {Michigan Math. J.}, FJOURNAL = {The Michigan Mathematical Journal}, VOLUME = 55, YEAR = 2007, NUMBER = 1, PAGES = {51--80}, ISSN = {0026-2285}, MRCLASS = {14G05 (11G10 14G40)}, MRNUMBER = {MR2320172}, } @article {MR2520769, AUTHOR = {Browning, T.~D. and Derenthal, U.}, TITLE = {Manin's conjecture for a cubic surface with {$D_5$} singularity}, JOURNAL = {Int. Math. Res. Not. IMRN}, FJOURNAL = {International Mathematics Research Notices. IMRN}, YEAR = {2009}, VOLUME = {2009}, NUMBER = {14}, PAGES = {2620--2647}, ISSN = {1073-7928}, MRCLASS = {14G25 (14G05)}, MRNUMBER = {MR2520769}, } @article {MR2853047, AUTHOR = {Le Boudec, P.}, TITLE = {Manin's conjecture for two quartic del {P}ezzo surfaces with {$3{\bf A}_1$} and {${\bf A}_1+{\bf A}_2$} singularity types}, JOURNAL = {Acta Arith.}, FJOURNAL = {Acta Arithmetica}, VOLUME = 151, YEAR = 2012, NUMBER = 2, PAGES = {109--163}, ISSN = {0065-1036}, CODEN = {AARIA9}, MRCLASS = {11D45 (14G05)}, MRNUMBER = 2853047, DOI = {10.4064/aa151-2-1}, URL = {http://dx.doi.org/10.4064/aa151-2-1}, } @article {MR2980925, AUTHOR = {Loughran, D.}, TITLE = {Manin's conjecture for a singular quartic del {P}ezzo surface}, JOURNAL = {J. Lond. Math. Soc. (2)}, FJOURNAL = {Journal of the London Mathematical Society. Second Series}, VOLUME = {86}, YEAR = {2012}, NUMBER = {2}, PAGES = {558--584}, ISSN = {0024-6107}, MRCLASS = {11D45 (11N37 14G05 14J26)}, MRNUMBER = {2980925}, DOI = {10.1112/jlms/jds017}, URL = {http://dx.doi.org/10.1112/jlms/jds017}, } @article {MR2874644, AUTHOR = {Bret{\`e}che, R. de la and Browning, T.~D. and Peyre, E.}, TITLE = {On {M}anin's conjecture for a family of {C}h\^atelet surfaces}, JOURNAL = {Ann. of Math. (2)}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {175}, YEAR = {2012}, NUMBER = {1}, PAGES = {297--343}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {11G35 (11G50 14Gxx)}, MRNUMBER = {2874644}, DOI = {10.4007/annals.2012.175.1.8}, URL = {http://dx.doi.org/10.4007/annals.2012.175.1.8}, } @article {MR2838351, AUTHOR = {Bret{\`e}che, R. de la and Browning, T.~D.}, TITLE = {Manin's conjecture for quartic del {P}ezzo surfaces with a conic fibration}, JOURNAL = {Duke Math. J.}, FJOURNAL = {Duke Mathematical Journal}, VOLUME = {160}, YEAR = {2011}, NUMBER = {1}, PAGES = {1--69}, ISSN = {0012-7094}, CODEN = {DUMJAO}, MRCLASS = {11D45 (14G05 14G25 14J26)}, MRNUMBER = {2838351}, DOI = {10.1215/00127094-1443466}, URL = {http://dx.doi.org/10.1215/00127094-1443466}, } @misc{arXiv:0710.1583, title = {{Manin's conjecture for a quintic del Pezzo surface with $\mathbf{A}_2$ singularity}}, author = {U. Derenthal}, eprint = {arXiv:0710.1583}, year = {arXiv:0710.1583, 2007} } @article {MR2961294, AUTHOR = {Le Boudec, P.}, TITLE = {Manin's conjecture for a quartic del {P}ezzo surface with {$\bold A_3$} singularity and four lines}, JOURNAL = {Monatsh. Math.}, FJOURNAL = {Monatshefte f\"ur Mathematik}, VOLUME = {167}, YEAR = {2012}, NUMBER = {3-4}, PAGES = {481--502}, ISSN = {0026-9255}, CODEN = {MNMTA2}, MRCLASS = {11D45 (11G05 14G05)}, MRNUMBER = {2961294}, DOI = {10.1007/s00605-011-0315-3}, URL = {http://dx.doi.org/10.1007/s00605-011-0315-3}, } @article {MR2961294, AUTHOR = {Le Boudec, P.}, TITLE = {Manin's conjecture for a quartic del {P}ezzo surface with {$\bold A_3$} singularity and four lines}, JOURNAL = {Monatsh. Math.}, FJOURNAL = {Monatshefte f\"ur Mathematik}, VOLUME = {167}, YEAR = {2012}, NUMBER = {3-4}, PAGES = {481--502}, ISSN = {0026-9255}, CODEN = {MNMTA2}, MRCLASS = {11D45 (11G05 14G05)}, MRNUMBER = {2961294}, DOI = {10.1007/s00605-011-0315-3}, URL = {http://dx.doi.org/10.1007/s00605-011-0315-3}, } @article{Compositio, title = "{Counting imaginary quadratic points via universal torsors}", author = "U. Derenthal and C. Frei", journal = "{Compositio. Math.}", volume = {150}, number = {10}, pages = {1631--1678}, year = {2014}, } @article {MR2753646, AUTHOR = {Derenthal, U. and Loughran, D.}, TITLE = {Singular del {P}ezzo surfaces that are equivariant compactifications}, JOURNAL = {Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)}, FJOURNAL = {Rossi\u\i skaya Akademiya Nauk. Sankt-Peterburgskoe Otdelenie. Matematicheski\u\i\ Institut im. V. A. Steklova. Zapiski Nauchnykh Seminarov (POMI)}, VOLUME = {377}, YEAR = {2010}, NUMBER = {Issledovaniya po Teorii Chisel. 10}, PAGES = {26--43, 241}, ISSN = {0373-2703}, MRCLASS = {14J26 (11G35 14G05 14M27)}, MRNUMBER = {2753646 (2012b:14068)}, MRREVIEWER = {Paul A. Hacking}, DOI = {10.1007/s10958-010-0174-9}, URL = {http://dx.doi.org/10.1007/s10958-010-0174-9}, } @article{arXiv:1212.3518, title = "{{Equivariant compactifications of two-dimensional algebraic groups}}", author = "U. Derenthal and D. Loughran", eprint = "arXiv:1212.3518", JOURNAL = {Proc. Edinb. Math. Soc., to appear}, year = "arXiv:1212.3518, 2012" } @incollection {MR2858922, AUTHOR = {Tanimoto, S. and Tschinkel, Yu.}, TITLE = {Height zeta functions of equivariant compactifications of semi-direct products of algebraic groups}, BOOKTITLE = {Zeta functions in algebra and geometry}, SERIES = {Contemp. Math.}, VOLUME = {566}, PAGES = {119--157}, PUBLISHER = {Amer. Math. Soc.}, ADDRESS = {Providence, RI}, YEAR = {2012}, MRCLASS = {11G35 (11G50)}, MRNUMBER = {2858922}, DOI = {10.1090/conm/566/11218}, URL = {http://dx.doi.org/10.1090/conm/566/11218}, } @article {MR1369408, AUTHOR = {Batyrev, V.~V. and Tschinkel, Yu.}, TITLE = {Rational points of bounded height on compactifications of anisotropic tori}, JOURNAL = {Internat. Math. Res. Notices}, FJOURNAL = {International Mathematics Research Notices}, YEAR = {1995}, NUMBER = {12}, PAGES = {591--635}, ISSN = {1073-7928}, MRCLASS = {14G05 (11G35 14M25)}, MRNUMBER = {1369408 (97a:14021)}, MRREVIEWER = {Emmanuel Peyre}, DOI = {10.1155/S1073792895000365}, URL = {http://dx.doi.org/10.1155/S1073792895000365}, } @article{MPCPS, title = {{Counting imaginary quadratic points via universal torsors, II}}, author = {U. Derenthal and C. Frei}, eprint = {arXiv:1304.3352}, JOURNAL = {Math. Proc. Cambridge Philos. Soc.}, VOLUME = 156, NUMBER = 3, PAGES = {383-407}, YEAR = 2014 } @article {MR2990624, AUTHOR = {Le Boudec, P.}, TITLE = {Manin's conjecture for a cubic surface with {$2\bold A_2+\bold A_1$} singularity type}, JOURNAL = {Math. Proc. Cambridge Philos. Soc.}, FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society}, VOLUME = {153}, YEAR = {2012}, NUMBER = {3}, PAGES = {419--455}, ISSN = {0305-0041}, CODEN = {MPCPCO}, MRCLASS = {11G05 (14Gxx)}, MRNUMBER = {2990624}, DOI = {10.1017/S030500411200031X}, URL = {http://dx.doi.org/10.1017/S030500411200031X}, } @misc{arXiv:1207.2685, title = {Affine congruences and rational points on a certain cubic surface}, author = {Le Boudec, P.}, eprint = {arXiv:1207.2685}, year = {arXiv:1207.2685, 2012} } @article {MR0211966, AUTHOR = {Cassels, J.~W.~S. and Guy, M.~J.~T.}, TITLE = {On the {H}asse principle for cubic surfaces}, JOURNAL = {Mathematika}, FJOURNAL = {Mathematika. A Journal of Pure and Applied Mathematics}, VOLUME = {13}, YEAR = {1966}, PAGES = {111--120}, ISSN = {0025-5793}, MRCLASS = {10.12 (14.40)}, MRNUMBER = {0211966 (35 \#2841)}, MRREVIEWER = {D. J. Lewis}, } @article {MR89f:11083, AUTHOR = {Coray, D.~F. and Tsfasman, M.~A.}, TITLE = {Arithmetic on singular {D}el {P}ezzo surfaces}, JOURNAL = {Proc. London Math. Soc. (3)}, FJOURNAL = {Proceedings of the London Mathematical Society. Third Series}, VOLUME = {57}, YEAR = {1988}, NUMBER = {1}, PAGES = {25--87}, ISSN = {0024-6115}, CODEN = {PLMTAL}, MRCLASS = {11G35 (11E04 11E95 11G25 14J17 14J20 14J26)}, MRNUMBER = {89f:11083}, MRREVIEWER = {Noriko Yui}, } @article {MR80f:14021, AUTHOR = {Bruce, J.~W. and Wall, C.~T.~C.}, TITLE = {On the classification of cubic surfaces}, JOURNAL = {J. London Math. Soc. (2)}, FJOURNAL = {The Journal of the London Mathematical Society. Second Series}, VOLUME = {19}, YEAR = {1979}, NUMBER = {2}, PAGES = {245--256}, ISSN = {0024-6107}, CODEN = {JLMSAK}, MRCLASS = {14J25 (14J17)}, MRNUMBER = {80f:14021}, MRREVIEWER = {I. Dolgachev}, } @article {MR95i:14046, AUTHOR = {Cox, D.~A.}, TITLE = {The homogeneous coordinate ring of a toric variety}, JOURNAL = {J. Algebraic Geom.}, FJOURNAL = {Journal of Algebraic Geometry}, VOLUME = {4}, YEAR = {1995}, NUMBER = {1}, PAGES = {17--50}, ISSN = {1056-3911}, MRCLASS = {14M25}, MRNUMBER = {95i:14046}, MRREVIEWER = {Mina Teicher}, } @article {MR2000f:11080, AUTHOR = {Heath-Brown, D.~R. and Moroz, B.~Z.}, TITLE = {The density of rational points on the cubic surface {$X\sb 0\sp 3=X\sb 1X\sb 2X\sb 3$}}, JOURNAL = {Math. Proc. Cambridge Philos. Soc.}, FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society}, VOLUME = {125}, YEAR = {1999}, NUMBER = {3}, PAGES = {385--395}, ISSN = {0305-0041}, CODEN = {MPCPCO}, MRCLASS = {11G50 (11P55 14G05)}, MRNUMBER = {2000f:11080}, MRREVIEWER = {Carlo Gasbarri}, } @article {MR98h:11083, AUTHOR = {Heath-Brown, D.~R.}, TITLE = {The density of rational points on cubic surfaces}, JOURNAL = {Acta Arith.}, FJOURNAL = {Acta Arithmetica}, VOLUME = {79}, YEAR = {1997}, NUMBER = {1}, PAGES = {17--30}, ISSN = {0065-1036}, CODEN = {AARIA9}, MRCLASS = {11G35 (14G05)}, MRNUMBER = {98h:11083}, MRREVIEWER = {Emmanuel Peyre}, } @incollection {MR2029868, AUTHOR = {Hassett, B. and Tschinkel, Yu.}, TITLE = {Universal torsors and {C}ox rings}, BOOKTITLE = {Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002)}, SERIES = {Progr. Math.}, VOLUME = {226}, PAGES = {149--173}, PUBLISHER = {Birkh\"auser Boston}, ADDRESS = {Boston, MA}, YEAR = {2004}, MRCLASS = {14G27 (11G50 14J26)}, MRNUMBER = {2 029 868}, } @book {MR833513, AUTHOR = {Manin, Yu.~I.}, TITLE = {Cubic forms}, SERIES = {North-Holland Mathematical Library}, VOLUME = {4}, EDITION = {Second}, PUBLISHER = {North-Holland Publishing Co.}, ADDRESS = {Amsterdam}, YEAR = {1986}, PAGES = {x+326}, ISBN = {0-444-87823-8}, MRCLASS = {11Gxx (14Gxx 14J20)}, MRNUMBER = {MR833513 (87d:11037)}, } @article {MR2019019, AUTHOR = {Peyre, E.}, TITLE = {Points de hauteur born\'ee, topologie ad\'elique et mesures de {T}amagawa}, NOTE = {Les XXII{\`e}mes Journ{\'e}es Arithmetiques (Lille, 2001)}, JOURNAL = {J. Th\'eor. Nombres Bordeaux}, FJOURNAL = {Journal de Th\'eorie des Nombres de Bordeaux}, VOLUME = {15}, YEAR = {2003}, NUMBER = {1}, PAGES = {319--349}, ISSN = {1246-7405}, MRCLASS = {14G05 (11G35 11G50 14G25)}, MRNUMBER = {2019019 (2004i:14021)}, MRREVIEWER = {Timothy D. Browning}, URL = {http://jtnb.cedram.org/item?id=JTNB_2003__15_1_319_0}, } @article {MR0073643, AUTHOR = {Skolem, Th.}, TITLE = {Einige {B}emerkungen \"uber die {A}uffindung der rationalen {P}unkte auf gewissen algebraischen {G}ebilden}, JOURNAL = {Math. Z.}, FJOURNAL = {Mathematische Zeitschrift}, VOLUME = {63}, YEAR = {1955}, PAGES = {295--312}, ISSN = {0025-5874}, MRCLASS = {10.2X}, MRNUMBER = {0073643 (17,464c)}, MRREVIEWER = {B. Segre}, } @article {Ashrafi2005, AUTHOR = {Ashrafi, N. and V{\'a}mos, P.}, TITLE = {On the unit sum number of some rings}, JOURNAL = {Q. J. Math.}, VOLUME = {56}, YEAR = {2005}, NUMBER = {1}, PAGES = {1--12}, } @article {Ashrafi2009, AUTHOR = {Ashrafi, N.}, TITLE = {A finer classification of the unit sum number of the ring of integers of quadratic fields and complex cubic fields}, JOURNAL = {Proc. Indian Acad. Sci. Math. Sci.}, VOLUME = {119}, YEAR = {2009}, NUMBER = {3}, PAGES = {267--274}, } @article {Ashrafi2008, AUTHOR = {Ashrafi, N.}, TITLE = {The unit sum number of some projective modules}, JOURNAL = {Glasg. Math. J.}, VOLUME = {50}, YEAR = {2008}, NUMBER = {1}, PAGES = {71--74}, } @article {Belcher1974, AUTHOR = {Belcher, P.}, TITLE = {Integers expressible as sums of distinct units}, JOURNAL = {Bull. Lond. Math. Soc.}, VOLUME = {6}, YEAR = {1974}, PAGES = {66--68}, } @article {Belcher1975, AUTHOR = {Belcher, P.}, TITLE = {A test for integers being sums of distinct units applied to cubic fields}, JOURNAL = {J. Lond. Math. Soc.}, VOLUME = {12}, YEAR = {1975/76}, NUMBER = {2}, PAGES = {141--148}, } @article {Brownawell1986, AUTHOR = {Brownawell, W.~D. and Masser, D.~W.}, TITLE = {Vanishing sums in function fields}, JOURNAL = {Math. Proc. Camb. Philos. Soc.}, VOLUME = {100}, YEAR = {1986}, NUMBER = {3}, PAGES = {427--434}, } @book {Conway2003, AUTHOR = {Conway, J.~H. and Smith, D.~A.}, TITLE = {On quaternions and octonions: their geometry, arithmetic and symmetry}, PUBLISHER = {A K Peters}, ADDRESS = {Natick, Massachusetts}, YEAR = {2003}, } @book {Cox1989, AUTHOR = {Cox, D.~A.}, TITLE = {Primes of the form {$x^2 + ny^2$}}, NOTE = {Fermat, class field theory and complex multiplication}, PUBLISHER = {John Wiley \& Sons Inc.}, ADDRESS = {New York}, YEAR = {1989}, } @article {Dieudonne1948, AUTHOR = {Dieudonn{\'e}, J.}, TITLE = {La th\'eorie de {G}alois des anneux simples et semi-simples}, JOURNAL = {Comment. Math. Helv.}, FJOURNAL = {Commentarii Mathematici Helvetici}, VOLUME = {21}, YEAR = {1948}, PAGES = {154--184}, } @article {Erdos1953, AUTHOR = {Erd{\"o}s, P.}, TITLE = {Arithmetical properties of polynomials}, JOURNAL = {J. London Math. Soc.}, VOLUME = {28}, YEAR = {1953}, PAGES = {416--425}, } @article {Everest1990, AUTHOR = {Everest, G.~R.}, TITLE = {Counting the values taken by sums of {$S$}-units}, JOURNAL = {J. Number Theory}, VOLUME = {35}, YEAR = {1990}, NUMBER = {3}, PAGES = {269--286}, } @article {Everest1999, AUTHOR = {Everest, G.~R. and Shparlinski, I.~E.}, TITLE = {Counting the values taken by algebraic exponential polynomials}, JOURNAL = {Proc. Amer. Math. Soc.}, VOLUME = {127}, YEAR = {1999}, NUMBER = {3}, PAGES = {665--675},} @article {Evertse1984, AUTHOR = {Evertse, J.-H.}, TITLE = {On equations in {$S$}-units and the {T}hue-{M}ahler equation}, JOURNAL = {Invent. Math.}, VOLUME = {75}, YEAR = {1984}, NUMBER = {3}, PAGES = {561--584}, } @article {Evertse1988, AUTHOR = {Evertse, J.-H. and Gy{\H{o}}ry, K.}, TITLE = {On the numbers of solutions of weighted unit equations}, JOURNAL = {Compositio Math.}, FJOURNAL = {Compositio Mathematica}, VOLUME = {66}, YEAR = {1988}, NUMBER = {3}, PAGES = {329--354}, } @article {Filipin2008, AUTHOR = {Filipin, A. and Tichy, R.~F. and Ziegler, V.}, TITLE = {The additive unit structure of pure quartic complex fields}, JOURNAL = {Funct. Approx. Comment. Math.}, VOLUME = {39}, YEAR = {2008}, NUMBER = {1}, PAGES = {113--131}, } @article {Filipin2008B, AUTHOR = {Filipin, A. and Tichy, R.~F. and Ziegler, V.}, TITLE = {On the quantitative unit sum number problem---an application of the subspace theorem}, JOURNAL = {Acta Arith.}, VOLUME = {133}, YEAR = {2008}, NUMBER = {4}, PAGES = {297--308}, } @article {Fisher1976, AUTHOR = {Fisher, J.~W. and Snider, R.~L.}, TITLE = {Rings generated by their units}, JOURNAL = {J. Algebra}, VOLUME = {42}, YEAR = {1976}, NUMBER = {2}, PAGES = {363--368}, } @article {Masser2006, AUTHOR = {Masser, D. and Vaaler, J.~D.}, TITLE = {Counting algebraic numbers with large height. {II}}, JOURNAL = {Trans. Amer. Math. Soc.}, VOLUME = {359}, YEAR = {2006}, NUMBER = {1}, PAGES = {427--445}, } @article {Widmer2010, AUTHOR = {Widmer, M.}, TITLE = {Counting primitive points of bounded height}, JOURNAL = {Trans. Amer. Math. Soc.}, VOLUME = {362}, YEAR = {2010}, PAGES = {4793-4829}, } @article {Blomer2018, AUTHOR = {Blomer, V.}, TITLE = {Higher order divisor problems}, JOURNAL = {Math. Z.}, VOLUME = {290}, YEAR = {2018}, PAGES = {937--952}, } @article {Jakhar2018, AUTHOR = {Jakhar, A. and Khanduja, S.~K. and Sangwan, N.}, TITLE = {On the compositum of integral closures of valuation rings}, JOURNAL = {J. Pure Appl. Algebra}, VOLUME = {222}, YEAR = {2018}, PAGES = {3560--3565}, } @article {Lenstra1977, AUTHOR = {Lenstra, H.~W.}, TITLE = {On {A}rtin's conjecture and {E}uclid's algorithm in global fields}, JOURNAL = {Invent. Math.}, VOLUME = {42}, YEAR = {1977}, PAGES = {201--224}, } @article {MR3556829, AUTHOR = {Bubboloni, D. and Sonn, J.}, TITLE = {Intersective {$S_n$} polynomials with few irreducible factors}, JOURNAL = {Manuscripta Math.}, FJOURNAL = {Manuscripta Mathematica}, VOLUME = 151, YEAR = 2016, NUMBER = {3-4}, PAGES = {477--492}, ISSN = {0025-2611,1432-1785}, } @book {MalleMatzat2018, AUTHOR = {Malle, G. and Matzat, B.H.}, TITLE = {Inverse Galois Theory}, PUBLISHER = {Springer}, EDITION = { 2nd edition}, YEAR = {2018}, } @incollection{HoffmanHumphreys, author = {Hoffman, P. N. and Humphreys, J. F.}, title = "{Projective Representations and Representation Groups}", booktitle = "{Projective Representations of the Symmetric Groups: Q-Functions and Shifted Tableaux}", publisher = {Oxford University Press}, year = {1992}, }