@article{DujellaPeral2020,
  author = {Dujella, Andrej and Peral, Juan Carlos},
  title = {High rank elliptic curves induced by rational {Diophantine} triples},
  journal = {Glasnik Matematički. Series III},
  volume = {55}, number = {2}, year = {2020}, pages = {237--252},
  doi = {10.3336/gm.55.2.05},
  eprint = {2005.10706v1}, archivePrefix = {arXiv}, primaryClass = {math.NT}
}

@article{DujellaPeral2024,
  author = {Dujella, Andrej and Peral, Juan Carlos},
  title = {An elliptic curve over {$\mathbb{Q}(u)$} with torsion {$\mathbb{Z}/4\mathbb{Z}$} and rank 6},
  journal = {Rad Hrvatske akademije znanosti i umjetnosti. Matematičke znanosti},
  volume = {28}, year = {2024}, pages = {185--192},
  doi = {10.21857/y54jof4o2m},
  eprint = {2207.08206v2}, archivePrefix = {arXiv}, primaryClass = {math.NT},
  note = {Mathematical-series volume 28, academy volume 558}
}

@incollection{Elkies2007,
  author = {Elkies, Noam D.},
  title = {Elliptic curves and surfaces of high rank, {I, II, III}},
  booktitle = {Explicit Methods in Number Theory},
  series = {Oberwolfach Reports}, volume = {4},
  year = {2007}, pages = {1992--2006}, publisher = {European Mathematical Society},
  doi = {10.4171/OWR/2007/34},
  eprint = {0709.2908v1}, archivePrefix = {arXiv}, primaryClass = {math.NT},
  note = {Contribution to the workshop report, whose DOI is given here; preprint entitled Three lectures on elliptic surfaces and curves of high rank}
}

@misc{Elkies2026,
  author = {Elkies, Noam D.},
  title = {An elliptic {K3} surface {$X/\mathbb{Q}(t)$} with {Mordell--Weil} rank 17, {I}: Formulas for {$X$} and base changes of rank 18 and 19},
  year = {2026}, eprint = {2608.25406v1}, archivePrefix = {arXiv},
  howpublished = {Preprint, \href{https://arxiv.org/abs/2608.25406v1}{arXiv:2608.25406v1}}, primaryClass = {math.NT}
}

@article{GT2010,
  author = {Green, Benjamin and Tao, Terence}, title = {Linear equations in primes},
  journal = {Annals of Mathematics}, series = {2}, volume = {171}, number = {3},
  year = {2010}, pages = {1753--1850}, doi = {10.4007/annals.2010.171.1753},
  eprint = {math/0606088v2}, archivePrefix = {arXiv}, primaryClass = {math.NT}
}

@article{GT2012,
  author = {Green, Ben and Tao, Terence},
  title = {The {M\"obius} function is strongly orthogonal to nilsequences},
  journal = {Annals of Mathematics}, series = {2}, volume = {175}, number = {2},
  year = {2012}, pages = {541--566}, doi = {10.4007/annals.2012.175.2.3},
  eprint = {0807.1736v4}, archivePrefix = {arXiv}, primaryClass = {math.NT}
}

@article{GTZ2012,
  author = {Green, Ben and Tao, Terence and Ziegler, Tamar},
  title = {An inverse theorem for the {Gowers} {$U^{s+1}[N]$}-norm},
  journal = {Annals of Mathematics}, series = {2}, volume = {176}, number = {2},
  year = {2012}, pages = {1231--1372}, doi = {10.4007/annals.2012.176.2.11},
  eprint = {1009.3998v5}, archivePrefix = {arXiv}, primaryClass = {math.CO},
  note = {See also the authors' \href{https://terrytao.wordpress.com/wp-content/uploads/2024/04/erratum-4.pdf}{2024 erratum} and the corrected preprint \href{https://arxiv.org/abs/1009.3998v5}{arXiv:1009.3998v5} (2026)}
}

@book{IrelandRosen,
  author = {Ireland, Kenneth and Rosen, Michael},
  title = {A Classical Introduction to Modern Number Theory}, edition = {Second},
  series = {Graduate Texts in Mathematics}, volume = {84},
  publisher = {Springer-Verlag}, address = {New York}, year = {1990},
  doi = {10.1007/978-1-4757-2103-4}
}

@article{HSW2014,
  author = {Harpaz, Yonatan and Skorobogatov, Alexei N. and Wittenberg, Olivier},
  title = {The {Hardy--Littlewood} conjecture and rational points},
  journal = {Compositio Mathematica}, volume = {150}, number = {12},
  year = {2014}, pages = {2095--2111}, doi = {10.1112/S0010437X14007568},
  eprint = {1304.3333v3}, archivePrefix = {arXiv}, primaryClass = {math.NT}
}

@misc{Kai,
  author = {Kai, Wataru}, title = {Linear patterns of prime elements in number fields},
  year = {2023}, eprint = {2306.16983v5}, archivePrefix = {arXiv},
  howpublished = {Preprint, \href{https://arxiv.org/abs/2306.16983v5}{arXiv:2306.16983v5}}, primaryClass = {math.NT}, note = {Revised 12 March 2026}
}

@misc{KoymansPagano2025,
  author = {Koymans, Peter and Pagano, Carlo},
  title = {Elliptic curves of rank one over number fields},
  year = {2025}, eprint = {2505.16910v1}, archivePrefix = {arXiv},
  howpublished = {Preprint, \href{https://arxiv.org/abs/2505.16910v1}{arXiv:2505.16910v1}}, primaryClass = {math.NT}
}

@misc{Lu2026,
  author = {Lu, Junyu}, title = {Prescribed ideal classes and elliptic points in {Ennola} cubic fields},
  year = {2026}, eprint = {2610.04254v1}, archivePrefix = {arXiv},
  howpublished = {Preprint, \href{https://arxiv.org/abs/2610.04254v1}{arXiv:2610.04254v1}}, primaryClass = {math.NT}
}

@article{MazurRubin2010,
  author = {Mazur, Barry and Rubin, Karl},
  title = {Ranks of twists of elliptic curves and {Hilbert}'s tenth problem},
  journal = {Inventiones Mathematicae}, volume = {181}, number = {3},
  year = {2010}, pages = {541--575}, doi = {10.1007/s00222-010-0252-0},
  eprint = {0904.3709v3}, archivePrefix = {arXiv}, primaryClass = {math.NT}
}

@article{Mestre1991,
  author = {Mestre, Jean-François},
  title = {Courbes elliptiques de rang {$\geq 12$} sur {$\mathbb{Q}(t)$}},
  journal = {Comptes Rendus de l'Académie des Sciences. Série I. Mathématique},
  volume = {313}, year = {1991}, pages = {171--174}
}

@misc{Savoie2026,
  author = {Savoie, Ben},
  title = {Infinitely many elliptic curves over {$\mathbb{Q}(i)$} of exact ranks 4 and 6 with {$j$}-invariant 1728},
  year = {2026}, eprint = {2608.13202v1}, archivePrefix = {arXiv},
  howpublished = {Preprint, \href{https://arxiv.org/abs/2608.13202v1}{arXiv:2608.13202v1}}, primaryClass = {math.NT}
}

@article{Silverman1983,
  author = {Silverman, Joseph H.},
  title = {Heights and the specialization map for families of abelian varieties},
  journal = {Journal für die reine und angewandte Mathematik},
  volume = {342}, year = {1983}, pages = {197--211}, doi = {10.1515/crll.1983.342.197}
}

@book{SilvermanAEC,
  author = {Silverman, Joseph H.}, title = {The Arithmetic of Elliptic Curves},
  edition = {Second}, series = {Graduate Texts in Mathematics}, volume = {106},
  publisher = {Springer}, address = {Dordrecht}, year = {2009},
  doi = {10.1007/978-0-387-09494-6}
}

@misc{Zywina2025,
  author = {Zywina, David},
  title = {There are infinitely many elliptic curves over the rationals of rank 2},
  year = {2025}, eprint = {2502.01957v1}, archivePrefix = {arXiv},
  howpublished = {Preprint, \href{https://arxiv.org/abs/2502.01957v1}{arXiv:2502.01957v1}}, primaryClass = {math.NT}
}

@misc{ZywinaRankOne2025,
  author = {Zywina, David}, title = {Rank one elliptic curves and rank stability},
  year = {2025}, eprint = {2505.16960v1}, archivePrefix = {arXiv},
  howpublished = {Preprint, \href{https://arxiv.org/abs/2505.16960v1}{arXiv:2505.16960v1}}, primaryClass = {math.NT}
}

@misc{Zywina2026,
  author = {Zywina, David},
  title = {On the infinitude of elliptic curves over a number field with prescribed small rank},
  year = {2026}, eprint = {2602.10865v1}, archivePrefix = {arXiv},
  howpublished = {Preprint, \href{https://arxiv.org/abs/2602.10865v1}{arXiv:2602.10865v1}}, primaryClass = {math.NT}
}

@article{TaoZiegler2008,
  author = {Tao, Terence and Ziegler, Tamar},
  title = {The primes contain arbitrarily long polynomial progressions},
  journal = {Acta Mathematica}, volume = {201}, number = {2}, year = {2008},
  pages = {213--305}, doi = {10.1007/s11511-008-0032-5},
  eprint = {math/0610050v2}, archivePrefix = {arXiv}, primaryClass = {math.NT},
  note = {See the 2013 erratum}
}

@article{TaoZieglerErratum2013,
  author = {Tao, Terence and Ziegler, Tamar},
  title = {Erratum to ``{T}he primes contain arbitrarily long polynomial progressions''},
  journal = {Acta Mathematica}, volume = {210}, number = {2}, year = {2013},
  pages = {403--404}, doi = {10.1007/s11511-013-0097-7}
}


@article{Mordell1922,
  author  = {Mordell, Louis J.},
  title   = {On the rational solutions of the indeterminate equations
             of the third and fourth degrees},
  journal = {Proceedings of the Cambridge Philosophical Society},
  volume  = {21},
  year    = {1922},
  pages   = {179--192}
}

@article{Weil1929,
  author  = {Weil, Andr{\'e}},
  title   = {L'arithm{\'e}tique sur les courbes alg{\'e}briques},
  journal = {Acta Mathematica},
  volume  = {52},
  year    = {1929},
  pages   = {281--315},
  doi     = {10.1007/BF02592688}
}

@book{SilvermanTate2015,
  author    = {Silverman, Joseph H. and Tate, John T.},
  title     = {Rational Points on Elliptic Curves},
  edition   = {Second},
  series    = {Undergraduate Texts in Mathematics},
  publisher = {Springer},
  address   = {Cham},
  year      = {2015},
  doi       = {10.1007/978-3-319-18588-0}
}

@book{Cassels1991,
  author    = {Cassels, J. W. S.},
  title     = {Lectures on Elliptic Curves},
  series    = {London Mathematical Society Student Texts},
  volume    = {24},
  publisher = {Cambridge University Press},
  address   = {Cambridge},
  year      = {1991},
  doi       = {10.1017/CBO9781139172530}
}

@article{PPVW2019,
  author  = {Park, Jennifer and Poonen, Bjorn and Voight, John
             and Wood, Melanie Matchett},
  title   = {A heuristic for boundedness of ranks of elliptic curves},
  journal = {Journal of the European Mathematical Society},
  volume  = {21},
  number  = {9},
  year    = {2019},
  pages   = {2859--2903},
  doi     = {10.4171/JEMS/893}
}

@article{BhargavaSkinner2014,
  author  = {Bhargava, Manjul and Skinner, Christopher},
  title   = {A positive proportion of elliptic curves over {$\mathbb Q$}
             have rank one},
  journal = {Journal of the Ramanujan Mathematical Society},
  volume  = {29},
  number  = {2},
  year    = {2014},
  pages   = {221--242},
  eprint  = {1401.0233v1},
  archivePrefix = {arXiv},
  primaryClass  = {math.NT}
}

@misc{Savoie2025,
  author = {Savoie, Ben},
  title  = {Infinitely many elliptic curves over {$\mathbb Q(i)$}
            with rank 2 and {$j$}-invariant 1728},
  year   = {2025},
  eprint = {2506.17605v2},
  archivePrefix = {arXiv},
  primaryClass  = {math.NT},
  howpublished  = {Preprint, \href{https://arxiv.org/abs/2506.17605v2}{arXiv:2506.17605v2}}
}

@article{RedeiReichardt1934,
  author  = {R{\'e}dei, L{\'a}szl{\'o} and Reichardt, Hans},
  title   = {Die {A}nzahl der durch 4 teilbaren {I}nvarianten der {K}lassengruppe
             eines beliebigen quadratischen {Z}ahlk{\"o}rpers},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume  = {170},
  year    = {1934},
  pages   = {69--74},
  doi     = {10.1515/crll.1934.170.69}
}

@article{Redei1934,
  author  = {R{\'e}dei, L{\'a}szl{\'o}},
  title   = {Arithmetischer {B}eweis des {S}atzes {\"u}ber die {A}nzahl der durch
             vier teilbaren {I}nvarianten der absoluten {K}lassengruppe
             im quadratischen {Z}ahlk{\"o}rper},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume  = {171},
  year    = {1934},
  pages   = {55--60},
  doi     = {10.1515/crll.1934.171.55}
}

@article{HeathBrownMonsky1994,
  author  = {Heath-Brown, D. R.},
  title   = {The size of {Selmer} groups for the congruent number problem, {II}},
  journal = {Inventiones Mathematicae},
  volume  = {118},
  year    = {1994},
  pages   = {331--370},
  doi     = {10.1007/BF01231536},
  note    = {With an appendix by P. Monsky, pp. 365--370}
}

@article{XiongZaharescu2009,
  author  = {Xiong, Maosheng and Zaharescu, Alexandru},
  title   = {{Selmer} groups and {Tate--Shafarevich} groups
             for the congruent number problem},
  journal = {Commentarii Mathematici Helvetici},
  volume  = {84},
  number  = {1},
  year    = {2009},
  pages   = {21--56},
  doi     = {10.4171/CMH/151}
}

@misc{KaneKlagsbrun2017,
  author = {Kane, Daniel and Klagsbrun, Zev},
  title  = {On the joint distribution of {$\mathrm{Sel}_\phi(E/\mathbb{Q})$} and
            {$\mathrm{Sel}_{\hat\phi}(E'/\mathbb{Q})$} in quadratic twist families},
  year   = {2017},
  eprint = {1702.02687v1},
  archivePrefix = {arXiv},
  primaryClass  = {math.NT},
  howpublished  = {Preprint, \href{https://arxiv.org/abs/1702.02687v1}{arXiv:1702.02687v1}}
}

@article{SchinzelSierpinski1958,
  author  = {Schinzel, Andrzej and Sierpi{\'n}ski, Wac{\l}aw},
  title   = {Sur certaines hypoth{\`e}ses concernant les nombres premiers},
  journal = {Acta Arithmetica},
  volume  = {4},
  number  = {3},
  year    = {1958},
  pages   = {185--208},
  doi     = {10.4064/aa-4-3-185-208},
  note = {Correction: Acta Arithmetica 5 (1959), no. 2, 259,
          \href{https://doi.org/10.4064/aa-5-2-259}{doi:10.4064/aa-5-2-259}}
}

@article{ColliotTheleneSansuc1982,
  author  = {Colliot-Th{\'e}l{\`e}ne, Jean-Louis and Sansuc, Jean-Jacques},
  title   = {Sur le principe de {Hasse} et l'approximation faible,
             et sur une hypoth{\`e}se de {Schinzel}},
  journal = {Acta Arithmetica},
  volume  = {41},
  number  = {1},
  year    = {1982},
  pages   = {33--53},
  doi     = {10.4064/aa-41-1-33-53}
}

@article{ColliotTheleneSwinnertonDyer1994,
  author  = {Colliot-Th{\'e}l{\`e}ne, Jean-Louis and Swinnerton-Dyer, Peter},
  title   = {{Hasse} principle and weak approximation for pencils
             of {Severi--Brauer} and similar varieties},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume  = {453},
  year    = {1994},
  pages   = {49--112},
  doi     = {10.1515/crll.1994.453.49}
}

@incollection{SwinnertonDyer1995,
  author    = {Swinnerton-Dyer, Peter},
  title     = {Rational points on certain intersections of two quadrics},
  booktitle = {Abelian Varieties},
  editor    = {Barth, Wolf and Hulek, Klaus and Lange, Herbert},
  publisher = {Walter de Gruyter},
  address   = {Berlin},
  year      = {1995},
  pages     = {273--292},
  doi       = {10.1515/9783110889437.273}
}

@article{CSS1998GenusOne,
  author  = {Colliot-Th{\'e}l{\`e}ne, Jean-Louis
             and Skorobogatov, Alexei N. and Swinnerton-Dyer, Peter},
  title   = {{Hasse} principle for pencils of curves of genus one
             whose {Jacobians} have rational {$2$}-division points},
  journal = {Inventiones Mathematicae},
  volume  = {134},
  number  = {3},
  year    = {1998},
  pages   = {579--650},
  doi     = {10.1007/s002220050274}
}

@article{SkorobogatovSwinnertonDyer2005,
  author  = {Skorobogatov, Alexei and Swinnerton-Dyer, Peter},
  title   = {{$2$}-descent on elliptic curves and rational points
             on certain {Kummer} surfaces},
  journal = {Advances in Mathematics},
  volume  = {198},
  number  = {2},
  year    = {2005},
  pages   = {448--483},
  doi     = {10.1016/j.aim.2005.06.005}
}

@article{OguisoShioda1991,
  author  = {Oguiso, Keiji and Shioda, Tetsuji},
  title   = {The {Mordell--Weil} lattice of a rational elliptic surface},
  journal = {Commentarii Mathematici Universitatis Sancti Pauli},
  volume  = {40},
  number  = {1},
  year    = {1991},
  pages   = {83--99}
}

@article{Shioda1992,
  author  = {Shioda, Tetsuji},
  title   = {Some remarks on elliptic curves over function fields},
  journal = {Ast{\'e}risque},
  volume  = {209},
  year    = {1992},
  pages   = {99--114},
  url     = {https://www.numdam.org/item/AST_1992__209__99_0/},
  note    = {Journ{\'e}es Arithm{\'e}tiques, 1991 (Geneva)}
}


@article{Salgado2012,
  author = {Salgado, Cec{\'i}lia},
  title = {On the rank of the fibers of rational elliptic surfaces},
  journal = {Algebra \& Number Theory},
  volume = {6},
  number = {7},
  year = {2012},
  pages = {1289--1313},
  doi = {10.2140/ant.2012.6.1289},
  eprint = {1307.5912v1},
  archivePrefix = {arXiv},
  primaryClass = {math.NT}
}


@article{LoughranSalgado2022,
  author = {Loughran, Daniel and Salgado, Cec{\'i}lia},
  title = {Rank jumps on elliptic surfaces and the {Hilbert} property},
  journal = {Annales de l'Institut Fourier},
  volume = {72},
  number = {2},
  year = {2022},
  pages = {617--638},
  doi = {10.5802/aif.3457},
  eprint = {1907.01987v2},
  archivePrefix = {arXiv},
  primaryClass = {math.NT}
}


@article{SchuttShioda2010,
  author = {Sch{\"u}tt, Matthias and Shioda, Tetsuji},
  title = {Elliptic surfaces},
  journal = {Advanced Studies in Pure Mathematics},
  volume = {60},
  year = {2010},
  pages = {51--160},
  eprint = {0907.0298v3},
  archivePrefix = {arXiv},
  primaryClass = {math.AG}
}


@incollection{Tate1975,
  author = {Tate, John},
  title = {Algorithm for determining the type of a singular fiber in an elliptic pencil},
  booktitle = {Modular Functions of One Variable IV},
  editor = {Birch, Bryan J. and Kuyk, Willem},
  series = {Lecture Notes in Mathematics},
  volume = {476},
  publisher = {Springer},
  address = {Berlin, Heidelberg},
  year = {1975},
  pages = {33--52},
  doi = {10.1007/BFb0097582}
}


@article{Feng1996,
  author  = {Feng, Keqin},
  title   = {Non-congruent numbers, odd graphs and
             the {Birch--Swinnerton-Dyer} conjecture},
  journal = {Acta Arithmetica},
  volume  = {75},
  number  = {1},
  year    = {1996},
  pages   = {71--83},
  doi     = {10.4064/aa-75-1-71-83}
}


@misc{KlingSavoie2024,
  author = {Kling, Anthony and Savoie, Ben},
  title = {A graph-theoretic approach to computing {Selmer} groups
           of elliptic curves {$y^2=x^3+bx$} over {$\mathbb{Q}(i)$}},
  year = {2024},
  eprint = {2410.22714v2},
  archivePrefix = {arXiv},
  primaryClass = {math.NT},
  howpublished = {\href{https://arxiv.org/abs/2410.22714v2}
                  {arXiv:2410.22714v2}},
  note = {Version 2, 21 June 2025. Published online in
          Journal of Number Theory, 25 September 2026,
          doi:10.1016/j.jnt.2026.08.014}
}


@article{ByeonJeong2016,
  author = {Byeon, Dongho and Jeong, Keunyoung},
  title = {Infinitely many elliptic curves of rank exactly two},
  journal = {Proceedings of the Japan Academy, Series A, Mathematical Sciences},
  volume = {92}, number = {5}, year = {2016}, pages = {64--66},
  doi = {10.3792/pjaa.92.64}
}


@article{Jeong2019,
  author = {Jeong, Keunyoung},
  title = {Infinitely many elliptic curves of rank exactly two {II}},
  journal = {Proceedings of the Japan Academy, Series A, Mathematical Sciences},
  volume = {95}, number = {6}, year = {2019}, pages = {53--57},
  doi = {10.3792/pjaa.95.53},
  eprint = {1805.00639v2}, archivePrefix = {arXiv}, primaryClass = {math.NT}
}


@misc{ZywinaNoJumps2025,
  author = {Zywina, David},
  title = {An elliptic surface with infinitely many fibers for which the rank does not jump},
  year = {2025}, eprint = {2502.01026}, archivePrefix = {arXiv},
  howpublished = {Preprint, \href{https://arxiv.org/abs/2502.01026}{arXiv:2502.01026}},
  primaryClass = {math.NT}
}


@article{BenderSwinnertonDyer2001,
  author = {Bender, A. O. and Swinnerton-Dyer, Peter},
  title = {Solubility of certain pencils of curves of genus 1,
           and of the intersection of two quadrics in {$\mathbb{P}^4$}},
  journal = {Proceedings of the London Mathematical Society},
  series = {3}, volume = {83}, number = {2}, year = {2001},
  pages = {299--329}, doi = {10.1112/plms/83.2.299}
}


@incollection{ColliotThelene2001,
  author = {Colliot-Th{\'e}l{\`e}ne, Jean-Louis},
  title = {{Hasse} principle for pencils of curves of genus one
           whose {Jacobians} have a rational {$2$}-division point:
           close variation on a paper of {Bender} and {Swinnerton-Dyer}},
  booktitle = {Rational Points on Algebraic Varieties},
  editor = {Peyre, Emmanuel and Tschinkel, Yuri},
  series = {Progress in Mathematics}, volume = {199},
  publisher = {Birkh{\"a}user}, address = {Basel},
  year = {2001}, pages = {117--161},
  doi = {10.1007/978-3-0348-8368-9_5}
}


@article{BrowningMatthiesenSkorobogatov2014,
  author = {Browning, Tim D. and Matthiesen, Lilian
            and Skorobogatov, Alexei N.},
  title = {Rational points on pencils of conics and quadrics
           with many degenerate fibres},
  journal = {Annals of Mathematics}, series = {2},
  volume = {180}, number = {1}, year = {2014}, pages = {381--402},
  doi = {10.4007/annals.2014.180.1.8}
}

@article{DujellaKazalickiPeral2021,
  author = {Dujella, Andrej and Kazalicki, Matija and Peral, Juan Carlos},
  title = {Elliptic curves with torsion groups {$\mathbb{Z}/8\mathbb{Z}$}
           and {$\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$}},
  journal = {Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas},
  volume = {115}, year = {2021},
  doi = {10.1007/s13398-021-01112-5},
  note = {Article 169},
  eprint = {2105.06215v2}, archivePrefix = {arXiv}, primaryClass = {math.NT}
}