@article{Kraus98, author = {Kraus, Alain}, title = {Sur l'\'equation $a^3+b^3=c^p$}, journal = {Experiment. Math.}, volume = {7}, number = {1}, year = {1998}, pages = {1--13} } @article{ChenSiksek09, author = {Chen, Imin and Siksek, Samir}, title = {Perfect powers expressible as sums of two cubes}, journal = {J. Algebra}, volume = {322}, number = {3}, year = {2009}, pages = {638--656} } @article{Freitas16, author = {Freitas, Nuno}, title = {On the {F}ermat-type equation $x^3+y^3=z^p$}, journal = {Comment. Math. Helv.}, volume = {91}, number = {2}, year = {2016}, pages = {295--304} } @article{BCDF-II, author = {Billerey, Nicolas and Chen, Imin and Dieulefait, Luis and Freitas, Nuno}, title = {On {D}armon's program for the generalized {F}ermat equation, {II}}, journal = {Math. Comp.}, volume = {94}, number = {354}, year = {2025}, pages = {1977--2003}, note = {cited by the numbering of \href{https://arxiv.org/abs/2308.07062v2}{arXiv:2308.07062v2}} } @article{BCDF19, author = {Billerey, Nicolas and Chen, Imin and Dieulefait, Luis and Freitas, Nuno}, title = {A multi-{F}rey approach to {F}ermat equations of signature $(r,r,p)$}, journal = {Trans. Amer. Math. Soc.}, volume = {371}, number = {12}, year = {2019}, pages = {8651--8677} } @article{BCDDF23, author = {Billerey, Nicolas and Chen, Imin and Demb\'el\'e, Lassina and Dieulefait, Luis and Freitas, Nuno}, title = {Some extensions of the modular method and {F}ermat equations of signature $(13,13,n)$}, journal = {Publ. Mat.}, volume = {67}, number = {2}, year = {2023}, pages = {715--741} } @article{BennettLucaMulholland11, author = {Bennett, Michael A. and Luca, Florian and Mulholland, Jamie}, title = {Twisted extensions of the cubic case of {F}ermat's {L}ast {T}heorem}, journal = {Ann. Sci. Math. Qu\'ebec}, volume = {35}, number = {1}, year = {2011}, pages = {1--15} } @misc{FreitasMocanu25, author = {Freitas, Nuno and Mocanu, Diana}, title = {Local points on twists of {$X(p)$} with applications}, year = {2025}, note = {preprint, \href{https://arxiv.org/abs/2509.04294v4}{arXiv:2509.04294v4}} } @incollection{Tate94, author = {Tate, John}, title = {The non-existence of certain {G}alois extensions of {$\mathbf{Q}$} unramified outside $2$}, booktitle = {Arithmetic geometry ({T}empe, {AZ}, 1993)}, series = {Contemp. Math.}, volume = {174}, publisher = {Amer. Math. Soc., Providence, RI}, year = {1994}, pages = {153--156} } @article{MoonTaguchi03, author = {Moon, Hyunsuk and Taguchi, Yuichiro}, title = {Refinement of {T}ate's discriminant bound and non-existence theorems for mod $p$ {G}alois representations}, journal = {Doc. Math.}, volume = {Extra Vol.}, year = {2003}, pages = {641--654} } @article{BCDF-I, author = {Billerey, Nicolas and Chen, Imin and Dieulefait, Luis and Freitas, Nuno}, title = {On {D}armon's program for the generalized {F}ermat equation, {I}}, journal = {J. Reine Angew. Math.}, volume = {822}, year = {2025}, pages = {107--146}, note = {cited by the numbering of \href{https://arxiv.org/abs/2205.15861v5}{arXiv:2205.15861v5}} } @book{KatzESDE, author = {Katz, Nicholas M.}, title = {Exponential sums and differential equations}, series = {Annals of Mathematics Studies}, volume = {124}, publisher = {Princeton University Press}, year = {1990} } @article{deJong95, author = {de Jong, A. J.}, title = {{\'E}tale fundamental groups of non-{A}rchimedean analytic spaces}, journal = {Compos. Math.}, volume = {97}, number = {1--2}, year = {1995}, pages = {89--118} } @article{Berkovich93, author = {Berkovich, Vladimir G.}, title = {{\'E}tale cohomology for non-{A}rchimedean analytic spaces}, journal = {Publ. Math. Inst. Hautes \'Etudes Sci.}, volume = {78}, year = {1993}, pages = {5--161} } @article{FontaineLaffaille, author = {Fontaine, Jean-Marc and Laffaille, Guy}, title = {Construction de repr\'esentations $p$-adiques}, journal = {Ann. Sci. \'Ec. Norm. Sup\'er. (4)}, volume = {15}, number = {4}, year = {1982}, pages = {547--608} } @article{Tsuji99, author = {Tsuji, Takeshi}, title = {$p$-adic \'etale cohomology and crystalline cohomology in the semi-stable reduction case}, journal = {Invent. Math.}, volume = {137}, number = {2}, year = {1999}, pages = {233--411} } @article{Mokrane93, author = {Mokrane, Abdellah}, title = {La suite spectrale des poids en cohomologie de {H}yodo--{K}ato}, journal = {Duke Math. J.}, volume = {72}, number = {2}, year = {1993}, pages = {301--337} } @article{Saito97, author = {Saito, Takeshi}, title = {Modular forms and $p$-adic {H}odge theory}, journal = {Invent. Math.}, volume = {129}, number = {3}, year = {1997}, pages = {607--620} } @article{Shin11, author = {Shin, Sug Woo}, title = {Galois representations arising from some compact {S}himura varieties}, journal = {Ann. of Math. (2)}, volume = {173}, number = {3}, year = {2011}, pages = {1645--1741} } @article{ChenevierHarris13, author = {Chenevier, Ga\"etan and Harris, Michael}, title = {Construction of automorphic {G}alois representations, {II}}, journal = {Camb. J. Math.}, volume = {1}, number = {1}, year = {2013}, pages = {53--73} } @article{Caraiani12, author = {Caraiani, Ana}, title = {Local-global compatibility and the action of monodromy on nearby cycles}, journal = {Duke Math. J.}, volume = {161}, number = {12}, year = {2012}, pages = {2311--2413} } @book{Rogawski90, author = {Rogawski, Jonathan D.}, title = {Automorphic representations of unitary groups in three variables}, series = {Annals of Mathematics Studies}, volume = {123}, publisher = {Princeton University Press}, year = {1990} } @article{AdlerLansky10, author = {Adler, Jeffrey D. and Lansky, Joshua M.}, title = {Depth-zero base change for ramified {$U(2,1)$}}, journal = {Trans. Amer. Math. Soc.}, volume = {362}, number = {10}, year = {2010}, pages = {5569--5599} } @article{Casselman80, author = {Casselman, William}, title = {The unramified principal series of $p$-adic groups. {I}. {T}he spherical function}, journal = {Compositio Math.}, volume = {40}, number = {3}, year = {1980}, pages = {387--406} } @article{ClozelThorne17, author = {Clozel, Laurent and Thorne, Jack A.}, title = {Level-raising and symmetric power functoriality, {III}}, journal = {Duke Math. J.}, volume = {166}, number = {2}, year = {2017}, pages = {325--402} } @article{Mok15, author = {Mok, Chung Pang}, title = {Endoscopic classification of representations of quasi-split unitary groups}, journal = {Mem. Amer. Math. Soc.}, volume = {235}, number = {1108}, year = {2015} } @article{Darmon00, author = {Darmon, Henri}, title = {Rigid local systems, {H}ilbert modular forms, and {F}ermat's last theorem}, journal = {Duke Math. J.}, volume = {102}, number = {3}, year = {2000}, pages = {413--449} } @misc{GolfieriPacetti, author = {Golfieri Madriaga, Franco and Pacetti, Ariel}, title = {Hypergeometric motives and the generalized {F}ermat equation}, year = {2025}, note = {preprint, \href{https://arxiv.org/abs/2412.08804v2}{arXiv:2412.08804v2}} } @misc{GPRV26, author = {Golfieri Madriaga, Franco and Pacetti, Ariel and Rodriguez-Villegas, Fernando}, title = {On rank 2 hypergeometric motives}, year = {2026}, note = {preprint, \href{https://arxiv.org/abs/2603.17978v1}{arXiv:2603.17978v1}} } @article{LaishramShorey12, author = {Laishram, Shanta and Shorey, Tarlok N.}, title = {Baker's explicit $abc$-conjecture and applications}, journal = {Acta Arith.}, volume = {155}, number = {4}, year = {2012}, pages = {419--429} } @article{DarmonGranville95, author = {Darmon, Henri and Granville, Andrew}, title = {On the equations {$z^m=F(x,y)$} and {$Ax^p+By^q=Cz^r$}}, journal = {Bull. Lond. Math. Soc.}, volume = {27}, number = {6}, year = {1995}, pages = {513--543} } @article{FreitasKraus22, author = {Freitas, Nuno and Kraus, Alain}, title = {On the symplectic type of isomorphisms of the $p$-torsion of elliptic curves}, journal = {Mem. Amer. Math. Soc.}, volume = {277}, number = {1361}, year = {2022} } @misc{Zhou25, author = {Zhou, Zhong-Peng}, title = {The inter-universal {T}eichm\"uller theory and new {D}iophantine results over the rational numbers. {I}}, year = {2025}, note = {preprint, \href{https://arxiv.org/abs/2503.14510v1}{arXiv:2503.14510v1}} } @article{SengunSiksek18, author = {\c{S}eng\"un, Mehmet Haluk and Siksek, Samir}, title = {On the asymptotic {F}ermat's last theorem over number fields}, journal = {Comment. Math. Helv.}, volume = {93}, number = {2}, year = {2018}, pages = {359--375} } @article{BeukersHeckman89, author = {Beukers, Frits and Heckman, Gert}, title = {Monodromy for the hypergeometric function {${}_nF_{n-1}$}}, journal = {Invent. Math.}, volume = {95}, number = {2}, year = {1989}, pages = {325--354} } @article{Fedorov18, author = {Fedorov, Roman}, title = {Variations of {H}odge structures for hypergeometric differential operators and parabolic {H}iggs bundles}, journal = {Int. Math. Res. Not. IMRN}, year = {2018}, number = {18}, pages = {5583--5608} } @article{DettweilerSabbah13, author = {Dettweiler, Michael and Sabbah, Claude}, title = {Hodge theory of the middle convolution}, journal = {Publ. Res. Inst. Math. Sci.}, volume = {49}, number = {4}, year = {2013}, pages = {761--800} } @incollection{Deligne87, author = {Deligne, Pierre}, title = {Un th\'eor\`eme de finitude pour la monodromie}, booktitle = {Discrete groups in geometry and analysis ({N}ew {H}aven, {C}onn., 1984)}, series = {Progr. Math.}, volume = {67}, publisher = {Birkh\"auser Boston}, year = {1987}, pages = {1--19} } @incollection{Deligne89, author = {Deligne, Pierre}, title = {Le groupe fondamental de la droite projective moins trois points}, booktitle = {Galois groups over {$\mathbf Q$} ({B}erkeley, {CA}, 1987)}, series = {Math. Sci. Res. Inst. Publ.}, volume = {16}, publisher = {Springer}, year = {1989}, pages = {79--297} } @article{DeligneWeilII, author = {Deligne, Pierre}, title = {La conjecture de {W}eil. {II}}, journal = {Publ. Math. Inst. Hautes \'Etudes Sci.}, volume = {52}, year = {1980}, pages = {137--252} } @book{SGA1, author = {Grothendieck, Alexander}, title = {Rev\^etements \'etales et groupe fondamental ({SGA}~1)}, series = {Lecture Notes in Math.}, volume = {224}, publisher = {Springer}, year = {1971} } @book{SGA7II, author = {Deligne, Pierre and Katz, Nicholas}, title = {Groupes de monodromie en g\'eom\'etrie alg\'ebrique ({SGA}~7~{II})}, series = {Lecture Notes in Math.}, volume = {340}, publisher = {Springer}, year = {1973} } @article{BCDT01, author = {Breuil, Christophe and Conrad, Brian and Diamond, Fred and Taylor, Richard}, title = {On the modularity of elliptic curves over {$\mathbf Q$}: wild 3-adic exercises}, journal = {J. Amer. Math. Soc.}, volume = {14}, number = {4}, year = {2001}, pages = {843--939} } @article{Feit78, author = {Feit, Walter}, title = {Some lattices over {$Q(\sqrt{-3})$}}, journal = {J. Algebra}, volume = {52}, number = {1}, year = {1978}, pages = {248--263} } @article{Herzig09, author = {Herzig, Florian}, title = {The weight in a {S}erre-type conjecture for tame $n$-dimensional {G}alois representations}, journal = {Duke Math. J.}, volume = {149}, number = {1}, year = {2009}, pages = {37--116} } @article{EGH13, author = {Emerton, Matthew and Gee, Toby and Herzig, Florian}, title = {Weight cycling and {S}erre-type conjectures for unitary groups}, journal = {Duke Math. J.}, volume = {162}, number = {9}, year = {2013}, pages = {1649--1722} } @article{BLGG18, author = {Barnet-Lamb, Thomas and Gee, Toby and Geraghty, David}, title = {Serre weights for {$U(n)$}}, journal = {J. Reine Angew. Math.}, volume = {735}, year = {2018}, pages = {199--224} } @misc{PARI, author = {{The PARI Group}}, title = {{PARI/GP} version 2.17.3}, publisher = {Univ. Bordeaux}, year = {2025}, note = {\url{https://pari.math.u-bordeaux.fr/}} } @misc{Companion, author = {Jaswal, Manvir}, title = {Programs and data for ``{T}he generalized {F}ermat equation $x^3+y^3=z^n$\kern-0.08em''}, year = {2026}, note = {programs, outputs and commands for Appendix~A, Remark~4.18 and the Supplement; Zenodo, \url{https://doi.org/10.5281/zenodo.23197806}} } @article{Ochiai99, author = {Ochiai, Tadashi}, title = {$l$-independence of the trace of monodromy}, journal = {Math. Ann.}, volume = {315}, number = {2}, year = {1999}, pages = {321--340} } @article{KhareWintenberger09, author = {Khare, Chandrashekhar and Wintenberger, Jean-Pierre}, title = {Serre's modularity conjecture. {I}, {II}}, journal = {Invent. Math.}, volume = {178}, number = {3}, year = {2009}, pages = {485--504 and 505--586} } @article{GanTakeda11, author = {Gan, Wee Teck and Takeda, Shuichiro}, title = {The local {L}anglands conjecture for {${\rm GSp}(4)$}}, journal = {Ann. of Math. (2)}, volume = {173}, number = {3}, year = {2011}, pages = {1841--1882} } @article{GanTakedaRT11, author = {Gan, Wee Teck and Takeda, Shuichiro}, title = {Theta correspondences for {${\rm GSp}(4)$}}, journal = {Represent. Theory}, volume = {15}, year = {2011}, pages = {670--718} } @article{Mok14, author = {Mok, Chung Pang}, title = {Galois representations attached to automorphic forms on {${\rm GL}_2$} over {CM} fields}, journal = {Compos. Math.}, volume = {150}, number = {4}, year = {2014}, pages = {523--567}, note = {Cited with the numbering of \href{https://arxiv.org/abs/1109.5392v4}{arXiv:1109.5392v4}} } @article{GeeTaibi19, author = {Gee, Toby and Ta\"ibi, Olivier}, title = {Arthur's multiplicity formula for {${\rm GSp}_4$} and restriction to {${\rm Sp}_4$}}, journal = {J. \'Ec. polytech. Math.}, volume = {6}, year = {2019}, pages = {469--535} } @book{Arthur13, author = {Arthur, James}, title = {The endoscopic classification of representations: orthogonal and symplectic groups}, series = {American Mathematical Society Colloquium Publications}, volume = {61}, publisher = {American Mathematical Society}, year = {2013} } @book{MoeglinWaldspurger16, author = {M{\oe}glin, Colette and Waldspurger, Jean-Loup}, title = {Stabilisation de la formule des traces tordue}, series = {Progress in Mathematics}, volume = {316--317}, publisher = {Birkh\"auser}, year = {2016} } @misc{AGIKMS, author = {Atobe, Hiraku and Gan, Wee Teck and Ichino, Atsushi and Kaletha, Tasho and M\'inguez, Alberto and Shin, Sug Woo}, title = {Local intertwining relations and co-tempered {$A$}-packets of classical groups}, year = {2024}, note = {Preprint, \href{https://arxiv.org/abs/2410.13504v3}{arXiv:2410.13504v3}} } @article{Schmidt05, author = {Schmidt, Ralf}, title = {Iwahori-spherical representations of {${\rm GSp}(4)$} and {S}iegel modular forms of degree 2 with square-free level}, journal = {J. Math. Soc. Japan}, volume = {57}, number = {1}, year = {2005}, pages = {259--293} } @book{Freitag83, author = {Freitag, Eberhard}, title = {Siegelsche {M}odulfunktionen}, series = {Grundlehren der mathematischen Wissenschaften}, volume = {254}, publisher = {Springer}, year = {1983} } @incollection{HulekSankaran, author = {Hulek, Klaus and Sankaran, G. K.}, title = {The geometry of {S}iegel modular varieties}, booktitle = {Higher dimensional birational geometry ({K}yoto, 1997)}, series = {Adv. Stud. Pure Math.}, volume = {35}, publisher = {Math. Soc. Japan}, year = {2002}, pages = {89--156}, note = {arXiv:math/9810153} } @article{Todd36, author = {Todd, J. A.}, title = {On a quartic primal with forty-five nodes, in space of four dimensions}, journal = {Quart. J. Math. Oxford Ser.}, volume = {7}, year = {1936}, pages = {168--174} } @article{BruinNasserden18, author = {Bruin, Nils and Nasserden, Brett}, title = {Arithmetic aspects of the {B}urkhardt quartic threefold}, journal = {J. Lond. Math. Soc. (2)}, volume = {98}, number = {3}, year = {2018}, pages = {536--556} } @incollection{Wallach84, author = {Wallach, Nolan R.}, title = {On the constant term of a square integrable automorphic form}, booktitle = {Operator algebras and group representations, {V}ol. {II} ({N}eptun, 1980)}, series = {Monogr. Stud. Math.}, volume = {18}, publisher = {Pitman}, year = {1984}, pages = {227--237} } @article{HerzigTilouine13, author = {Herzig, Florian and Tilouine, Jacques}, title = {Conjecture de type de {S}erre et formes compagnons pour {${\rm GSp}_4$}}, journal = {J. Reine Angew. Math.}, volume = {676}, year = {2013}, pages = {1--32} } @article{BennettBruniFreitas18, author = {Bennett, Michael A. and Bruni, Carmen and Freitas, Nuno}, title = {Sums of two cubes as twisted perfect powers, revisited}, journal = {Algebra Number Theory}, volume = {12}, number = {4}, year = {2018}, pages = {959--999} } @article{CEG22, author = {Calegari, Frank and Emerton, Matthew and Gee, Toby}, title = {Globally realizable components of local deformation rings}, journal = {J. Inst. Math. Jussieu}, volume = {21}, number = {2}, year = {2022}, pages = {533--602}, note = {cited by the numbering of \href{https://arxiv.org/abs/1807.03529v3}{arXiv:1807.03529v3}} } @article{BLGGT14, author = {Barnet-Lamb, Thomas and Gee, Toby and Geraghty, David and Taylor, Richard}, title = {Potential automorphy and change of weight}, journal = {Ann. of Math. (2)}, volume = {179}, number = {2}, year = {2014}, pages = {501--609} } @article{BellovinGee19, author = {Bellovin, Rebecca and Gee, Toby}, title = {{$G$}-valued local deformation rings and global lifts}, journal = {Algebra Number Theory}, volume = {13}, number = {2}, year = {2019}, pages = {333--378}, note = {in \href{https://arxiv.org/abs/1708.04885v3}{arXiv:1708.04885v3} the results of \S3.3 are numbered one higher} } @article{Caraiani14, author = {Caraiani, Ana}, title = {Monodromy and local-global compatibility for {$l=p$}}, journal = {Algebra Number Theory}, volume = {8}, number = {7}, year = {2014}, pages = {1597--1646} } @article{CHT08, author = {Clozel, Laurent and Harris, Michael and Taylor, Richard}, title = {Automorphy for some {$l$}-adic lifts of automorphic mod {$l$} {G}alois representations}, journal = {Publ. Math. Inst. Hautes \'Etudes Sci.}, volume = {108}, year = {2008}, pages = {1--181} } @article{PatrikisTaylor15, author = {Patrikis, Stefan and Taylor, Richard}, title = {Automorphy and irreducibility of some {$l$}-adic representations}, journal = {Compos. Math.}, volume = {151}, number = {2}, year = {2015}, pages = {207--229} } @article{Kisin08, author = {Kisin, Mark}, title = {Potentially semi-stable deformation rings}, journal = {J. Amer. Math. Soc.}, volume = {21}, number = {2}, year = {2008}, pages = {513--546} } @article{Abrashkin15, author = {Abrashkin, Victor}, title = {Ramification estimate for {F}ontaine--{L}affaille {G}alois modules}, journal = {J. Algebra}, volume = {427}, year = {2015}, pages = {319--328} } @article{Abrashkin89, author = {Abrashkin, Victor}, title = {Modular representations of the {G}alois group of a local field and a generalization of a conjecture of {S}hafarevich}, journal = {Izv. Akad. Nauk SSSR Ser. Mat.}, volume = {53}, number = {6}, year = {1989}, pages = {1135--1182}, note = {English translation: Math. USSR-Izv. \textbf{35} (1990), no.~3, 469--518} } @article{Fontaine85, author = {Fontaine, Jean-Marc}, title = {Il n'y a pas de vari\'et\'e ab\'elienne sur {$\mathbf{Z}$}}, journal = {Invent. Math.}, volume = {81}, number = {3}, year = {1985}, pages = {515--538} } @incollection{Fontaine93, author = {Fontaine, Jean-Marc}, title = {Sch\'emas propres et lisses sur {$\mathbf{Z}$}}, booktitle = {Proceedings of the Indo-French Conference on Geometry (Bombay, 1989)}, publisher = {Hindustan Book Agency}, address = {Delhi}, year = {1993}, pages = {43--56} } @incollection{Hattori19, author = {Hattori, Shin}, title = {Integral {$p$}-adic {H}odge theory and ramification of crystalline representations}, booktitle = {An excursion into $p$-adic {H}odge theory: from foundations to recent trends}, series = {Panor. Synth\`eses}, volume = {54}, publisher = {Soc. Math. France}, address = {Paris}, year = {2019}, pages = {159--203} } @article{Poitou77, author = {Poitou, Georges}, title = {Sur les petits discriminants}, journal = {S\'eminaire Delange--Pisot--Poitou. Th\'eorie des nombres}, volume = {18}, number = {1}, year = {1976--1977}, pages = {Expos\'e 6, 18 pp.} } @article{Odlyzko90, author = {Odlyzko, A. M.}, title = {Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results}, journal = {S\'em. Th\'eor. Nombres Bordeaux (2)}, volume = {2}, number = {1}, year = {1990}, pages = {119--141} } @book{Cohen00, author = {Cohen, Henri}, title = {Advanced topics in computational number theory}, series = {Graduate Texts in Mathematics}, volume = {193}, publisher = {Springer-Verlag}, address = {New York}, year = {2000} } @book{SerreCL, author = {Serre, Jean-Pierre}, title = {Corps locaux}, edition = {Second}, publisher = {Hermann}, address = {Paris}, year = {1968} } @book{SerreAbelian, author = {Serre, Jean-Pierre}, title = {Abelian {$l$}-adic representations and elliptic curves}, publisher = {W. A. Benjamin}, address = {New York--Amsterdam}, year = {1968} } @incollection{Serre77, author = {Serre, Jean-Pierre}, title = {Modular forms of weight one and {G}alois representations}, booktitle = {Algebraic number fields: {$L$}-functions and {G}alois properties (Proc. Sympos., Univ. Durham, 1975)}, publisher = {Academic Press}, address = {London}, year = {1977}, pages = {193--268} } @article{Serre87, author = {Serre, Jean-Pierre}, title = {Sur les repr\'esentations modulaires de degr\'e {$2$} de {$\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$}}, journal = {Duke Math. J.}, volume = {54}, number = {1}, year = {1987}, pages = {179--230} } @article{KhareWintenbergerAnnals09, author = {Khare, Chandrashekhar and Wintenberger, Jean-Pierre}, title = {On {S}erre's conjecture for 2-dimensional mod {$p$} representations of {$\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$}}, journal = {Ann. of Math. (2)}, volume = {169}, number = {1}, year = {2009}, pages = {229--253} } @article{Calegari04, author = {Calegari, Frank}, title = {Semistable abelian varieties over {$\mathbf{Q}$}}, journal = {Manuscripta Math.}, volume = {113}, number = {4}, year = {2004}, pages = {507--529} } @article{Dieulefait04, author = {Dieulefait, Luis V.}, title = {Existence of families of {G}alois representations and new cases of the {F}ontaine--{M}azur conjecture}, journal = {J. Reine Angew. Math.}, volume = {577}, year = {2004}, pages = {147--151} } @misc{Wintenberger04, author = {Wintenberger, Jean-Pierre}, title = {Sur les repr\'esentations $p$-adiques g\'eom\'etriques de conducteur $1$ et de dimension $2$ de {$G_{\mathbb{Q}}$}}, year = {2004}, note = {preprint, \href{https://arxiv.org/abs/math/0406576v2}{arXiv:math/0406576v2}} } @article{BrumerKramer01, author = {Brumer, Armand and Kramer, Kenneth}, title = {Non-existence of certain semistable abelian varieties}, journal = {Manuscripta Math.}, volume = {106}, number = {3}, year = {2001}, pages = {291--304} } @article{Schoof05, author = {Schoof, Ren\'e}, title = {Abelian varieties over {$\mathbf{Q}$} with bad reduction in one prime only}, journal = {Compos. Math.}, volume = {141}, number = {4}, year = {2005}, pages = {847--868} } @misc{GhitzaYamauchi25, author = {Ghitza, Alexandru and Yamauchi, Takuya}, title = {The non-existence of some {G}alois representations of moderate dimension in small characteristic}, year = {2025}, note = {preprint, \href{https://arxiv.org/abs/2509.00635v2}{arXiv:2509.00635v2}} } @article{Moon00, author = {Moon, Hyunsuk}, title = {Finiteness results on certain mod {$p$} {G}alois representations}, journal = {J. Number Theory}, volume = {84}, number = {1}, year = {2000}, pages = {156--165} } @article{Sengun09, author = {{\c{S}}eng{\"u}n, Mehmet Haluk}, title = {The nonexistence of certain representations of the absolute {G}alois group of quadratic fields}, journal = {Proc. Amer. Math. Soc.}, volume = {137}, number = {1}, year = {2009}, pages = {27--35} } @article{FreitasNaskreckiStoll20, author = {Freitas, Nuno and Naskr\k{e}cki, Bartosz and Stoll, Michael}, title = {The generalized {F}ermat equation with exponents 2, 3, {$n$}}, journal = {Compos. Math.}, volume = {156}, number = {1}, year = {2020}, pages = {77--113} } @article{DahmenSiksek14, author = {Dahmen, Sander R. and Siksek, Samir}, title = {Perfect powers expressible as sums of two fifth or seventh powers}, journal = {Acta Arith.}, volume = {164}, number = {1}, year = {2014}, pages = {65--100} } @misc{BestDahmenFreitas25, author = {Best, Alex J. and Dahmen, Sander R. and Freitas, Nuno}, title = {On the generalized {F}ermat equation $x^{13}+y^{13}=z^n$}, year = {2025}, note = {Preprint, \href{https://arxiv.org/abs/2510.12092}{arXiv:2510.12092}} } @article{Edixhoven92, author = {Edixhoven, Bas}, title = {N\'eron models and tame ramification}, journal = {Compos. Math.}, volume = {81}, number = {3}, year = {1992}, pages = {291--306} } @book{DiamondShurman, author = {Diamond, Fred and Shurman, Jerry}, title = {A first course in modular forms}, series = {Graduate Texts in Mathematics}, volume = {228}, publisher = {Springer-Verlag}, address = {New York}, year = {2005} } @book{ATLAS, author = {Conway, J. H. and Curtis, R. T. and Norton, S. P. and Parker, R. A. and Wilson, R. A.}, title = {Atlas of finite groups}, publisher = {Oxford University Press}, address = {Eynsham}, year = {1985} } @misc{GAP, author = {{The GAP Group}}, title = {{GAP} -- {G}roups, {A}lgorithms, and {P}rogramming, version 4.15.1}, year = {2025}, note = {\url{https://www.gap-system.org}} } @misc{LMFDB, author = {{The LMFDB Collaboration}}, title = {The {$L$}-functions and modular forms database}, year = {2026}, note = {\url{https://www.lmfdb.org}; number fields of degree $5$ unramified outside $\{3,5\}$} } @article{Johansson17, author = {Johansson, Fredrik}, title = {Arb: efficient arbitrary-precision midpoint-radius interval arithmetic}, journal = {IEEE Trans. Comput.}, volume = {66}, number = {8}, year = {2017}, pages = {1281--1292} } @misc{OdlyzkoTables, author = {Odlyzko, A. M.}, title = {Discriminant bounds}, year = {1976}, note = {unpublished tables dated 29 November 1976; available at \url{https://www-users.cse.umn.edu/~odlyzko/unpublished/index.html}} } @article{KrausOesterle92, author = {Kraus, Alain and Oesterl\'e, Joseph}, title = {Sur une question de {B}. {M}azur}, journal = {Math. Ann.}, volume = {293}, year = {1992}, pages = {259--275}, doi = {10.1007/BF01444715} } @article{Mazur78, author = {Mazur, Barry}, title = {Rational isogenies of prime degree (with an appendix by {D}. {G}oldfeld)}, journal = {Invent. Math.}, volume = {44}, number = {2}, year = {1978}, pages = {129--162}, doi = {10.1007/BF01390348} } @article{Kenku82, author = {Kenku, M. A.}, title = {On the number of {$\mathbf Q$}-isomorphism classes of elliptic curves in each {$\mathbf Q$}-isogeny class}, journal = {J. Number Theory}, volume = {15}, number = {2}, year = {1982}, pages = {199--202}, doi = {10.1016/0022-314X(82)90025-7} } @article{Ribet90, author = {Ribet, Kenneth A.}, title = {On modular representations of {${\rm Gal}(\overline{\mathbf Q}/{\mathbf Q})$} arising from modular forms}, journal = {Invent. Math.}, volume = {100}, year = {1990}, pages = {431--476}, doi = {10.1007/BF01231195} } @incollection{Ribet94, author = {Ribet, Kenneth A.}, title = {Report on mod {$l$} representations of {${\rm Gal}(\overline{\mathbf Q}/{\mathbf Q})$}}, booktitle = {Motives ({S}eattle, {WA}, 1991)}, series = {Proc. Sympos. Pure Math.}, volume = {55, Part 2}, publisher = {Amer. Math. Soc., Providence, RI}, year = {1994}, pages = {639--676}, doi = {10.1090/pspum/055.2/1265566} } @article{Serre72, author = {Serre, Jean-Pierre}, title = {Propri\'et\'es galoisiennes des points d'ordre fini des courbes elliptiques}, journal = {Invent. Math.}, volume = {15}, number = {4}, year = {1972}, pages = {259--331}, doi = {10.1007/BF01405086} } @book{SilvermanAT, author = {Silverman, Joseph H.}, title = {Advanced topics in the arithmetic of elliptic curves}, series = {Graduate Texts in Mathematics}, volume = {151}, publisher = {Springer-Verlag, New York}, year = {1994}, doi = {10.1007/978-1-4612-0851-8} } @article{BLGG13, author = {Barnet-Lamb, Thomas and Gee, Toby and Geraghty, David}, title = {Serre weights for rank two unitary groups}, journal = {Math. Ann.}, volume = {356}, number = {4}, year = {2013}, pages = {1551--1598}, doi = {10.1007/s00208-012-0893-y} } @incollection{Bruin00, author = {Bruin, Nils}, title = {On powers as sums of two cubes}, booktitle = {Algorithmic number theory ({L}eiden, 2000)}, series = {Lecture Notes in Comput. Sci.}, volume = {1838}, publisher = {Springer, Berlin}, year = {2000}, pages = {169--184}, doi = {10.1007/10722028_9} } @book{SilvermanAEC, author = {Silverman, Joseph H.}, title = {The arithmetic of elliptic curves}, edition = {Second}, series = {Graduate Texts in Mathematics}, volume = {106}, publisher = {Springer, Dordrecht}, year = {2009}, doi = {10.1007/978-0-387-09494-6} } @article{Schaefer95, author = {Schaefer, Edward F.}, title = {2-descent on the {J}acobians of hyperelliptic curves}, journal = {J. Number Theory}, volume = {51}, number = {2}, year = {1995}, pages = {219--232}, doi = {10.1006/jnth.1995.1044} } @article{Stoll01, author = {Stoll, Michael}, title = {Implementing 2-descent for {J}acobians of hyperelliptic curves}, journal = {Acta Arith.}, volume = {98}, number = {3}, year = {2001}, pages = {245--277}, doi = {10.4064/aa98-3-4} } @book{CasselsFlynn96, author = {Cassels, J. W. S. and Flynn, E. V.}, title = {Prolegomena to a middlebrow arithmetic of curves of genus 2}, series = {London Mathematical Society Lecture Note Series}, volume = {230}, publisher = {Cambridge University Press, Cambridge}, year = {1996}, doi = {10.1017/CBO9780511526084} } @article{Katz81, author = {Katz, Nicholas M.}, title = {Galois properties of torsion points on abelian varieties}, journal = {Invent. Math.}, volume = {62}, number = {3}, year = {1981}, pages = {481--502}, doi = {10.1007/BF01394256} } @article{Coleman85, author = {Coleman, Robert F.}, title = {Effective {C}habauty}, journal = {Duke Math. J.}, volume = {52}, number = {3}, year = {1985}, pages = {765--770}, doi = {10.1215/S0012-7094-85-05240-8} } @incollection{McCallumPoonen12, author = {McCallum, William and Poonen, Bjorn}, title = {The method of {C}habauty and {C}oleman}, booktitle = {Explicit methods in number theory}, series = {Panor. Synth\`eses}, volume = {36}, publisher = {Soc. Math. France, Paris}, year = {2012}, pages = {99--117} } @misc{KhareWintenberger04, author = {Khare, Chandrashekhar and Wintenberger, Jean-Pierre}, title = {On {S}erre's reciprocity conjecture for 2-dimensional mod {$p$} representations of {$\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$}}, year = {2004}, note = {Preprint, \href{https://arxiv.org/abs/math/0412076v1}{arXiv:math/0412076v1}} } @article{BaiZhang14, author = {Bai, Meng and Zhang, Zhongfeng}, title = {On the {D}iophantine equation $x^3+y^3=2z^{2n}$}, journal = {J. Southwest China Normal Univ. Nat. Sci. Ed.}, volume = {39}, number = {6}, year = {2014}, pages = {14--17}, doi = {10.13718/j.cnki.xsxb.2014.06.004}, note = {in Chinese} } @misc{Wilcox24, author = {Wilcox, Ashleigh}, title = {A systematic approach to {D}iophantine equations: two thousand solved examples}, year = {2024}, note = {preprint, \href{https://arxiv.org/abs/2404.08719v2}{arXiv:2404.08719v2}} } @article{Greenberg16, author = {Greenberg, Ralph}, title = {Galois representations with open image}, journal = {Ann. Math. Qu\'ebec}, volume = {40}, number = {1}, year = {2016}, pages = {83--119} } @article{Shafarevich63, author = {Shafarevich, Igor R.}, title = {Extensions with prescribed ramification points}, journal = {Publ. Math. Inst. Hautes \'Etudes Sci.}, volume = {18}, year = {1963}, pages = {71--92}, note = {in Russian} } @book{Koch02, author = {Koch, Helmut}, title = {Galois theory of {$p$}-extensions}, series = {Springer Monographs in Mathematics}, publisher = {Springer-Verlag}, address = {Berlin}, year = {2002} } @article{BLGGTlgc1, author = {Barnet-Lamb, Thomas and Gee, Toby and Geraghty, David and Taylor, Richard}, title = {Local-global compatibility for {$l=p$}, {I}}, journal = {Ann. Fac. Sci. Toulouse Math. (6)}, volume = {21}, number = {1}, year = {2012}, pages = {57--92} } @article{BLGGTlgc2, author = {Barnet-Lamb, Thomas and Gee, Toby and Geraghty, David and Taylor, Richard}, title = {Local-global compatibility for {$l=p$}, {II}}, journal = {Ann. Sci. \'Ec. Norm. Sup\'er. (4)}, volume = {47}, number = {1}, year = {2014}, pages = {165--179} } @article{Berger02, author = {Berger, Laurent}, title = {Repr\'esentations {$p$}-adiques et \'equations diff\'erentielles}, journal = {Invent. Math.}, volume = {148}, number = {2}, year = {2002}, pages = {219--284} } @incollection{Fontaine94, author = {Fontaine, Jean-Marc}, title = {Repr\'esentations {$p$}-adiques semi-stables}, booktitle = {P\'eriodes {$p$}-adiques ({B}ures-sur-{Y}vette, 1988)}, series = {Ast\'erisque}, volume = {223}, publisher = {Soc. Math. France}, year = {1994}, pages = {113--184} } @incollection{Fontaine94pst, author = {Fontaine, Jean-Marc}, title = {Repr\'esentations {$\ell$}-adiques potentiellement semi-stables}, booktitle = {P\'eriodes {$p$}-adiques ({B}ures-sur-{Y}vette, 1988)}, series = {Ast\'erisque}, volume = {223}, publisher = {Soc. Math. France}, year = {1994}, pages = {321--347} } @article{ColmezFontaine00, author = {Colmez, Pierre and Fontaine, Jean-Marc}, title = {Construction des repr\'esentations {$p$}-adiques semi-stables}, journal = {Invent. Math.}, volume = {140}, number = {1}, year = {2000}, pages = {1--43} } @book{Dickson01, author = {Dickson, Leonard Eugene}, title = {Linear groups, with an exposition of the {G}alois field theory}, publisher = {B. G. Teubner}, address = {Leipzig}, year = {1901} } @article{Faber23, author = {Faber, Xander}, title = {Finite {$p$}-irregular subgroups of {$\mathrm{PGL}_2(k)$}}, journal = {La Matematica}, volume = {2}, year = {2023}, pages = {479--522} } @misc{DieulefaitGolfieri25, author = {Dieulefait, Luis and Golfieri Madriaga, Franco}, title = {On {F}ermat's {L}ast {T}heorem over the {$\mathbb{Z}_3$}-extension of {$\mathbb{Q}$} and other fields}, year = {2025}, note = {preprint, \href{https://arxiv.org/abs/2507.16883v1}{arXiv:2507.16883v1}} } @article{FreitasSanchez25, author = {Freitas, Nuno and S\'anchez-Rodr\'iguez, Ignasi}, title = {Comparing {G}alois representations in the residually reducible case}, journal = {Math. Comp.}, year = {2026}, doi = {10.1090/mcom/4274}, note = {to appear; cited by the numbering of \href{https://arxiv.org/abs/2510.12956v1}{arXiv:2510.12956v1}} } @article{FreitasKrausSiksek20, author = {Freitas, Nuno and Kraus, Alain and Siksek, Samir}, title = {Class field theory, {D}iophantine analysis and the asymptotic {F}ermat's {L}ast {T}heorem}, journal = {Adv. Math.}, volume = {363}, year = {2020}, pages = {106964} } @article{FreitasKrausSiksek21, author = {Freitas, Nuno and Kraus, Alain and Siksek, Samir}, title = {On asymptotic {F}ermat over the {$\mathbb{Z}_2$}-extension of {$\mathbb{Q}$}}, journal = {Ann. Math. Blaise Pascal}, volume = {28}, number = {1}, year = {2021}, pages = {1--6} } @article{Abrashkin17, author = {Abrashkin, Victor}, title = {Groups of automorphisms of local fields of period $p$ and nilpotent class $