@article{SilverbergZarhin1995,
  author = {Silverberg, Alice and Zarhin, Yuri G.},
  title = {Semistable reduction and torsion subgroups of abelian varieties},
  journal = {Annales de l'Institut Fourier},
  volume = {45},
  number = {2},
  year = {1995},
  pages = {403--420},
  doi = {10.5802/aif.1459},
  url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1459/},
  note = {\bibdoi{10.5802/aif.1459}}
}


@misc{PointwiseTwoConverse,
  author = {{OpenAI}},
  title = {{A pointwise $2$-converse for elliptic curves with rational two-torsion}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026/paper.pdf}{OAI:A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026}},
  year = {2026},
  note = {Theorem~1.1}
}
@misc{UnrestrictedTwoConverse,
  author = {{OpenAI}},
  title = {{Goldfeld's analytic density conjecture and the $2$-converse for elliptic curves}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf}{OAI:Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026}},
  year = {2026},
  note = {Theorem~1.1}
}
@misc{OddRegularFontaineMazurAtTwo,
  author = {{OpenAI}},
  title = {{Fontaine--Mazur modularity at the prime 2}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026/paper.pdf}{OAI:Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026}},
  year = {2026},
  note = {Theorem~1.1 and Section~2.1}
}

@book{Silverman2009,
  author = {Silverman, Joseph H.},
  title = {The Arithmetic of Elliptic Curves},
  edition = {Second},
  series = {Graduate Texts in Mathematics},
  volume = {106},
  publisher = {Springer},
  address = {New York},
  year = {2009},
  doi = {10.1007/978-0-387-09494-6},
  url = {https://doi.org/10.1007/978-0-387-09494-6},
  note = {\bibdoi{10.1007/978-0-387-09494-6}}
}
@article{Cassels1962,
  author = {Cassels, J. W. S.},
  title = {Arithmetic on curves of genus 1. {IV}. {Proof} of the {Hauptvermutung}},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume = {211},
  year = {1962},
  pages = {95--112},
  doi = {10.1515/crll.1962.211.95},
  url = {https://doi.org/10.1515/crll.1962.211.95},
  note = {\bibdoi{10.1515/crll.1962.211.95}}
}
@article{PoonenStoll1999,
  author = {Poonen, Bjorn and Stoll, Michael},
  title = {The {Cassels--Tate} pairing on polarized abelian varieties},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {150},
  number = {3},
  year = {1999},
  pages = {1109--1149},
  doi = {10.2307/121064},
  url = {https://math.mit.edu/~poonen/papers/sha.pdf},
  note = {\bibdoi{10.2307/121064}}
}
@book{BoschLutkebohmertRaynaud1990,
  author = {Bosch, Siegfried and L{\"u}tkebohmert, Werner and Raynaud, Michel},
  title = {N{\'e}ron Models},
  series = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
  volume = {21},
  publisher = {Springer},
  address = {Berlin},
  year = {1990},
  doi = {10.1007/978-3-642-51438-8},
  url = {https://doi.org/10.1007/978-3-642-51438-8},
  note = {\bibdoi{10.1007/978-3-642-51438-8}}
}
@book{Davenport2000,
  author = {Davenport, Harold},
  title = {Multiplicative Number Theory},
  edition = {Third},
  series = {Graduate Texts in Mathematics},
  volume = {74},
  publisher = {Springer-Verlag},
  address = {New York},
  year = {2000},
  note = {Revised and with a preface by Hugh L. Montgomery. \href{https://link.springer.com/book/9780387950976}{Source}},
  isbn = {978-0-387-95097-6},
  url = {https://link.springer.com/book/9780387950976}
}
@book{Neukirch1999,
  author = {Neukirch, J{\"u}rgen},
  title = {Algebraic Number Theory},
  series = {Grundlehren der mathematischen Wissenschaften},
  volume = {322},
  publisher = {Springer},
  address = {Berlin},
  year = {1999},
  note = {Translated from the German by Norbert Schappacher. \bibdoi{10.1007/978-3-662-03983-0}},
  doi = {10.1007/978-3-662-03983-0},
  url = {https://doi.org/10.1007/978-3-662-03983-0}
}
@article{GreenTao2010,
  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},
  url = {https://annals.math.princeton.edu/2010/171-3/p08},
  note = {\bibdoi{10.4007/annals.2010.171.1753}}
}
@article{GreenTao2012,
  author = {Green, Ben and Tao, Terence},
  title = {The {M{\"o}bius} 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},
  url = {https://annals.math.princeton.edu/2012/175-2/p03},
  note = {\bibdoi{10.4007/annals.2012.175.2.3}}
}
@article{GreenTaoZiegler2012,
  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},
  url = {https://annals.math.princeton.edu/2012/176-2/p11},
  note = {\bibdoi{10.4007/annals.2012.176.2.11}}
}
@misc{GreenTaoZiegler2024Erratum,
  author = {Green, Ben and Tao, Terence and Ziegler, Tamar},
  title = {Erratum for {An inverse theorem for the Gowers {$U^{s+1}[N]$}-norm}},
  year = {2024},
  month = apr,
  note = {Six-page author erratum. \href{https://terrytao.wordpress.com/wp-content/uploads/2024/04/erratum-4.pdf}{Source}},
  url = {https://terrytao.wordpress.com/wp-content/uploads/2024/04/erratum-4.pdf}
}
@book{Voight2021,
  author = {Voight, John},
  title = {Quaternion Algebras},
  series = {Graduate Texts in Mathematics},
  volume = {288},
  publisher = {Springer},
  address = {Cham},
  year = {2021},
  doi = {10.1007/978-3-030-56694-4},
  url = {https://doi.org/10.1007/978-3-030-56694-4},
  note = {\bibdoi{10.1007/978-3-030-56694-4}}
}
@article{Pazuki2012,
  author = {Pazuki, Fabien},
  title = {Theta height and {Faltings} height},
  journal = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France},
  volume = {140},
  number = {1},
  year = {2012},
  pages = {19--49},
  doi = {10.24033/bsmf.2623},
  url = {https://www.numdam.org/item/BSMF_2012__140_1_19_0/},
  note = {\bibdoi{10.24033/bsmf.2623}}
}
@article{Javanpeykar2014,
  author = {Javanpeykar, Ariyan},
  title = {Polynomial bounds for {Arakelov} invariants of {Belyi} curves},
  journal = {Algebra \& Number Theory},
  volume = {8},
  number = {1},
  year = {2014},
  pages = {89--140},
  note = {With an appendix by Peter Bruin. \bibdoi{10.2140/ant.2014.8.89}},
  doi = {10.2140/ant.2014.8.89},
  url = {https://msp.org/ant/2014/8-1/p04.xhtml}
}
@article{GaudronRemond2014,
  author = {Gaudron, {\'E}ric and R{\'e}mond, Ga{\"e}l},
  title = {Polarisations et isog{\'e}nies},
  journal = {Duke Mathematical Journal},
  volume = {163},
  number = {11},
  year = {2014},
  pages = {2057--2108},
  doi = {10.1215/00127094-2782528},
  url = {https://lmbp.uca.fr/~gaudron/art13.pdf},
  note = {\bibdoi{10.1215/00127094-2782528}}
}
@article{Faltings1983,
  author = {Faltings, Gerd},
  title = {Endlichkeitss{\"a}tze f{\"u}r abelsche {Variet{\"a}ten} {\"u}ber {Zahlk{\"o}rpern}},
  journal = {Inventiones Mathematicae},
  volume = {73},
  number = {3},
  year = {1983},
  pages = {349--366},
  doi = {10.1007/BF01388432},
  url = {https://doi.org/10.1007/BF01388432},
  note = {\bibdoi{10.1007/BF01388432}. Erratum: ibid. 75 (1984), 381, \bibdoi{10.1007/BF01388572}}
}
@article{SerreTate1968,
  author = {Serre, Jean-Pierre and Tate, John},
  title = {Good reduction of abelian varieties},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {88},
  number = {3},
  year = {1968},
  pages = {492--517},
  doi = {10.2307/1970722},
  url = {https://annals.math.princeton.edu/1968/88-3/p05},
  note = {\bibdoi{10.2307/1970722}}
}
@book{Serre1979,
  author = {Serre, Jean-Pierre},
  title = {Local Fields},
  series = {Graduate Texts in Mathematics},
  volume = {67},
  publisher = {Springer-Verlag},
  address = {New York},
  year = {1979},
  note = {Translated from the French by Marvin Jay Greenberg. \bibdoi{10.1007/978-1-4757-5673-9}},
  doi = {10.1007/978-1-4757-5673-9},
  url = {https://doi.org/10.1007/978-1-4757-5673-9}
}
@book{Serre1997,
  author = {Serre, Jean-Pierre},
  title = {Galois Cohomology},
  series = {Springer Monographs in Mathematics},
  publisher = {Springer-Verlag},
  address = {Berlin},
  year = {1997},
  note = {Translated from the French by Patrick Ion. \bibdoi{10.1007/978-3-642-59141-9}},
  doi = {10.1007/978-3-642-59141-9},
  url = {https://doi.org/10.1007/978-3-642-59141-9}
}
@book{MilneADT2006,
  author = {Milne, James S.},
  title = {Arithmetic Duality Theorems},
  edition = {Second},
  publisher = {BookSurge},
  address = {Charleston, SC},
  year = {2006},
  url = {https://www.jmilne.org/math/Books/adt.html},
  note = {\href{https://www.jmilne.org/math/Books/adt.html}{Source}}
}
@misc{MilneAV2008,
  author = {Milne, James S.},
  title = {{Abelian Varieties}},
  year = {2008},
  note = {Course notes, version 2.0, March 16. \href{https://www.jmilne.org/math/CourseNotes/AV.pdf}{Source}},
  url = {https://www.jmilne.org/math/CourseNotes/AV.pdf}
}
@misc{MilneCFT2020,
  author = {Milne, James S.},
  title = {{Class Field Theory}},
  year = {2020},
  note = {Course notes, version 4.03, August 6. \href{https://www.jmilne.org/math/CourseNotes/CFT.pdf}{Source}},
  url = {https://www.jmilne.org/math/CourseNotes/CFT.pdf}
}
@misc{MilneEtale2013,
  author = {Milne, James S.},
  title = {{Lectures on {\'E}tale Cohomology}},
  year = {2013},
  note = {Course notes, version 2.21, March 22. \href{https://www.jmilne.org/math/CourseNotes/LEC.pdf}{Source}},
  url = {https://www.jmilne.org/math/CourseNotes/LEC.pdf}
}
@book{Mumford1970,
  author = {Mumford, David},
  title = {Abelian Varieties},
  series = {Tata Institute of Fundamental Research Studies in Mathematics},
  volume = {5},
  publisher = {Oxford University Press},
  address = {London},
  year = {1970}
}
@book{Marker2002,
  author = {Marker, David},
  title = {Model Theory: An Introduction},
  series = {Graduate Texts in Mathematics},
  volume = {217},
  publisher = {Springer},
  address = {New York},
  year = {2002},
  doi = {10.1007/b98860},
  url = {https://doi.org/10.1007/b98860},
  note = {\bibdoi{10.1007/b98860}}
}


@article{Hilbert1900,
  author = {Hilbert, David},
  title = {{Mathematische Probleme. Vortrag, gehalten auf dem internationalen Mathematiker-Kongre{\ss} zu Paris 1900}},
  journal = {Nachrichten von der K{\"o}niglichen Gesellschaft der Wissenschaften zu G{\"o}ttingen, Mathematisch-Physikalische Klasse},
  volume = {1900},
  number = {3},
  year = {1900},
  pages = {253--297},
  url = {https://gdz.sub.uni-goettingen.de/id/PPN252457811_1900},
  language = {German},
  note = {\href{https://gdz.sub.uni-goettingen.de/id/PPN252457811_1900}{Source}}
}

@article{DavisPutnamRobinson1961,
  author = {Davis, Martin and Putnam, Hilary and Robinson, Julia},
  title = {The decision problem for exponential {Diophantine} equations},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {74},
  number = {3},
  year = {1961},
  pages = {425--436},
  doi = {10.2307/1970289},
  url = {https://doi.org/10.2307/1970289},
  note = {\bibdoi{10.2307/1970289}}
}

@article{Davis1973,
  author = {Davis, Martin},
  title = {{Hilbert}'s tenth problem is unsolvable},
  journal = {The American Mathematical Monthly},
  volume = {80},
  number = {3},
  year = {1973},
  pages = {233--269},
  url = {https://math.umd.edu/~mcl/Pubs/713/Diophantine.pdf},
  note = {\href{https://math.umd.edu/~mcl/Pubs/713/Diophantine.pdf}{Source}}
}

@article{Matiyasevich1970,
  author = {Matiyasevich, Yu. V.},
  title = {Diofantovost' perechislimykh mnozhestv},
  journal = {Doklady Akademii Nauk SSSR},
  volume = {191},
  number = {2},
  year = {1970},
  pages = {279--282},
  url = {https://www.mathnet.ru/eng/dan35274},
  language = {Russian},
  note = {English title: The Diophantineness of enumerable sets. \href{https://www.mathnet.ru/eng/dan35274}{Source}}
}

@article{Robinson1949,
  author = {Robinson, Julia},
  title = {Definability and decision problems in arithmetic},
  journal = {The Journal of Symbolic Logic},
  volume = {14},
  number = {2},
  year = {1949},
  pages = {98--114},
  doi = {10.2307/2266510},
  url = {https://doi.org/10.2307/2266510},
  note = {\bibdoi{10.2307/2266510}}
}

@article{Poonen2009,
  author = {Poonen, Bjorn},
  title = {Characterizing integers among rational numbers with a universal-existential formula},
  journal = {American Journal of Mathematics},
  volume = {131},
  number = {3},
  year = {2009},
  pages = {675--682},
  doi = {10.1353/ajm.0.0057},
  url = {https://doi.org/10.1353/ajm.0.0057},
  eprint = {math/0703907},
  archivePrefix = {arXiv},
  primaryClass = {math.NT},
  note = {\bibdoi{10.1353/ajm.0.0057}}
}

@article{Koenigsmann2016,
  author = {Koenigsmann, Jochen},
  title = {Defining {$\mathbb{Z}$} in {$\mathbb{Q}$}},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {183},
  number = {1},
  year = {2016},
  pages = {73--93},
  doi = {10.4007/annals.2016.183.1.2},
  url = {https://doi.org/10.4007/annals.2016.183.1.2},
  eprint = {1011.3424},
  archivePrefix = {arXiv},
  primaryClass = {math.NT},
  note = {\bibdoi{10.4007/annals.2016.183.1.2}}
}

@article{PoonenSubrings,
  author = {Poonen, Bjorn},
  title = {{Hilbert}'s tenth problem and {Mazur}'s conjecture for large subrings of {$\mathbb{Q}$}},
  journal = {Journal of the American Mathematical Society},
  volume = {16},
  number = {4},
  year = {2003},
  pages = {981--990},
  doi = {10.1090/S0894-0347-03-00433-8},
  url = {https://doi.org/10.1090/S0894-0347-03-00433-8},
  eprint = {math/0306277},
  archivePrefix = {arXiv},
  primaryClass = {math.NT},
  note = {\bibdoi{10.1090/S0894-0347-03-00433-8}}
}

@article{PoonenSurvey,
  author = {Poonen, Bjorn},
  title = {Undecidability in number theory},
  journal = {Notices of the American Mathematical Society},
  volume = {55},
  number = {3},
  year = {2008},
  pages = {344--350},
  url = {https://math.mit.edu/~poonen/papers/h10_notices.pdf},
  note = {\href{https://math.mit.edu/~poonen/papers/h10_notices.pdf}{Source}}
}

@incollection{Deligne1971,
  author    = {Deligne, Pierre},
  title     = {Travaux de {Shimura}},
  booktitle = {S{\'e}minaire Bourbaki, Vol. 1970/71,
               Expos{\'e}s 382--399},
  series    = {Lecture Notes in Mathematics},
  volume    = {244},
  publisher = {Springer-Verlag},
  year      = {1971},
  pages     = {123--165},
  note      = {Expos{\'e} 389. \href{https://www.numdam.org/item/SB_1970-1971__13__123_0/}{Source}},
  url       = {https://www.numdam.org/item/SB_1970-1971__13__123_0/}
}

@incollection{Deligne1979,
  author    = {Deligne, Pierre},
  title     = {Vari{\'e}t{\'e}s de {Shimura}:
               interpr{\'e}tation modulaire, et techniques de
               construction de mod{\`e}les canoniques},
  booktitle = {Automorphic Forms, Representations and {$L$}-Functions,
               Part 2},
  series    = {Proceedings of Symposia in Pure Mathematics},
  volume    = {33},
  publisher = {American Mathematical Society},
  year      = {1979},
  pages     = {247--289},
  url       = {https://publications.ias.edu/deligne/paper/380},
  note = {\href{https://publications.ias.edu/deligne/paper/380}{Source}}
}

@incollection{Deligne1971Modular,
  author    = {Deligne, Pierre},
  title     = {Formes modulaires et repr{\'e}sentations {$\ell$}-adiques},
  booktitle = {S{\'e}minaire Bourbaki, Vol. 1968/69,
               Expos{\'e}s 347--363},
  series    = {Lecture Notes in Mathematics},
  volume    = {179},
  publisher = {Springer-Verlag},
  year      = {1971},
  pages     = {139--172},
  note      = {Expos{\'e} 355. \href{https://www.numdam.org/item/SB_1968-1969__11__139_0/}{Source}},
  url       = {https://www.numdam.org/item/SB_1968-1969__11__139_0/}
}

@incollection{Fontaine1994,
  author    = {Fontaine, Jean-Marc},
  title     = {Repr{\'e}sentations {$p$}-adiques semi-stables},
  booktitle = {P{\'e}riodes {$p$}-adiques:
               S{\'e}minaire de Bures, 1988},
  editor    = {Fontaine, Jean-Marc},
  series    = {Ast{\'e}risque},
  number    = {223},
  publisher = {Soci{\'e}t{\'e} Math{\'e}matique de France},
  year      = {1994},
  pages     = {113--184},
  note      = {Expos{\'e} III. \href{https://www.numdam.org/item/AST_1994__223__113_0/}{Source}},
  url       = {https://www.numdam.org/item/AST_1994__223__113_0/}
}
@incollection{Faltings1989,
  author    = {Faltings, Gerd},
  title     = {Crystalline cohomology and {$p$}-adic
               {Galois} representations},
  booktitle = {Algebraic Analysis, Geometry, and Number Theory},
  editor    = {Igusa, Jun-Ichi},
  publisher = {Johns Hopkins University Press},
  address   = {Baltimore, MD},
  year      = {1989},
  pages     = {25--80}
}

@article{Faltings1988,
  author  = {Faltings, Gerd},
  title   = {{$p$}-adic {Hodge} theory},
  journal = {Journal of the American Mathematical Society},
  volume  = {1},
  number  = {1},
  year    = {1988},
  pages   = {255--299},
  doi     = {10.1090/S0894-0347-1988-0924705-1},
  note = {\bibdoi{10.1090/S0894-0347-1988-0924705-1}}
}

@article{Carayol1986,
  author  = {Carayol, Henri},
  title   = {Sur les repr{\'e}sentations {$l$}-adiques associ{\'e}es
             aux formes modulaires de {Hilbert}},
  journal = {Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure},
  series  = {4},
  volume  = {19},
  number  = {3},
  year    = {1986},
  pages   = {409--468},
  doi     = {10.24033/asens.1512},
  url     = {https://numdam.org/articles/10.24033/asens.1512/},
  note = {\bibdoi{10.24033/asens.1512}}
}

@book{Cox2013,
  author    = {Cox, David A.},
  title     = {Primes of the Form {$x^2+ny^2$}:
               Fermat, Class Field Theory, and Complex Multiplication},
  edition   = {Second},
  publisher = {John Wiley \& Sons},
  year      = {2013},
  isbn      = {978-1-118-39018-4},
  doi       = {10.1002/9781118400722},
  note = {\bibdoi{10.1002/9781118400722}}
}
@article{Siegel1935,
  author  = {Siegel, Carl Ludwig},
  title   = {{{\"U}ber die Classenzahl quadratischer Zahlk{\"o}rper}},
  journal = {Acta Arithmetica},
  volume  = {1},
  number  = {1},
  year    = {1935},
  pages   = {83--86},
  doi     = {10.4064/aa-1-1-83-86},
  note = {\bibdoi{10.4064/aa-1-1-83-86}}
}

@article{Zarhin1974,
  author  = {Zarhin, Yu. G.},
  title   = {A remark on endomorphisms of abelian varieties over
             function fields of finite characteristic},
  journal = {Mathematics of the USSR-Izvestiya},
  volume  = {8},
  number  = {3},
  year    = {1974},
  pages   = {477--480},
  doi     = {10.1070/IM1974v008n03ABEH002115},
  url     = {https://www.mathnet.ru/eng/im1933},
  note = {\bibdoi{10.1070/IM1974v008n03ABEH002115}}
}

@article{Bost1996,
  author  = {Bost, Jean-Beno{\^\i}t},
  title   = {Intrinsic heights of stable varieties and abelian varieties},
  journal = {Duke Mathematical Journal},
  volume  = {82},
  number  = {1},
  year    = {1996},
  pages   = {21--70},
  doi     = {10.1215/S0012-7094-96-08202-2},
  note = {\bibdoi{10.1215/S0012-7094-96-08202-2}}
}
@incollection{Bost1996Bourbaki,
  author    = {Bost, Jean-Beno{\^\i}t},
  title     = {{P{\'e}riodes et isog{\'e}nies des vari{\'e}t{\'e}s
               ab{\'e}liennes sur les corps de nombres
               (d'apr{\`e}s D. Masser et G. W{\"u}stholz)}},
  booktitle = {S{\'e}minaire Bourbaki, Vol. 1994/95,
               Expos{\'e}s 790--804},
  series    = {Ast{\'e}risque},
  number    = {237},
  publisher = {Soci{\'e}t{\'e} Math{\'e}matique de France},
  year      = {1996},
  pages     = {115--161},
  note      = {Expos{\'e} 795. \href{https://archive.numdam.org/item/SB_1994-1995__37__115_0/}{Source}},
  url       = {https://archive.numdam.org/item/SB_1994-1995__37__115_0/}
}

@book{BombieriGubler2006,
  author    = {Bombieri, Enrico and Gubler, Walter},
  title     = {Heights in Diophantine Geometry},
  series    = {New Mathematical Monographs},
  volume    = {4},
  publisher = {Cambridge University Press},
  year      = {2006},
  isbn      = {978-0-521-84615-8},
  doi       = {10.1017/CBO9780511542879},
  note = {\bibdoi{10.1017/CBO9780511542879}}
}

@article{GaudronRemondPeriods2014,
  author = {Gaudron, {\'E}ric and R{\'e}mond, Ga{\"e}l},
  title = {Th{\'e}or{\`e}me des p{\'e}riodes et degr{\'e}s minimaux d'isog{\'e}nies},
  journal = {Commentarii Mathematici Helvetici},
  volume = {89},
  number = {2},
  year = {2014},
  pages = {343--403},
  doi = {10.4171/CMH/322},
  url = {https://lmbp.uca.fr/~gaudron/art11.pdf},
  note = {\bibdoi{10.4171/CMH/322}}
}

@article{vonKaenel2021,
  author = {von K{\"a}nel, Rafael},
  title = {The effective {Shafarevich} conjecture for abelian varieties of {$\mathrm{GL}_2$}-type},
  journal = {Forum of Mathematics, Sigma},
  volume = {9},
  year = {2021},
  pages = {e39, 1--29},
  doi = {10.1017/fms.2021.29},
  url = {https://doi.org/10.1017/fms.2021.29},
  note = {\bibdoi{10.1017/fms.2021.29}}
}

@misc{Alpoge2021,
  author = {Alp{\"o}ge, Levent},
  title = {Modularity and effective {Mordell} {I}},
  year = {2021},
  eprint = {2109.07917},
  archivePrefix = {arXiv},
  primaryClass = {math.NT},
  url = {https://arxiv.org/abs/2109.07917},
  note = {arXiv:2109.07917v2, November 24, 2021. \href{https://arxiv.org/abs/2109.07917}{Source}}
}

@article{GarciaFritzPastenVidaux2025,
  author = {Garcia-Fritz, Natalia and Pasten, Hector and Vidaux, Xavier},
  title = {Effectivity for existence of rational points is undecidable},
  journal = {Journal of Number Theory},
  volume = {276},
  year = {2025},
  pages = {81--97},
  doi = {10.1016/j.jnt.2025.01.023},
  url = {https://doi.org/10.1016/j.jnt.2025.01.023},
  eprint = {2311.01958},
  archivePrefix = {arXiv},
  primaryClass = {math.NT},
  note = {\bibdoi{10.1016/j.jnt.2025.01.023}}
}


@incollection{PoonenRankOne2002,
  author = {Poonen, Bjorn},
  title = {Using elliptic curves of rank one towards the undecidability of
           {Hilbert}'s tenth problem over rings of algebraic integers},
  booktitle = {Algorithmic Number Theory: 5th International Symposium,
               {ANTS-V}, Sydney, Australia, July 2002, Proceedings},
  editor = {Fieker, Claus and Kohel, David R.},
  series = {Lecture Notes in Computer Science},
  volume = {2369},
  pages = {33--42},
  publisher = {Springer-Verlag},
  address = {Berlin},
  year = {2002},
  url = {https://math.mit.edu/~poonen/papers/ants5.pdf},
  note = {\href{https://math.mit.edu/~poonen/papers/ants5.pdf}{Source}}
}

@article{CornelissenZahidi2007,
  author = {Cornelissen, Gunther and Zahidi, Karim},
  title = {Elliptic divisibility sequences and undecidable problems
           about rational points},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume = {613},
  year = {2007},
  pages = {1--33},
  eprint = {math/0412473},
  archivePrefix = {arXiv},
  primaryClass = {math.NT},
  url = {https://arxiv.org/abs/math/0412473v3},
  note = {Author manuscript, version 3, June 23, 2006. \href{https://arxiv.org/abs/math/0412473v3}{Source}}
}

@article{Scholze2013DeRham,
  author = {Scholze, Peter},
  title = {{$p$}-adic {Hodge} theory for rigid-analytic varieties},
  journal = {Forum of Mathematics, Pi},
  volume = {1},
  year = {2013},
  eid = {e1},
  pages = {e1, 1--77},
  doi = {10.1017/fmp.2013.1},
  url = {https://www.math.uni-bonn.de/people/scholze/pAdicHodgeTheory.pdf},
  note = {Corollary 1.8; see also the 2016 corrigendum. \bibdoi{10.1017/fmp.2013.1}}
}
@article{Scholze2016DeRhamCorrigendum,
  author = {Scholze, Peter},
  title = {{$p$}-adic {Hodge} theory for rigid-analytic varieties---corrigendum},
  journal = {Forum of Mathematics, Pi},
  volume = {4},
  year = {2016},
  eid = {e6},
  pages = {e6},
  doi = {10.1017/fmp.2016.4},
  url = {https://people.math.uni-bonn.de/scholze/pAdicHodgeErratum.pdf},
  note = {\bibdoi{10.1017/fmp.2016.4}}
}

@article{GuitartMolina2009,
  author = {Guitart, Xavier and Molina, Santiago},
  title = {Parametrization of {Abelian} {$K$}-surfaces with quaternionic multiplication},
  journal = {Comptes Rendus. Math\'ematique},
  volume = {347},
  number = {23--24},
  pages = {1325--1330},
  year = {2009},
  doi = {10.1016/j.crma.2009.09.025},
  url = {https://www.numdam.org/item/10.1016/j.crma.2009.09.025.pdf},
  note = {\bibdoi{10.1016/j.crma.2009.09.025}}
}

@incollection{Ribet1992,
  author = {Ribet, Kenneth A.},
  title = {Abelian varieties over {$\mathbf{Q}$} and modular forms},
  booktitle = {Algebra and Topology 1992},
  publisher = {Korea Advanced Institute of Science and Technology},
  address = {Taejon},
  pages = {53--79},
  year = {1992},
  note = {Preprint arXiv:alg-geom/9208002. \href{https://arxiv.org/abs/alg-geom/9208002}{Source}},
  url = {https://arxiv.org/abs/alg-geom/9208002}
}

@phdthesis{Pyle1995,
  author = {Pyle, Elisabeth Eve},
  title = {Abelian Varieties over {$\mathbf{Q}$} with Large Endomorphism Algebras and Their Simple Components over {$\overline{\mathbf{Q}}$}},
  school = {University of California, Berkeley},
  year = {1995},
  url = {https://math.berkeley.edu/~ribet/pyle_thesis.pdf}
}

@article{Quer2000,
  author = {Quer, Jordi},
  title = {{$\mathbf{Q}$}-curves and abelian varieties of {$\mathrm{GL}_2$}-type},
  journal = {Proceedings of the London Mathematical Society},
  series = {3},
  volume = {81},
  number = {2},
  pages = {285--317},
  year = {2000},
  doi = {10.1112/S0024611500012570}
}

@article{Quer2001Fields,
  author = {Quer, Jordi},
  title = {Fields of definition of {$\mathbf{Q}$}-curves},
  journal = {Journal de Th\'eorie des Nombres de Bordeaux},
  volume = {13},
  number = {1},
  year = {2001},
  pages = {275--285},
  doi = {10.5802/jtnb.321},
  url = {https://www.numdam.org/article/JTNB_2001__13_1_275_0.pdf},
  note = {\bibdoi{10.5802/jtnb.321}}
}

@article{Williams1974,
  author = {Williams, Kenneth S.},
  title = {Mertens' theorem for arithmetic progressions},
  journal = {Journal of Number Theory},
  volume = {6},
  year = {1974},
  pages = {353--359},
  doi = {10.1016/0022-314X(74)90032-8},
  url = {https://people.math.carleton.ca/~williams/papers/pdf/057.pdf},
  note = {\bibdoi{10.1016/0022-314X(74)90032-8}}
}
@book{MontgomeryVaughan2007,
  author = {Montgomery, Hugh L. and Vaughan, Robert C.},
  title = {Multiplicative Number Theory I: Classical Theory},
  series = {Cambridge Studies in Advanced Mathematics},
  volume = {97},
  publisher = {Cambridge University Press},
  year = {2007},
  doi = {10.1017/CBO9780511618314},
  note = {\bibdoi{10.1017/CBO9780511618314}}
}
@article{Selberg1947,
  author = {Selberg, Atle},
  title = {On an elementary method in the theory of primes},
  journal = {Norske Videnskabers Selskab, Forhandlinger},
  volume = {19},
  number = {18},
  year = {1947},
  pages = {64--67}
}

@incollection{Ribet1977Nebentypus,
  author = {Ribet, Kenneth A.},
  title = {Galois representations attached to eigenforms with nebentypus},
  booktitle = {Modular Functions of One Variable V},
  editor = {Serre, Jean-Pierre and Zagier, Don Bernard},
  series = {Lecture Notes in Mathematics},
  volume = {601},
  pages = {18--52},
  publisher = {Springer},
  address = {Berlin, Heidelberg},
  year = {1977},
  doi = {10.1007/BFb0063943},
  url = {https://doi.org/10.1007/BFb0063943},
  note = {\bibdoi{10.1007/BFb0063943}}
}

@incollection{Zarhin2023Quaternion,
  author = {Zarhin, Yuri G.},
  title = {Abelian varieties, quaternion trick and endomorphisms},
  booktitle = {Birational Geometry, K{\"a}hler--Einstein Metrics and Degenerations},
  editor = {Cheltsov, Ivan and Chen, Xiuxiong and Katzarkov, Ludmil and Park, Jihun},
  series = {Springer Proceedings in Mathematics and Statistics},
  volume = {409},
  publisher = {Springer},
  address = {Cham},
  pages = {857--864},
  year = {2023},
  doi = {10.1007/978-3-031-17859-7_42},
  url = {https://doi.org/10.1007/978-3-031-17859-7_42},
  eprint = {2010.07995},
  archivePrefix = {arXiv},
  note = {Author version arXiv:2010.07995v5. \bibdoi{10.1007/978-3-031-17859-7_42}}
}


@article{Mazur1995,
  author = {Mazur, Barry},
  title = {Speculations about the topology of rational points: an up-date},
  journal = {Ast{\'e}risque},
  number = {228},
  year = {1995},
  pages = {165--181},
  url = {https://www.numdam.org/item/AST_1995__228__165_0/},
  note = {\href{https://www.numdam.org/item/AST_1995__228__165_0/}{Source}}
}

@incollection{CornelissenZahidi2000,
  author = {Cornelissen, Gunther and Zahidi, Karim},
  title = {Topology of {Diophantine} sets: remarks on {Mazur}'s conjectures},
  booktitle = {Hilbert's Tenth Problem: Relations with Arithmetic and Algebraic Geometry},
  editor = {Denef, Jan and Lipshitz, Leonard and Pheidas, Thanases and Van Geel, Jan},
  series = {Contemporary Mathematics},
  volume = {270},
  publisher = {American Mathematical Society},
  address = {Providence, RI},
  year = {2000},
  pages = {253--260},
  doi = {10.1090/conm/270/04377},
  eprint = {math/0006140},
  archivePrefix = {arXiv},
  primaryClass = {math.LO},
  url = {https://arxiv.org/abs/math/0006140},
  note = {Author preprint dated June 20, 2000. \bibdoi{10.1090/conm/270/04377}}
}

@article{EisentragerMillerParkShlapentokh2017,
  author = {Eisentr{\"a}ger, Kirsten and Miller, Russell and Park, Jennifer and Shlapentokh, Alexandra},
  title = {As easy as {$\mathbb{Q}$}: {Hilbert}'s tenth problem for subrings of the rationals and number fields},
  journal = {Transactions of the American Mathematical Society},
  volume = {369},
  number = {11},
  year = {2017},
  pages = {8291--8315},
  doi = {10.1090/tran/7075},
  eprint = {1601.07158},
  archivePrefix = {arXiv},
  primaryClass = {math.NT},
  url = {https://arxiv.org/abs/1601.07158},
  note = {\bibdoi{10.1090/tran/7075}}
}

@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},
  url = {https://escholarship.org/content/qt3nx8s5vn/qt3nx8s5vn.pdf},
  note = {\bibdoi{10.1007/s00222-010-0252-0}}
}

@article{KoymansPagano2026,
  author = {Koymans, Peter and Pagano, Carlo},
  title = {{Hilbert}'s tenth problem via additive combinatorics},
  journal = {Journal of the American Mathematical Society},
  year = {2026},
  doi = {10.1090/jams/1081},
  eprint = {2412.01768},
  archivePrefix = {arXiv},
  primaryClass = {math.NT},
  url = {https://doi.org/10.1090/jams/1081},
  note = {Published electronically August 10, 2026; arXiv:2412.01768v3 submitted November 24, 2025. \bibdoi{10.1090/jams/1081}}
}

@article{AlpogeBhargavaHoShnidman2026,
  author = {Alp{\"o}ge, Levent and Bhargava, Manjul and Ho, Wei and Shnidman, Ari},
  title = {Rank stability in quadratic extensions and {Hilbert}'s tenth problem for the ring of integers of a number field},
  journal = {Inventiones Mathematicae},
  volume = {243},
  year = {2026},
  pages = {1129--1139},
  doi = {10.1007/s00222-025-01392-3},
  url = {https://doi.org/10.1007/s00222-025-01392-3},
  note = {Version of record published online December 1, 2025. \bibdoi{10.1007/s00222-025-01392-3}}
}

@article{DemeyerVanGeel2006,
  author = {Demeyer, Jeroen and Van Geel, Jan},
  title = {An existential divisibility lemma for global fields},
  journal = {Monatshefte f{\"u}r Mathematik},
  volume = {147},
  number = {4},
  year = {2006},
  pages = {293--308},
  doi = {10.1007/s00605-005-0342-z},
  url = {https://backoffice.biblio.ugent.be/download/328999/454769},
  note = {Author manuscript dated July 11, 2005. \bibdoi{10.1007/s00605-005-0342-z}}
}

@article{Mokrani2020,
  author = {Mokrani, Youcef},
  title = {Adaptation of {Monsky} matrices for {$\theta$}-congruent numbers},
  journal = {International Journal of Number Theory},
  volume = {16},
  number = {2},
  year = {2020},
  pages = {377--396},
  doi = {10.1142/S1793042120500207},
  url = {https://doi.org/10.1142/S1793042120500207},
  note = {Published online September 5, 2019. \bibdoi{10.1142/S1793042120500207}}
}

@misc{WeiGuo2022,
  author = {Wei, Tao and Guo, Xuejun},
  title = {The rank of {$2$}-{Selmer} group associate to {$\theta$}-congruent numbers},
  year = {2022},
  eprint = {2210.01678},
  archivePrefix = {arXiv},
  primaryClass = {math.NT},
  url = {https://arxiv.org/abs/2210.01678},
  note = {arXiv:2210.01678v1, October 4, 2022. \href{https://arxiv.org/abs/2210.01678}{Source}}
}

@article{LongMaclachlanReid2006,
  author = {Long, D. D. and Maclachlan, C. and Reid, A. W.},
  title = {Arithmetic {Fuchsian} groups of genus zero},
  journal = {Pure and Applied Mathematics Quarterly},
  volume = {2},
  number = {2},
  year = {2006},
  pages = {569--599},
  doi = {10.4310/PAMQ.2006.v2.n2.a9},
  url = {https://web.ma.utexas.edu/users/areid/genus0_final.pdf},
  note = {\bibdoi{10.4310/PAMQ.2006.v2.n2.a9}}
}

@phdthesis{Nualart2015,
  author = {Nualart Riera, Joan},
  title = {On the Hyperbolic Uniformization of {Shimura} Curves with an {Atkin--Lehner} Quotient of Genus 0},
  school = {Universitat de Barcelona},
  year = {2015},
  month = oct,
  url = {https://diposit.ub.edu/items/ece138ce-1523-4d59-90ae-b08826c454e8},
  note = {Title-page date October 2015; repository release February 3, 2016. \href{https://diposit.ub.edu/items/ece138ce-1523-4d59-90ae-b08826c454e8}{Source}}
}

@article{DarmonGranville1995,
  author = {Darmon, Henri and Granville, Andrew},
  title = {On the equations {$z^m=F(x,y)$} and {$Ax^p+By^q=Cz^r$}},
  journal = {Bulletin of the London Mathematical Society},
  volume = {27},
  number = {6},
  year = {1995},
  pages = {513--543},
  doi = {10.1112/blms/27.6.513},
  url = {https://www.math.mcgill.ca/darmon/pub/Articles/Research/12.Granville/pub12.pdf},
  note = {\bibdoi{10.1112/blms/27.6.513}}
}

@article{MurtyPasten2013,
  author = {Murty, M. Ram and Pasten, Hector},
  title = {Modular forms and effective {Diophantine} approximation},
  journal = {Journal of Number Theory},
  volume = {133},
  number = {11},
  year = {2013},
  pages = {3739--3754},
  doi = {10.1016/j.jnt.2013.05.006},
  url = {https://mast.queensu.ca/~murty/Murty-Pasten-1.pdf},
  note = {\bibdoi{10.1016/j.jnt.2013.05.006}}
}

@misc{vonKaenel2013Preprint,
  author = {von K{\"a}nel, Rafael},
  title = {Modularity and integral points on moduli schemes},
  year = {2013},
  eprint = {1310.7263},
  archivePrefix = {arXiv},
  primaryClass = {math.NT},
  url = {https://arxiv.org/abs/1310.7263v1},
  note = {arXiv:1310.7263v1, October 27, 2013; revised version 2, March 23, 2014. \href{https://arxiv.org/abs/1310.7263v1}{Source}}
}

@misc{vonKaenelKret2023,
  author = {von K{\"a}nel, Rafael and Kret, Arno},
  title = {Integral points on coarse {Hilbert} moduli schemes},
  year = {2023},
  eprint = {2307.06944},
  archivePrefix = {arXiv},
  primaryClass = {math.NT},
  url = {https://arxiv.org/abs/2307.06944},
  note = {arXiv:2307.06944v1, July 13, 2023. \href{https://arxiv.org/abs/2307.06944}{Source}}
}

@article{Wu2020,
  author = {Wu, Chenyan},
  title = {Virtual {Abelian} varieties of {$\mathrm{GL}_2$}-type},
  journal = {Mathematical Research Letters},
  volume = {27},
  number = {3},
  year = {2020},
  pages = {903--944},
  doi = {10.4310/MRL.2020.v27.n3.a14},
  eprint = {1507.03069},
  archivePrefix = {arXiv},
  primaryClass = {math.AG},
  url = {https://arxiv.org/abs/1507.03069},
  note = {Author version arXiv:1507.03069v3, January 25, 2022. \bibdoi{10.4310/MRL.2020.v27.n3.a14}}
}

@article{CornelissenShlapentokh2009,
  author = {Cornelissen, Gunther and Shlapentokh, Alexandra},
  title = {Defining the integers in large rings of a number field using one universal quantifier},
  journal = {Journal of Mathematical Sciences (New York)},
  volume = {158},
  number = {5},
  year = {2009},
  pages = {713--726},
  doi = {10.1007/s10958-009-9404-4},
  eprint = {0708.3075},
  archivePrefix = {arXiv},
  primaryClass = {math.LO},
  url = {https://arxiv.org/abs/0708.3075v2},
  note = {Originally published in Zapiski Nauchnykh Seminarov POMI 358 (2008), 199--223; author version arXiv:0708.3075v2, February 14, 2008. \href{https://arxiv.org/abs/0708.3075v2}{Author version}. \bibdoi{10.1007/s10958-009-9404-4}}
}

@article{HeathBrown1994,
  author = {Heath-Brown, D. R.},
  title = {The size of {Selmer} groups for the congruent number problem, {II}},
  journal = {Inventiones Mathematicae},
  volume = {118},
  pages = {331--370},
  year = {1994},
  doi = {10.1007/BF01231536},
  url = {https://doi.org/10.1007/BF01231536},
  note = {With an appendix by Paul Monsky, pp.~365--370. \bibdoi{10.1007/BF01231536}}
}