@incollection{Goldfeld1979,
  author = {Goldfeld, Dorian},
  title = {Conjectures on elliptic curves over quadratic fields},
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  editor = {Nathanson, Melvyn B.},
  series = {Lecture Notes in Mathematics},
  volume = {751},
  pages = {108--118},
  publisher = {Springer},
  address = {Berlin},
  year = {1979},
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}

@misc{Smith2025,
  author = {Smith, Alexander},
  title = {The {Birch} and {Swinnerton-Dyer} conjecture implies {Goldfeld}'s conjecture},
  year = {2025},
  note = {arXiv:2503.17619, version 1},
  eprint = {2503.17619},
  archivePrefix = {arXiv},
  primaryClass = {math.NT},
  doi = {10.48550/arXiv.2503.17619},
  url = {https://arxiv.org/abs/2503.17619v1}
}

@article{CaiShuTian2014,
  author = {Cai, Li and Shu, Jie and Tian, Ye},
  title = {Explicit {Gross--Zagier} and {Waldspurger} formulae},
  journal = {Algebra \& Number Theory},
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  number = {10},
  pages = {2523--2572},
  year = {2014},
  doi = {10.2140/ant.2014.8.2523},
  url = {https://msp.org/ant/2014/8-10/ant-v8-n10-p05-p.pdf}
}

@incollection{Kato2004,
  author = {Kato, Kazuya},
  title = {{$p$}-adic {Hodge} theory and values of zeta functions of modular forms},
  booktitle = {Cohomologie {$p$}-adiques et applications arithm\'etiques ({III})},
  editor = {Berthelot, Pierre and Fontaine, Jean-Marc and Illusie, Luc and Kato, Kazuya and Rapoport, Michael},
  series = {Ast\'erisque},
  number = {295},
  pages = {117--290},
  publisher = {Soci\'et\'e Math\'ematique de France},
  year = {2004},
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}

@article{BurungaleTian2026,
  author = {Burungale, Ashay A. and Tian, Ye},
  title = {A rank zero {$p$}-converse to a theorem of {Gross--Zagier}, {Kolyvagin} and {Rubin}},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {203},
  number = {1},
  pages = {1--13},
  year = {2026},
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  note = {Theorem numbering follows arXiv:2506.03465v2 (\url{https://arxiv.org/abs/2506.03465v2})},
  url = {https://annals.math.princeton.edu/2026/203-1/p01}
}

@article{Cassels1965,
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  pages = {180--199},
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@book{Neukirch1999,
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}

@article{RadziwillSoundararajan2015,
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@article{Iwaniec1990,
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@article{HeathBrown1995,
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@article{ErdosKac1940,
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@article{GrossZagier1986,
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  pages = {225--320},
  year = {1986},
  doi = {10.1007/BF01388809},
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@incollection{Kolyvagin1990,
  author = {Kolyvagin, V. A.},
  title = {Euler systems},
  booktitle = {The {Grothendieck} Festschrift, Volume {II}},
  editor = {Cartier, Pierre and Illusie, Luc and Katz, Nicholas M. and Laumon, G{\'e}rard and Manin, Yuri I. and Ribet, Kenneth A.},
  series = {Progress in Mathematics},
  volume = {87},
  pages = {435--483},
  publisher = {Birkh\"auser},
  address = {Boston},
  year = {1990},
  mrnumber = {1106906},
  note = {Linked chapter is the 2007 reprint},
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}

@article{Waldspurger1980,
  author = {Waldspurger, Jean-Loup},
  title = {Correspondance de {Shimura}},
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  volume = {59},
  number = {1},
  pages = {1--132},
  year = {1980}
}

@article{Waldspurger1981,
  author = {Waldspurger, Jean-Loup},
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  journal = {Journal de Math\'ematiques Pures et Appliqu\'ees},
  series = {9},
  volume = {60},
  number = {4},
  pages = {375--484},
  year = {1981},
  url = {https://mate.dm.uba.ar/~nsirolli/files/waldspurger.pdf}
}

@inproceedings{Waldspurger1983,
  author = {Waldspurger, Jean-Loup},
  title = {Correspondances de {Shimura}},
  booktitle = {Proceedings of the International Congress of Mathematicians, Warszawa, 1983},
  editor = {Ciesielski, Zbigniew and Olech, Czes{\l}aw},
  volume = {1},
  pages = {525--531},
  publisher = {PWN--Polish Scientific Publishers and Elsevier Science Publishers},
  address = {Warszawa and Amsterdam},
  year = {1984},
  url = {https://www.imj-prg.fr/wp-content/uploads/2020/prix/waldspurger1983.pdf}
}

@article{Weil1964,
  author = {Weil, Andr\'e},
  title = {Sur certains groupes d'op\'erateurs unitaires},
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  year = {1964},
  doi = {10.1007/BF02391012},
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@incollection{Deligne1969,
  author = {Deligne, Pierre},
  title = {Formes modulaires et repr\'esentations {$\ell$}-adiques},
  booktitle = {S\'eminaire {Bourbaki}, 1968/69},
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  volume = {179},
  pages = {139--172},
  publisher = {Springer},
  address = {Berlin},
  year = {1971},
  note = {Expos\'e 355, February 1969},
  url = {https://www.numdam.org/item/SB_1968-1969__11__139_0/}
}

@article{Carayol1986,
  author = {Carayol, Henri},
  title = {Sur les repr\'esentations {$\ell$}-adiques associ\'ees aux formes modulaires de {Hilbert}},
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  series = {4},
  volume = {19},
  number = {3},
  pages = {409--468},
  year = {1986},
  doi = {10.24033/asens.1512},
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}

@article{JetchevKane2011,
  author = {Jetchev, Dimitar and Kane, Ben},
  title = {Equidistribution of {Heegner} points and ternary quadratic forms},
  journal = {Mathematische Annalen},
  volume = {350},
  number = {3},
  pages = {501--532},
  year = {2011},
  doi = {10.1007/s00208-010-0568-5},
  eprint = {0908.3905},
  archivePrefix = {arXiv},
  note = {Section numbering follows arXiv:0908.3905v1},
  url = {https://arxiv.org/abs/0908.3905v1}
}

@article{FerreroWashington1979,
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  number = {2},
  pages = {377--395},
  year = {1979},
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  url = {https://doi.org/10.2307/1971116}
}

@article{Rohrlich1984,
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  number = {3},
  pages = {409--423},
  year = {1984},
  doi = {10.1007/BF01388636},
  url = {https://doi.org/10.1007/BF01388636}
}

@article{Rohrlich1988,
  author = {Rohrlich, David E.},
  title = {{$L$}-functions and division towers},
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  volume = {281},
  number = {4},
  pages = {611--632},
  year = {1988},
  doi = {10.1007/BF01456842},
  url = {https://doi.org/10.1007/BF01456842}
}

@article{Rohrlich1989,
  author = {Rohrlich, David E.},
  title = {Nonvanishing of {$L$}-functions for {$\mathrm{GL}(2)$}},
  journal = {Inventiones Mathematicae},
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  pages = {381--403},
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}

@article{Tate1966,
  author = {Tate, John},
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  volume = {9},
  pages = {415--440},
  year = {1966},
  note = {Expos\'e 306},
  url = {https://www.numdam.org/item/SB_1964-1966__9__415_0/}
}

@article{Cassels1962,
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  volume = {211},
  pages = {95--112},
  year = {1962},
  doi = {10.1515/crll.1962.211.95},
  url = {https://doi.org/10.1515/crll.1962.211.95}
}

@article{Kramer1981,
  author = {Kramer, Kenneth},
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  journal = {Transactions of the American Mathematical Society},
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  number = {1},
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  year = {1981},
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}

@article{DokchitserDokchitser2011,
  author = {Dokchitser, Tim and Dokchitser, Vladimir},
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  volume = {658},
  pages = {39--64},
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  doi = {10.1515/CRELLE.2011.060},
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  archivePrefix = {arXiv},
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}

@article{DokchitserDokchitser2010,
  author = {Dokchitser, Tim and Dokchitser, Vladimir},
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@article{CesnaviciusImai2016,
  author = {{\v C}esnavi{\v c}ius, K{\k e}stutis and Imai, Naoki},
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  journal = {Compositio Mathematica},
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  number = {11},
  pages = {2255--2268},
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  doi = {10.1112/S0010437X16007624},
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}

@book{IwaniecKowalski2004,
  author = {Iwaniec, Henryk and Kowalski, Emmanuel},
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  publisher = {American Mathematical Society},
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  url = {https://doi.org/10.1090/coll/053}
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@book{FriedlanderIwaniec2010,
  author = {Friedlander, John and Iwaniec, Henryk},
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  year = {2010},
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  url = {https://doi.org/10.1090/coll/057}
}

@book{Serre1968,
  author = {Serre, Jean-Pierre},
  title = {Abelian {$\ell$}-adic representations and elliptic curves},
  publisher = {W. A. Benjamin},
  address = {New York},
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  note = {McGill University lecture notes, written with the collaboration of Willem Kuyk and John Labute},
  url = {https://www.math.mcgill.ca/darmon/courses/18-19/gs/serre-mcgill.pdf}
}

@article{Serre1972,
  author = {Serre, Jean-Pierre},
  title = {Propri\'et\'es galoisiennes des points d'ordre fini des courbes elliptiques},
  journal = {Inventiones Mathematicae},
  volume = {15},
  number = {4},
  pages = {259--331},
  year = {1972},
  doi = {10.1007/BF01405086},
  url = {https://wstein.org/sage_summer/bsd_comp/Serre-properties_galoisiennes_des_points_dordre_fini_des_courbes_elliptiques.pdf}
}

@book{Nekovar2006,
  author = {Nekov{\'a}{\v r}, Jan},
  title = {Selmer complexes},
  series = {Ast\'erisque},
  volume = {310},
  publisher = {Soci\'et\'e Math\'ematique de France},
  year = {2006},
  isbn = {978-2-85629-227-3},
  doi = {10.24033/ast.717},
  url = {https://www.numdam.org/item/AST_2006__310__R1_0/}
}

@article{BreuilConradDiamondTaylor2001,
  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},
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  number = {4},
  pages = {843--939},
  year = {2001},
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  url = {https://www.ams.org/jams/2001-14-04/S0894-0347-01-00370-8/S0894-0347-01-00370-8.pdf}
}

@article{AtkinLi1978,
  author = {Atkin, A. O. L. and Li, Wen Ch'ing},
  title = {Twists of newforms and pseudo-eigenvalues of {$W$}-operators},
  journal = {Inventiones Mathematicae},
  volume = {48},
  number = {3},
  pages = {221--243},
  year = {1978},
  doi = {10.1007/BF01390245},
  url = {https://doi.org/10.1007/BF01390245}
}

@article{Deuring1941,
  author = {Deuring, Max},
  title = {Die {Typen} der {Multiplikatorenringe} elliptischer {Funktionenk\"orper}},
  journal = {Abhandlungen aus dem Mathematischen Seminar der Hansischen Universit\"at},
  volume = {14},
  pages = {197--272},
  year = {1941},
  doi = {10.1007/BF02940746},
  url = {https://doi.org/10.1007/BF02940746}
}

@article{BirchSwinnertonDyer1965,
 author={Birch, Bryan J. and Swinnerton-Dyer, H. P. F.},
 title={Notes on elliptic curves. {II}},
 journal={Journal f{\"u}r die reine und angewandte Mathematik},volume={218},pages={79--108},year={1965},
 note={\href{https://doi.org/10.1515/crll.1965.218.79}{doi:10.1515/crll.1965.218.79}}
}

@article{Skinner2020,
 author={Skinner, Christopher},
 title={A converse to a theorem of {Gross}, {Zagier}, and {Kolyvagin}},
 journal={Annals of Mathematics},volume={191},number={2},pages={329--354},year={2020},
 doi={10.4007/annals.2020.191.2.1},
 note={\href{https://annals.math.princeton.edu/wp-content/uploads/annals-v191-n2-p01-s.pdf}{primary text}}
}

@misc{KellerYin2024,
 author={Keller, Timo and Yin, Mulun},
 title={{$p$}-converse theorems for elliptic curves of potentially good ordinary reduction at {Eisenstein} primes},
 year={2024},note={\href{https://arxiv.org/abs/2410.23241v1}{arXiv:2410.23241v1}}
}

@article{Howard2012,
 author={Howard, Benjamin},title={The {Heegner} point {Kolyvagin} system},
 journal={Compositio Mathematica},volume={140},number={6},pages={1439--1472},year={2004},
 note={\href{https://doi.org/10.1112/S0010437X04000569}{doi:10.1112/S0010437X04000569};
 text at \href{https://arxiv.org/abs/1202.6340v1}{arXiv:1202.6340v1}}
}

@book{MilneADT2006,
 author={Milne, J. S.},title={Arithmetic Duality Theorems},
 edition={Second},publisher={BookSurge},year={2006},
 note={\href{https://www.jmilne.org/math/Books/ADTnot.pdf}{author's electronic edition}}
}

@article{GelbartJacquet1978,
 author={Gelbart, Stephen and Jacquet, Herv{\'e}},
 title={A relation between automorphic representations of {$\mathrm{GL}(2)$} and {$\mathrm{GL}(3)$}},
 journal={Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure},
 series={4},volume={11},number={4},pages={471--542},year={1978},
 note={\href{https://doi.org/10.24033/asens.1355}{doi:10.24033/asens.1355}}
}

@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},pages={349--366},year={1983},
 note={\href{https://doi.org/10.1007/BF01388432}{doi:10.1007/BF01388432}}
}

@article{Bogomolov1980,
 author={Bogomolov, Fedor},
 title={Sur l'alg{\'e}bricit{\'e} des repr{\'e}sentations {$l$}-adiques},
 journal={Comptes Rendus de l'Acad{\'e}mie des Sciences de Paris, S{\'e}rie A--B},
 volume={290},number={15},pages={A701--A703},year={1980}
}

@article{Monsky1996,
  author = {Monsky, Paul},
  title = {Generalizing the {Birch--Stephens} theorem. {I}. {Modular} curves},
  journal = {Mathematische Zeitschrift},
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  pages = {415--420},
  year = {1996},
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  url = {https://link.springer.com/article/10.1007/PL00004518}
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@article{HoffsteinLuo1997,
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  note = {With an appendix by David E. Rohrlich},
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}

@article{Bogomolov1981,
  author = {Bogomolov, Fedor A.},
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  journal = {Mathematics of the USSR-Izvestiya},
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  number = {1},
  pages = {55--72},
  year = {1981},
  doi = {10.1070/IM1981v017n01ABEH001329},
  url = {https://www.mathnet.ru/eng/im1843},
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@article{BFH1990,
  author = {Bump, Daniel and Friedberg, Solomon and Hoffstein, Jeffrey},
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  number = {3},
  pages = {543--618},
  year = {1990},
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}

@book{BoschLutkebohmertRaynaud1990,
  author = {Bosch, Siegfried and L\"utkebohmert, Werner and Raynaud, Michel},
  title = {N\'eron Models},
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  address = {Berlin--Heidelberg},
  year = {1990},
  doi = {10.1007/978-3-642-51438-8},
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@book{MilneEC2021,
  author = {Milne, J. S.},
  title = {Elliptic Curves},
  edition = {Second},
  publisher = {World Scientific},
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  url = {https://www.jmilne.org/math/Books/EC2.pdf},
  note = {Author's electronic version}
}

@article{Drinfeld1973,
  author = {Drinfeld, V. G.},
  title = {Two theorems on modular curves},
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  number = {2},
  pages = {155--156},
  year = {1973},
  doi = {10.1007/BF01078890},
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  note = {Russian original: Funktsional. Anal. i Prilozhen. 7(2), 83--84}
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@misc{MilneCFT2020,
  author = {Milne, J. S.},
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