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}

@article{adleman1997quantum,
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  number    = {5},
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  year      = {1997},
  doi       = {10.1137/S0097539795293639}
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@book{andrews1998theory,
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  year      = {1998},
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@inproceedings{babai1984complexity,
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  booktitle = {25th Annual Symposium on Foundations of Computer Science},
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  year      = {1984},
  doi       = {10.1109/SFCS.1984.715919}
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@article{babai1992bounded,
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  doi       = {10.1137/0405008}
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@article{bach1990explicit,
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@article{bhargava2014congruent,
  author    = {Manjul Bhargava and Christopher Skinner and Wei Zhang},
  title     = {A majority of elliptic curves over $\mathbb{Q}$ satisfy the {B}irch and {S}winnerton-{D}yer conjecture},
  journal   = {arXiv preprint arXiv:1407.1826},
  year      = {2014},
  doi       = {10.48550/arXiv.1407.1826}
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@article{bringmann2022class,
  author    = {Kathrin Bringmann and Ben Kane},
  title     = {Class numbers and representations by ternary quadratic forms with congruence conditions},
  journal   = {Mathematics of Computation},
  volume    = {91},
  number    = {333},
  pages     = {295--329},
  year      = {2022},
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}

@article{BSD1965,
  author    = {Bryan John Birch and Harold Peter Francis Swinnerton-Dyer},
  title     = {Notes on elliptic curves. {II}},
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  doi       = {10.1515/crll.1965.218.79}
}

@article{coates2018two,
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  journal   = {arXiv preprint arXiv:1303.5218},
  year      = {2013},
  doi       = {10.48550/arXiv.1303.5218}
}

@article{cai1989enumerative,
  author    = {Jin-Yi Cai and Lane A. Hemachandra},
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  number    = {1},
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  year      = {1989},
  doi       = {10.1016/0890-5401(89)90063-1}
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@article{coates1977conjecture,
  author    = {John Coates and Andrew Wiles},
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  journal   = {Inventiones Mathematicae},
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  number    = {3},
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  year      = {1977},
  doi       = {10.1007/BF01402975}
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@book{cohen1993course,
  author    = {Henri Cohen},
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  address   = {Berlin},
  year      = {1993},
  series    = {Graduate Texts in Mathematics},
  volume    = {138},
  doi       = {10.1007/978-3-662-02945-9}
}

@inproceedings{cook1971complexity,
  author    = {Stephen Arthur Cook},
  title     = {The complexity of theorem-proving procedures},
  booktitle = {Proceedings of the Third Annual ACM Symposium on Theory of Computing},
  pages     = {151--158},
  year      = {1971},
  publisher = {ACM},
  address   = {New York, NY, USA},
  doi       = {10.1145/800157.805047}
}

@article{deligne1974weil,
  author    = {Pierre Deligne},
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  journal   = {Publications Math\'{e}matiques de l'IH\'{E}S},
  volume    = {43},
  pages     = {273--307},
  year      = {1974},
  doi       = {10.1007/BF02684373}
}

@article{duke1988hyperbolic,
  author    = {William Duke},
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  journal   = {Inventiones Mathematicae},
  volume    = {92},
  number    = {1},
  pages     = {73--90},
  year      = {1988},
  doi       = {10.1007/BF01393993}
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@article{duke1990representation,
  author    = {William Duke and Rainer Schulze-Pillot},
  title     = {Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids},
  journal   = {Inventiones Mathematicae},
  volume    = {99},
  number    = {1},
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  year      = {1990},
  doi       = {10.1007/BF01234411}
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@article{gill1977,
  author    = {John Gill},
  title     = {Computational complexity of probabilistic {T}uring machines},
  journal   = {SIAM Journal on Computing},
  volume    = {6},
  number    = {4},
  pages     = {675--695},
  year      = {1977},
  doi       = {10.1137/0206049}
}

@article{hafner1989rigorous,
  author    = {James L. Hafner and Kevin S. McCurley},
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  year      = {1989},
  doi       = {10.1090/S0894-0347-1989-1002631-0}
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@inproceedings{hart2010congruent,
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  booktitle = {Algorithmic Number Theory (ANTS-IX)},
  series    = {Lecture Notes in Computer Science},
  volume    = {6197},
  pages     = {186--200},
  publisher = {Springer},
  address   = {Berlin},
  year      = {2010},
  doi       = {10.1007/978-3-642-14518-6_17}
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@article{heathbrown1994size,
  author    = {D. Roger Heath-Brown},
  title     = {The size of {S}elmer groups for the congruent number problem, {II}},
  journal   = {Inventiones Mathematicae},
  volume    = {118},
  number    = {1},
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  year      = {1994},
  note      = {With an appendix by P. Monsky},
  doi       = {10.1007/BF01231536}
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@article{iwaniec1987fourier,
  author    = {Henryk Iwaniec},
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  journal   = {Inventiones Mathematicae},
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  number    = {2},
  pages     = {385--401},
  year      = {1987},
  doi       = {10.1007/BF01389423}
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@article{kannan1979polynomial,
  author    = {Ravindran Kannan and Achim Bachem},
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  journal   = {SIAM Journal on Computing},
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  year      = {1979},
  doi       = {10.1137/0208040}
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@incollection{karp1972reducibility,
  author    = {Richard M. Karp},
  title     = {Reducibility among Combinatorial Problems},
  booktitle = {Complexity of Computer Computations},
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  pages     = {85--103},
  publisher = {Plenum},
  address   = {New York, USA},
  year      = {1972},
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}

@book{koblitz1993introduction,
  author    = {Neal Koblitz},
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  edition   = {2nd},
  publisher = {Springer-Verlag},
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  year      = {1993},
  series    = {Graduate Texts in Mathematics},
  volume    = {97},
  doi       = {10.1007/978-1-4612-0909-6}
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@article{koiran1996hilbert,
  author    = {Pascal Koiran},
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  journal   = {Journal of Complexity},
  volume    = {12},
  number    = {4},
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  year      = {1996},
  doi       = {10.1006/jcom.1996.0019}
}

@article{lautemann1983bpp,
  author    = {Clemens Lautemann},
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  number    = {4},
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  year      = {1983},
  doi       = {10.1016/0020-0190(83)90044-3}
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@article{manders1978np,
  author    = {Kenneth L. Manders and Leonard Adleman},
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  number    = {2},
  pages     = {168--184},
  year      = {1978},
  doi       = {10.1016/0022-0000(78)90044-2}
}

@article{monsky1990mock,
  author    = {Paul Monsky},
  title     = {Mock {H}eegner points and congruent numbers},
  journal   = {Mathematische Zeitschrift},
  volume    = {204},
  number    = {1},
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  year      = {1990},
  doi       = {10.1007/BF02570859}
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@article{ono1997tate,
  author    = {Ken Ono},
  title     = {Tate--{S}hafarevich groups of the congruent number elliptic curves},
  journal   = {Acta Arithmetica},
  volume    = {81},
  number    = {3},
  pages     = {247--252},
  year      = {1997},
  doi       = {10.4064/aa-81-3-247-252}
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@misc{openai2026exactbsd,
  author       = {{OpenAI}},
  title        = {Exact {B}irch--{S}winnerton-{D}yer Formula from Low {S}elmer Corank},
  howpublished = {Preprint, Result Family 002 of the OpenAI mathematics manuscript collection, \url{https://github.com/openai/math/blob/main/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf}},
  year         = {2026},
  month        = {October},
  note         = {Dated October 7, 2026; no DOI assigned}
}

@misc{openai2026selmerconverse,
  author       = {{OpenAI}},
  title        = {The {S}elmer converse for elliptic curves at every prime},
  howpublished = {Preprint, Result Family 002 of the OpenAI mathematics manuscript collection, \url{https://github.com/openai/math/blob/main/preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026/main.pdf}},
  year         = {2026},
  month        = {October},
  note         = {Dated October 7, 2026; no DOI assigned}
}

@misc{openai2026twoprimary,
  author       = {{OpenAI}},
  title        = {The two-primary {B}irch--{S}winnerton-{D}yer formula in {S}elmer corank at most one},
  howpublished = {Preprint, Result Family 002 of the OpenAI mathematics manuscript collection, \url{https://github.com/openai/math/blob/main/preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026/paper.pdf}},
  year         = {2026},
  month        = {October},
  note         = {Dated October 6, 2026; no DOI assigned}
}

@article{rubin1987tate,
  author    = {Karl Rubin},
  title     = {Tate--{S}hafarevich groups and {$L$}-functions of elliptic curves with complex multiplication},
  journal   = {Inventiones Mathematicae},
  volume    = {89},
  number    = {3},
  pages     = {527--559},
  year      = {1987},
  doi       = {10.1007/BF01388984}
}

@article{rubin1991main,
  author    = {Karl Rubin},
  title     = {The ``main conjectures'' of {I}wasawa theory for imaginary quadratic fields},
  journal   = {Inventiones Mathematicae},
  volume    = {103},
  number    = {1},
  pages     = {25--68},
  year      = {1991},
  doi       = {10.1007/BF01239508}
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@article{shimura1973,
  author    = {Goro Shimura},
  title     = {On modular forms of half integral weight},
  journal   = {Annals of Mathematics},
  volume    = {97},
  number    = {3},
  pages     = {440--481},
  year      = {1973},
  doi       = {10.2307/1970831}
}

@article{shor1997polynomial,
  author    = {Peter W. Shor},
  title     = {Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer},
  journal   = {SIAM Journal on Computing},
  volume    = {26},
  number    = {5},
  pages     = {1484--1509},
  year      = {1997},
  doi       = {10.1137/S0097539795293172}
}

@article{siegel1935analytische,
  author    = {Carl Ludwig Siegel},
  title     = {\"{U}ber die analytische {T}heorie der quadratischen {F}ormen},
  journal   = {Annals of Mathematics},
  volume    = {36},
  number    = {3},
  pages     = {527--606},
  year      = {1935},
  doi       = {10.2307/1968644}
}

@article{simon2002,
  author    = {Denis Simon},
  title     = {Computing the rank of elliptic curves over number fields},
  journal   = {LMS Journal of Computation and Mathematics},
  volume    = {5},
  pages     = {7--17},
  year      = {2002},
  doi       = {10.1112/S1461157000000668}
}

@article{smith2017selmer,
  author    = {Alexander Smith},
  title     = {$2^\infty$-{S}elmer groups, $2^\infty$-class groups, and {G}oldfeld's conjecture},
  journal   = {arXiv preprint arXiv:1702.02325},
  year      = {2017},
  doi       = {10.48550/arXiv.1702.02325}
}

@article{stockmeyer1985,
  author    = {Larry Stockmeyer},
  title     = {On approximation algorithms for \#{P}},
  journal   = {SIAM Journal on Computing},
  volume    = {14},
  number    = {4},
  pages     = {849--861},
  year      = {1985},
  doi       = {10.1137/0214060}
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@article{toda1991pp,
  author    = {Seinosuke Toda},
  title     = {{PP} is as hard as the polynomial-time hierarchy},
  journal   = {SIAM Journal on Computing},
  volume    = {20},
  number    = {5},
  pages     = {865--877},
  year      = {1991},
  doi       = {10.1137/0220053}
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@article{tunnell1983,
  author    = {Jerrold Bruce Tunnell},
  title     = {A classical {D}iophantine problem and modular forms of weight $3/2$},
  journal   = {Inventiones Mathematicae},
  volume    = {72},
  number    = {2},
  pages     = {323--334},
  year      = {1983},
  doi       = {10.1007/BF01389327}
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@incollection{tseitin1983complexity,
  author    = {G. S. Tseitin},
  title     = {On the complexity of derivation in propositional calculus},
  booktitle = {Automation of Reasoning 2: Classical Papers on Computational Logic 1967--1970},
  editor    = {J{\"o}rg H. Siekmann and Graham Wrightson},
  publisher = {Springer},
  address   = {Berlin, Heidelberg},
  pages     = {466--483},
  year      = {1983},
  doi       = {10.1007/978-3-642-81955-1_28}
}

@article{toth2014counting,
  author    = {L{\'a}szl{\'o} T{\'o}th},
  title     = {Counting solutions of quadratic congruences in several variables revisited},
  journal   = {Journal of Integer Sequences},
  volume    = {17},
  pages     = {Article 14.11.6},
  year      = {2014},
  note      = {No DOI assigned}
}

@article{valiant1979complexity,
  author    = {Leslie Gabriel Valiant},
  title     = {The complexity of computing the permanent},
  journal   = {Theoretical Computer Science},
  volume    = {8},
  number    = {2},
  pages     = {189--201},
  year      = {1979},
  doi       = {10.1016/0304-3975(79)90044-6}
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@article{valiant1979enumeration,
  author    = {Leslie Gabriel Valiant},
  title     = {The complexity of enumeration and reliability problems},
  journal   = {SIAM Journal on Computing},
  volume    = {8},
  number    = {3},
  pages     = {410--421},
  year      = {1979},
  doi       = {10.1137/0208032}
}

@article{waldspurger1981,
  author    = {Jean-Loup Waldspurger},
  title     = {Sur les coefficients de {F}ourier des formes modulaires de poids demi-entier},
  journal   = {Journal de Math\'{e}matiques Pures et Appliqu\'{e}es},
  volume    = {60},
  number    = {4},
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  year      = {1981},
  note      = {No DOI assigned}
}

@inproceedings{watrous2001quantum,
  author    = {John Watrous},
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  publisher = {ACM},
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  year      = {2001},
  doi       = {10.1145/380752.380759}
}

@article{zanko1991p,
  author    = {Viktoria Zank\'{o}},
  title     = {\#{P}-completeness via many-one reductions},
  journal   = {International Journal of Foundations of Computer Science},
  volume    = {2},
  number    = {1},
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  year      = {1991},
  doi       = {10.1142/S0129054191000066}
}