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  primaryClass = {math.NT},
  url = {https://annals.math.princeton.edu/2026/203-1/p03},
  note = {Theorem 7.1.2 is cited in arXiv:2209.06366v1}
}

@article{Serre1987,
  author = {Serre, Jean-Pierre},
  title = {Sur les repr{\'e}sentations modulaires de degr{\'e} 2 de
    {$\operatorname{Gal}(\overline{\mathbf Q}/\mathbf Q)$}},
  journal = {Duke Mathematical Journal},
  volume = {54}, number = {1}, year = {1987}, pages = {179--230},
  doi = {10.1215/S0012-7094-87-05413-5}
}

@article{Wiles1995,
  author = {Wiles, Andrew},
  title = {Modular elliptic curves and {Fermat}'s last theorem},
  journal = {Annals of Mathematics},
  volume = {141}, number = {3}, year = {1995}, pages = {443--551},
  doi = {10.2307/2118559}
}

@article{SkinnerWiles1999,
  author = {Skinner, Christopher M. and Wiles, Andrew J.},
  title = {Residually reducible representations and modular forms},
  journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
  volume = {89}, year = {1999}, pages = {5--126},
  doi = {10.1007/BF02698855}
}

@article{SkinnerWiles2001,
  author = {Skinner, Christopher M. and Wiles, Andrew J.},
  title = {Nearly ordinary deformations of irreducible residual representations},
  journal = {Annales de la Facult{\'e} des Sciences de Toulouse. Math{\'e}matiques},
  series = {6}, volume = {10}, number = {1}, year = {2001}, pages = {185--215},
  url = {https://www.numdam.org/item/AFST_2001_6_10_1_185_0/}
}

@article{Allen2014,
  author = {Allen, Patrick B.},
  title = {Modularity of nearly ordinary 2-adic residually dihedral {Galois} representations},
  journal = {Compositio Mathematica},
  volume = {150}, number = {8}, year = {2014}, pages = {1235--1346},
  doi = {10.1112/S0010437X1300780X},
  url = {https://arxiv.org/abs/1301.1113v2},
  note = {Manuscript locators refer to arXiv:1301.1113v2}
}

@article{Kisin2009FM,
  author = {Kisin, Mark},
  title = {The {Fontaine--Mazur} conjecture for {$\mathrm{GL}_2$}},
  journal = {Journal of the American Mathematical Society},
  volume = {22}, number = {3}, year = {2009}, pages = {641--690},
  doi = {10.1090/S0894-0347-09-00628-6}
}

@misc{Emerton2011,
  author = {Emerton, Matthew},
  title = {Local-global compatibility in the {$p$}-adic {Langlands} programme for {$\mathrm{GL}_2/\mathbf Q$}},
  year = {2011},
  note = {Author draft, 23 March 2011},
  url = {https://math.uchicago.edu/~emerton/pdffiles/lg.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}
}

@article{BillereyMenares2016,
  author = {Billerey, Nicolas and Menares, Ricardo},
  title = {On the modularity of reducible mod {$l$} {Galois} representations},
  journal = {Mathematical Research Letters},
  volume = {23},
  number = {1},
  pages = {15--41},
  year = {2016},
  url = {https://lmbp.uca.fr/~billerey/Research/Published/Menares_Modularity.pdf}
}

@article{DeligneSerre1974,
  author = {Deligne, Pierre and Serre, Jean-Pierre},
  title = {Formes modulaires de poids {1}},
  journal = {Annales scientifiques de l'{\'{E}}cole Normale Sup{\'e}rieure},
  series = {4},
  volume = {7},
  number = {4},
  year = {1974},
  pages = {507--530},
  doi = {10.24033/asens.1277},
  url = {https://www.numdam.org/articles/10.24033/asens.1277/}
}

@incollection{Mazur1989,
  author = {Mazur, Barry},
  title = {Deforming {Galois} representations},
  booktitle = {Galois groups over {$\mathbb{Q}$}},
  editor = {Ihara, Yasutaka and Ribet, Kenneth A. and Serre, Jean-Pierre},
  series = {Mathematical Sciences Research Institute Publications},
  volume = {16},
  publisher = {Springer},
  address = {New York},
  year = {1989},
  pages = {385--437},
  doi = {10.1007/978-1-4613-9649-9_7},
  url = {https://doi.org/10.1007/978-1-4613-9649-9_7}
}


@incollection{Serre1964Demushkin,
  author = {Serre, Jean-Pierre},
  title = {Structure de certains pro-{$p$}-groupes},
  booktitle = {S{\'e}minaire Bourbaki: ann{\'e}es 1962/63--1963/64, expos{\'e}s 241--276},
  series = {S{\'e}minaire Bourbaki},
  number = {8},
  year = {1964},
  pages = {145--155},
  note = {Expos{\'e} 252, February 1963},
  publisher = {Soci{\'e}t{\'e} math{\'e}matique de France},
  url = {https://www.numdam.org/item/SB_1962-1964__8__145_0/}
}

@misc{OpenAIH10Q2026,
  author = {{OpenAI}},
  title = {{Hilbert's tenth problem over the rational numbers}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026/main.pdf}{OAI:Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026}},
  year = {2026},
  note = {Section~4}
}

@article{Ramakrishna2002,
  author = {Ramakrishna, Ravi},
  title = {Deforming {Galois} representations and the conjectures of {Serre} and {Fontaine--Mazur}},
  journal = {Annals of Mathematics},
  volume = {156},
  number = {1},
  year = {2002},
  pages = {115--154},
  doi = {10.2307/3597186},
  url = {https://annals.math.princeton.edu/2002/156-1/p04}
}

@article{Tang2019,
  author = {Tang, Shiang},
  title = {Algebraic monodromy groups of {$l$}-adic representations of {$\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)$}},
  journal = {Algebra \& Number Theory},
  volume = {13},
  number = {6},
  year = {2019},
  doi = {10.2140/ant.2019.13.1353},
  url = {https://msp.org/ant/2019/13-6/ant-v13-n6-p05-s.pdf}
}