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Serre modularity over real quadratic fields

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Luis V. Dieulefait, Nuno Freitas ↗, Ariel Pacetti

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

Assuming GRH for Dedekind zeta functions, we prove Serre’s conjecture over the real quadratic fields Q(d)\mathbf Q(\sqrt d) for \[ d\in\{2,3,5,6,7,10,11,13,17,21,33,37,41,57,65,73,89\}. \] More precisely, every continuous, absolutely irreducible, totally odd two-dimensional Galois representation over these fields in odd residual characteristic is modular. The result for d=17d=17 is unconditional; for d=33d=33, GRH can be replaced by the assertion that Dedekind zeta functions have no zeros with real part greater than 31/3231/32. The proof combines conductor reduction with local ramification estimates and global discriminant bounds. Its principal new ingredients are a modularity criterion for 22-adic representations with scalar reduction and integral trace arguments that force quadratic descent or bound coefficient fields. Uniform local lifting constructions allow these methods to treat arbitrary conductor.

Provenance statement

During the preparation of this article, the authors used ChatGPT 6-Astra and Sol 6.1 as an assistant in mathematical research and manuscript preparation. Its use included exploring proof strategies, checking mathematical arguments, investigating possible simplifications, assisting with computations and verification scripts, checking references, and drafting and revising portions of the exposition.

References

  1. M. Adrian and B. Liu, https://doi.org/10.1016/j.jnt.2015.08.002 Some results on simple supercuspidal representations of $\GL_n(F)$ , J. Number Theory 160 (2016), 117--147. Consulted preprint: The local Langlands correspondence for simple supercuspidal representations of $\GL_n(F)$ , arXiv:1310.2585v4 (https://arxiv.org/abs/1310.2585v4) (2014).arXivDOI
  2. P. B. Allen, https://doi.org/10.1112/S0010437X1300780X Modularity of nearly ordinary 22-adic residually dihedral Galois representations , Compos. Math. 150 (2014), 1235--1346; arXiv:1301.1113v2 (https://arxiv.org/abs/1301.1113v2).arXivDOI
  3. P. B. Allen, C. Khare and J. A. Thorne, https://doi.org/10.1073/pnas.2108064118 Modularity of PGL2(Fp)\mathrm{PGL}_2(\mathbf F_p)-representations over totally real fields , Proc. Natl. Acad. Sci. USA 118 (2021), no. 33, e2108064118; arXiv:2104.14732v2 (https://arxiv.org/abs/2104.14732v2).arXivDOI
  4. J. Arthur and L. Clozel, Simple algebras, base change, and the advanced theory of the trace formula , Annals of Mathematics Studies 120, Princeton University Press, Princeton, NJ, 1989.
  5. E. Artin and J. Tate, Class Field Theory , Benjamin, New York, 1967; corrected reprint, AMS Chelsea Publishing 366, American Mathematical Society, Providence, RI, 2009.
  6. T. Barnet-Lamb, T. Gee and D. Geraghty, Serre weights for rank two unitary groups , Math. Ann. 356 (2013), 1551--1598.
  7. T. Barnet-Lamb, T. Gee, D. Geraghty and R. Taylor, Potential automorphy and change of weight , Ann. of Math. 179 (2014), 501--609.
  8. S. Bloch and K. Kato, LL-functions and Tamagawa numbers of motives , in The Grothendieck Festschrift , Vol. I, Progress in Mathematics 86, Birkhäuser, Boston, MA, 1990, 333--400.
  9. G. Böckle, A. Iyengar and V. Paškūnas, Zariski density of crystalline points (https://doi.org/10.1073/pnas.2221042120), Proc. Natl. Acad. Sci. USA 120 (2023), no. 13, e2221042120; arXiv:2206.06944v1 (https://arxiv.org/abs/2206.06944v1) (consulted version).arXivDOI
  10. K. Buzzard, F. Diamond and F. Jarvis, https://doi.org/10.1215/00127094-2010-052 On Serre's conjecture for mod ℓ\ell Galois representations over totally real fields , Duke Math. J. 155 (2010), 105--161; arXiv:0810.2106v5 (https://arxiv.org/abs/0810.2106v5).arXivDOI
Show all 76 referencesShow fewer references
  1. A. Caraiani, M. Emerton, T. Gee and D. Savitt, Local geometry of moduli stacks of two-dimensional Galois representations , in Arithmetic Geometry: Proceedings of the International Colloquium, Mumbai 2020 (N. Fakhruddin, E. Ghate, A. Nair, C. S. Rajan and S. Varma, eds.), Hindustan Book Agency, New Delhi, 2024, 127--201; arXiv:2207.14337 (https://arxiv.org/abs/2207.14337) (consulted version).arXiv
  2. P. Colmez, G. Dospinescu and V. Paškūnas, https://doi.org/10.4310/CJM.2014.v2.n1.a1 The pp-adic local Langlands correspondence for GL2(Qp)\mathrm{GL}_2(\mathbf Q_p) , Cambridge J. Math. 2 (2014), 1--47.DOI
  3. B. Conrad, Ramified deformation problems (https://doi.org/10.1215/S0012-7094-99-09718-1), Duke Math. J. 97 (1999), no. 3, 439--513. Author's version (https://math.stanford.edu/~conrad/papers/ramdef.pdf).DOI
  4. B. Conrad, Global extensions approximating local extensions , Math 248A handout, Stanford University, undated, 5 pp. https://math.stanford.edu/~conrad/154Page/handouts/localglobal.pdf.link
  5. D. Cureton and J. Thorner, A quasi-Riemann hypothesis for Dedekind zeta functions , Hexagon 2610.00200v2 (2026). https://hexagonmath.org/2610.00200v2.hexagon
  6. C. S. Dalawat, Further remarks on local discriminants , J. Ramanujan Math. Soc. 25 (2010), no. 4, 393--417; arXiv:0909.2541v2 (https://arxiv.org/abs/0909.2541v2).arXiv
  7. P. Deligne and J.-P. Serre, Formes modulaires de poids 1 (https://doi.org/10.24033/asens.1277), Ann. Sci. École Norm. Sup. (4) 7 (1974), 507--530.DOI
  8. L. V. Dieulefait, Existence of families of Galois representations and new cases of the Fontaine--Mazur conjecture , J. Reine Angew. Math. 577 (2004), 147--151; arXiv:math/0304433 (https://arxiv.org/abs/math/0304433). https://doi.org/10.1515/crll.2004.2004.577.147.arXivDOI
  9. L. V. Dieulefait, The level 11 weight 22 case of Serre's conjecture , Rev. Mat. Iberoamericana 23 (2007), 1115--1124. https://doi.org/10.4171/RMI/525.DOI
  10. L. V. Dieulefait, Fully ordinary subfamilies and the Fontaine--Mazur conjecture in the residually reducible mixed case , unpublished draft, 24 October 2026, second version, 11 pp.; The precise input statements used here are specified in Section external:dyadic .
  11. L. V. Dieulefait, On the Fontaine--Mazur conjecture for 22-adic representations , unpublished manuscript, 30 October 2026, 30 pp.; The precise input statements used here are specified in Section external:dyadic .
  12. L. Dieulefait and T. Gee, Automorphy lifting for small ll , arXiv:1209.5105 (https://arxiv.org/abs/1209.5105); Appendix B to L. Dieulefait, Automorphy of $\operatorname{Sym}^5(\GL(2))$ and base change , J. Math. Pures Appl. 104 (2015), 619--656.arXiv
  13. L. Dieulefait and A. Pacetti, Connectedness of Hecke algebras and the Rayuela conjecture: a path to functoriality and modularity (https://doi.org/10.1017/CBO9781316106877.013), in Arithmetic and Geometry (L. Dieulefait, G. Faltings, D. R. Heath-Brown, Yu. V. Manin, B. Z. Moroz and J.-P. Wintenberger, eds.), LMS Lecture Note Ser. 420, Cambridge University Press, Cambridge, 2015, 193--216.DOI
  14. L. V. Dieulefait and A. M. Pacetti, A simplified proof of Serre's conjecture , Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 117 (2023), article 153. https://doi.org/10.1007/s13398-023-01478-8.DOI
  15. L. Dieulefait and A. Pacetti, Serre's conjectures for Q(13)\mathbf Q(\sqrt{13}) , arXiv:2610.38884v1 (https://arxiv.org/abs/2610.38884v1), preprint, 2026.arXiv
  16. A. Dotto and J. A. Thorne, On the modularity of residually reducible Galois representations of rank 22 , arXiv:2610.41764v1 (https://arxiv.org/abs/2610.41764v1), 2026.arXiv
  17. J. S. Ellenberg, Serre's conjecture over $\F_9$ , Ann. of Math. (2) 161 (2005), 1111--1142.
  18. M. Emerton, T. Gee, L. Pan and X. Zhu, Prescribed lifts of 22-dimensional representations , arXiv:2606.25485v1 (https://arxiv.org/abs/2606.25485v1), 2026.arXiv
  19. G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern , Invent. Math. 73 (1983), 349--366.
  20. J.-M. Fontaine, Sur certains types de représentations pp-adiques du groupe de Galois d'un corps local; construction d'un anneau de Barsotti--Tate , Ann. of Math. (2) 115 (1982), 529--577.
  21. J.-M. Fontaine, Il n'y a pas de variété abélienne sur Z\Z , Invent. Math. 81 (1985), 515--538. https://www.imo.universite-paris-saclay.fr/~fontaine/varabZ.pdf.link
  22. J.-M. Fontaine and G. Laffaille, https://doi.org/10.24033/asens.1437 Construction de représentations pp-adiques , Ann. Sci. École Norm. Sup. (4) 15 (1982), 547--608.DOI
  23. N. Freitas, B. V. Le Hung and S. Siksek, Elliptic curves over real quadratic fields are modular (https://doi.org/10.1007/s00222-014-0550-z), Invent. Math. 201 (2015), 159--206; arXiv:1310.7088v4 (https://arxiv.org/abs/1310.7088v4).arXivDOI
  24. T. Gee, On the weights of mod pp Hilbert modular forms , Invent. Math. 184 (2011), 1--46.
  25. T. Gee and M. Kisin, The Breuil--Mézard conjecture for potentially Barsotti--Tate representations , Forum Math. Pi 2 (2014), e1. https://people.math.harvard.edu/~kisin/dvifiles/bm.pdf.link
  26. T. Gee, T. Liu and D. Savitt, The weight part of Serre's conjecture for GL(2)\mathrm{GL}(2) , Forum Math. Pi 3 (2015), e2, 52 pp.
  27. R. Gow and G. McGuire, On the realization of subgroups of $\PGL(2,F)$, and their automorphism groups, as Galois groups over function fields , arXiv:2109.06693v3 (https://arxiv.org/abs/2109.06693v3), 13 November 2024.arXiv
  28. C. Khare and J. A. Thorne, Automorphy of some residually S5S_5 Galois representations , Math. Z. 286 (2017), 399--429. Final-version theorem numbering is used; see the author manuscript dated 27 April 2016 (https://www.dpmms.cam.ac.uk/~jat58/quintic.pdf), Theorem 7.2 and Corollary 7.3, p. 17. The original arXiv:1503.03796v1 (https://arxiv.org/abs/1503.03796v1) has different statements and numbering at this point.arXiv
  29. C. Khare and J.-P. Wintenberger, https://doi.org/10.4007/annals.2009.169.229 On Serre's conjecture for 22-dimensional mod pp representations of Gal(Q‾/Q)\mathrm{Gal}(\overline{\mathbf Q}/\mathbf Q) , Ann. of Math. (2) 169 (2009), 229--253.DOI
  30. C. Khare and J.-P. Wintenberger, Serre's modularity conjecture. I (https://doi.org/10.1007/s00222-009-0205-7), Invent. Math. 178 (2009), 485--504.DOI
  31. C. Khare and J.-P. Wintenberger, Serre's modularity conjecture. II (https://doi.org/10.1007/s00222-009-0206-6), Invent. Math. 178 (2009), 505--586.DOI
  32. W. Kim, https://doi.org/10.4310/MRL.2012.v19.n1.a10 The classification of pp-divisible groups over 22-adic discrete valuation rings , Math. Res. Lett. 19 (2012), 121--141; arXiv:1007.1904v3 (https://arxiv.org/abs/1007.1904v3).arXivDOI
  33. M. Kisin, Modularity of 22-adic Barsotti--Tate representations , Invent. Math. 178 (2009), 587--634.
  34. M. Kisin, Moduli of finite flat group schemes, and modularity (https://doi.org/10.4007/annals.2009.170.1085), Ann. of Math. (2) 170 (2009), 1085--1180.DOI
  35. The LMFDB Collaboration, The L-functions and Modular Forms Database , https://www.lmfdb.org, accessed 2 November 2026.link
  36. H. Moon and Y. Taguchi, https://doi.org/10.4171/DMS/3/18 Refinement of Tate's discriminant bound and non-existence theorems for mod pp Galois representations , Doc. Math., Extra Volume Kato (2003), 641--654.DOI
  37. A. M. Odlyzko, Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results , J. Théor. Nombres Bordeaux 2 (1990), 119--141. doi:10.5802/jtnb.22 (https://doi.org/10.5802/jtnb.22).DOI
  38. OpenAI, Fontaine--Mazur modularity at the prime 22 , preprint, 6 October 2026, version 4eaee8754825 . https://github.com/openai/math/tree/4eaee87548250b2682affda7d05d947c05554bae/preprints/Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026.link
  39. A. Pacetti, L. V. Dieulefait and N. Freitas, Computations for Serre modularity over real quadratic fields , computer programs and exact certificates, version 2026-11-03, ancillary code archive for the present paper.
  40. L. Pan, The Fontaine--Mazur conjecture in the residually reducible case (https://doi.org/10.1090/jams/991), J. Amer. Math. Soc. 35 (2022), 1031--1169; arXiv:1901.07166v2 (https://arxiv.org/abs/1901.07166v2).arXivDOI
  41. R. Paterson, https://doi.org/10.1017/S0305004124000197 The failure of Galois descent for pp-Selmer groups of elliptic curves , Math. Proc. Cambridge Philos. Soc. 177 (2024), 185--218.DOI
  42. G. Poitou, Sur les petits discriminants , Séminaire Delange--Pisot--Poitou, Théorie des nombres 18 , no. 1 (1976--1977), exposé 6, 1--18.
  43. M. Raynaud, https://doi.org/10.24033/bsmf.1779 Schémas en groupes de type (p,…,p)(p,\ldots,p) , Bull. Soc. Math. France 102 (1974), 241--280.DOI
  44. K. A. Ribet, https://doi.org/10.1007/978-3-0348-7919-4_15 Abelian varieties over $\Q$ and modular forms , in Modular Curves and Abelian Varieties (J. E. Cremona, J.-C. Lario, J. Quer and K. A. Ribet, eds.), Progress in Math. 224, Birkhäuser, Basel, 2004, 241--261; arXiv:alg-geom/9208002 (https://arxiv.org/abs/alg-geom/9208002).arXivDOI
  45. R. Schoof, Abelian varieties over cyclotomic fields with good reduction everywhere (https://doi.org/10.1007/s00208-002-0368-7), Math. Ann. 325 (2003), 413--448.DOI
  46. R. Schoof, https://doi.org/10.1112/S0010437X05001107 Abelian varieties over $\Q$ with bad reduction in one prime only , Compos. Math. 141 (2005), 847--868.DOI
  47. J.-P. Serre, Abelian ℓ\ell-adic representations and elliptic curves , Benjamin, New York, 1968.
  48. J.-P. Serre, Valeurs propres des opérateurs de Hecke modulo ℓ\ell , in Journées arithmétiques de Bordeaux, Astérisque 24--25 (1975), 109--117. https://numdam.org/item/AST_1975__24-25__109_0/.link
  49. J.-P. Serre, Local Fields (https://doi.org/10.1007/978-1-4757-5673-9), translated by M. J. Greenberg, Graduate Texts in Mathematics 67, Springer, New York, 1979.DOI
  50. J.-P. Serre, Sur les représentations modulaires de degré 22 de Gal(Q‾/Q)\mathrm{Gal}(\overline{\mathbf Q}/\mathbf Q) , Duke Math. J. 54 (1987), 179--230.
  51. C. Skinner, https://doi.org/10.4171/DM/272 A note on the pp-adic Galois representations attached to Hilbert modular forms , Doc. Math. 14 (2009), 241--258.DOI
  52. C. Skinner and A. Wiles, Residually reducible representations and modular forms (https://doi.org/10.1007/BF02698855), Publ. Math. IHES 89 (1999), 5--126.DOI
  53. C. Skinner and A. Wiles, Nearly ordinary deformations of irreducible residual representations (https://doi.org/10.5802/afst.988), Ann. Fac. Sci. Toulouse Math. (6) 10 (2001), 185--215.DOI
  54. A. Snowden, On two dimensional weight two odd representations of totally real fields , preprint, 2009, arXiv:0905.4266v1 (https://arxiv.org/abs/0905.4266v1).arXiv
  55. The Stacks Project Authors, The Stacks Project , online reference, https://stacks.math.columbia.edu/, accessed 3 November 2026.link
  56. R. Taylor, On the meromorphic continuation of degree two LL-functions , Doc. Math., Extra Volume Coates (2006), 729--779.
  57. R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras (https://doi.org/10.2307/2118560), Ann. of Math. (2) 141 (1995), 553--572.DOI
  58. J. A. Thorne, Automorphy of some residually dihedral Galois representations (https://doi.org/10.1007/s00208-015-1214-z), Math. Ann. 364 (2016), 589--648; arXiv:1504.00994 (https://arxiv.org/abs/1504.00994).arXivDOI
  59. J. A. Thorne, https://doi.org/10.1007/s00209-016-1681-2 A 22-adic automorphy lifting theorem for unitary groups over CM fields , Math. Z. 285 (2017), 1--38.DOI
  60. S.-N. Tung, https://doi.org/10.2140/ant.2021.15.2173 On the automorphy of 22-dimensional potentially semistable deformation rings of $G_{\Q_p}$ , Algebra Number Theory 15 (2021), 2173--2194; arXiv:1803.07451v4 (https://arxiv.org/abs/1803.07451v4).arXivDOI
  61. S.-N. Tung, https://doi.org/10.1007/s00209-020-02588-4 On the modularity of 22-adic potentially semi-stable deformation rings , Math. Z. 298 (2021), 107--159; https://arxiv.org/abs/1908.06174.DOI
  62. P. Wake and C. Wang-Erickson, Deformation conditions for pseudorepresentations (https://doi.org/10.1017/fms.2019.19), Forum Math. Sigma 7 (2019), e20. Author's version (https://sites.pitt.edu/~caw203/pdfs/PG4.pdf).DOI
  63. A. Wiles, https://doi.org/10.1007/BF01394275 On ordinary λ\lambda-adic representations associated to modular forms , Invent. Math. 94 (1988), 529--573.DOI
  64. A. Wiles, Modular elliptic curves and Fermat's Last Theorem (https://doi.org/10.2307/2118559), Ann. of Math. (2) 141 (1995), 443--551.DOI
  65. J.-P. Wintenberger, Le corps des normes de certaines extensions infinies de corps locaux; applications (https://doi.org/10.24033/asens.1440), Ann. Sci. École Norm. Sup. (4) 16 (1983), 59--89.DOI
  66. X. Zhang, On the Fontaine--Mazur conjecture for p=3p=3 , arXiv:2412.06812v5 (https://arxiv.org/abs/2412.06812v5), 7 October 2026.arXiv

Version history

  1. v1Submitted by Nuno FreitasInitial depositCurrentNov 03, 2026