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Serre modularity over real quadratic fields

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

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

Version 2 / Nov 04, 2026 / CC BY 4.0

Abstract

We prove Serre's modularity conjecture, in every odd residual characteristic and with arbitrary conductor, over twenty real quadratic fields $K=\Q(\sqrt d)$. The proofs for d=2,5,13,17,41d=2,5,13,17,41 use classical discriminant bounds. For d=33d=33 the proof uses the zero-free region ℜs>31/32\Re s>31/32 of Cureton and Thorner, and for d=6d=6 the region ℜs>0.9586\Re s>0.9586, which we derive from specified analytic inputs. For the remaining thirteen fields we assume GRH for Dedekind zeta functions. The main tools are conductor reductions at an odd split prime and at two, a weight-reduction step by congruences, a scalar dyadic modularity criterion, and integral trace orders. We also use a modularity theorem for the mixed ordinary--nonordinary case and, for the seven fields in which two splits, dyadic propagation results.

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.

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Version history

  1. v1 · 156 pagesSubmitted by Nuno FreitasInitial depositSupersededNov 03, 2026
  2. v2 · 235 pagesSubmitted by Nuno FreitasSubmitted as a new versionCurrentNov 11, 2026