Nuno Freitas

math.NT — Number Theory

Serre modularity over real quadratic fields

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

Assuming GRH for Dedekind zeta functions, we prove Serre’s conjecture over the real quadratic fields $\mathbf Q(\sqrt d)$ 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 $d=17$ is unconditional; for $d=33$, GRH can be replaced by the assertion that Dedekind zeta functions have no zeros with real part greater than $31/32$. The proof combines conductor reduction with local ramification estimates and global discriminant bounds. Its principal new ingredients are a modularity criterion for $2$-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.