Serre modularity over real quadratic fields

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.