Serre modularity over real quadratic fields
Source: hexagonmath.org/abs/2611.00071v1
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…
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