Density of Selmer ranks in families of even Galois representations

Peter Vang Uttenthal

Published 2023-01-24Version 1

This paper concerns an even, reducible residual Galois representation in even characteristic. By thickening the image with cohomology classes, all lifts of the representation are ensured to be irreducible. The global reciprocity law of Galois cohomology is applied to lift the representation to mod 8, and smooth quotients of the local deformation rings at the primes where the representation is ramified are found. By using the generic smoothness of the local deformation rings at trivial primes and the Wiles-Greenberg formula, a balanced global setting is created, in the sense that the Selmer group and the dual Selmer group have the same rank. The Selmer group is computed explicitly and shown to have rank three. Finally, the distribution over primes of the ranks of Selmer groups in a family of even representations obtained by allowing ramification at auxiliary primes is studied. The infinitude of primes for which the Selmer rank increases by one is proved, and the density of such primes is shown to be 1/192.