Density of Selmer ranks in families of even Galois representations, Wiles' formula, and global reciprocity

TL;DR

This paper studies the distribution of Selmer ranks in families of even Galois representations, using Wiles' formula and global reciprocity, revealing that primes increasing the rank are infinitely many with density 1/192.

Contribution

It provides a Galois cohomological analogue of a known theorem, explicitly computes Selmer groups at minimal level, and determines the density of primes increasing Selmer ranks.

Findings

Infinitely many primes increase the Selmer rank.

Density of such primes is 1/192.

Explicit computation of Selmer groups at minimal level.

Abstract

This paper concerns the distribution of Selmer ranks in a family of even Galois representations in even residual characteristic obtained by allowing ramification at auxiliary primes. The main result is a Galois cohomological analogue of a theorem of Friedlander, Iwaniec, Mazur and Rubin on the distribution of Selmer ranks in a family of twists of elliptic curves. The Selmer groups are constructed as prescribed by the Galois cohomological method for GL(2): At each ramified place, the local Selmer condition is the tangent space of a smooth quotient of the local deformation ring. By methods of global class field theory, the Selmer group at the minimal level is computed explicitly. 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. The proof combines Wiles' formula and the global reciprocity law. The result has implications for the algebraic structure of even deformation rings and the distribution of their presentations in families.