Wieferich Primes and a mod $p$ Leopoldt Conjecture

TL;DR

This paper explores a Galois cohomological analog of Wieferich primes and formulates a mod p version of the Leopoldt conjecture, connecting automorphic forms, Galois representations, and number field regulators.

Contribution

It introduces a new analog relating Galois cohomology, Wieferich primes, and a mod p version of the Leopoldt conjecture, extending classical number theory conjectures.

Findings

Proposes a Galois cohomological analog for Wieferich primes.

Formulates a mod p analog of the Leopoldt conjecture for almost all primes.

Connects automorphic forms with deformation theory of Galois representations.

Abstract

We consider questions in Galois cohomology which arise by considering mod $p$ Galois representations arising from automorphic forms. We consider a Galois cohomological analog for the standard heuristics about the distribution of Wieferich primes, i.e. prime $p$ such that $2^{p-1}$ is 1 mod $p^2$. Our analog relates to asking if in a compatible system of Galois representations, for almost all primes $p$, the residual mod $p$ representation arising from it has unobstructed deformation theory. This analog leads in particular to formulating a mod $p$ analog for almost all primes $p$ of the classical Leopoldt conjecture, which has been considered previously by G. Gras. Leopoldt conjectured that for a number field $F$, and a prime $p$, the $p$-adic regulator $R_{F,p}$ is non-zero. The mod $p$ analog is that for a fixed number field $F$, for almost all primes $p$, the $p$-adic regulator $R_{F,p}$ is a unit at $p$.