Congruences with Eisenstein series and mu-invariants

TL;DR

This paper investigates the behavior of mu-invariants in Hida families with reducible Galois representations, establishing bounds, conjecturing their equality, and proving main conjectures in specific cases.

Contribution

It provides a lower bound for mu-invariants in Hida families, proves the conjecture of their equality when certain conditions hold, and confirms the main conjecture for related Selmer groups.

Findings

Mu-invariants are unbounded along the family.

The lower bound often equals the p-adic zeta function.

Main conjecture proven when U_p-1 generates the Eisenstein ideal.

Abstract

We study the variation of mu-invariants in Hida families with residually reducible Galois representations. We prove a lower bound for these invariants which is often expressible in terms of the p-adic zeta function. This lower bound forces these mu-invariants to be unbounded along the family, and moreover, we conjecture that this lower bound is an equality. When U_p-1 generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the p-adic L-function is simply a power of p up to a unit (i.e. lambda=0). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms.