Congruences with Eisenstein series and -invariants

Abstract

We study the variation of$\unicode[STIX]{x1D707}$-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$\unicode[STIX]{x1D707}$-invariants to be unbounded along the family, and 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. $\unicode[STIX]{x1D706}=0$). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms.