Congruences with Eisenstein series and mu-invariants
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Citation (published version)Joël Bellaïche, Robert Pollack. 2019. "Congruences with Eisenstein series and mu-invariants." Compositio Mathematica, Volume 155, Issue 5, pp. 863 - 901. https://doi.org/10.1112/S0010437X19007127
We study the variation of -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 -adic zeta function. This lower bound forces these -invariants to be unbounded along the family, and we conjecture that this lower bound is an equality. When generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the -adic -function is simply a power of up to a unit (i.e. ). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms.