Congruences with Eisenstein series and μ-invariants

Bellaïche, Joël; Pollack, Robert

doi:10.1112/S0010437X19007127

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Congruences with Eisenstein series and μ-invariants

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Pollack, Robert
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1112/S0010437X19007127
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arXiv:1806.04240.pdf
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Citation

Bellaïche, J., & Pollack, R. (2019). Congruences with Eisenstein series and μ-invariants. Compositio Mathematica, 155(5), 863-901. doi:10.1112/S0010437X19007127.

Cite as: https://hdl.handle.net/21.11116/0000-0004-8AA3-8

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.