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Title: Congruences with Eisenstein series and -invariants
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.  more » « less
Award ID(s):
1702178
NSF-PAR ID:
10124930
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Compositio Mathematica
Volume:
155
Issue:
5
ISSN:
0010-437X
Page Range / eLocation ID:
863 to 901
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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