# Eisenstein and CM congruence modules defined over a real quadratic filed

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Speaker:
Tomomi Ozawa
Affiliation:
University Paris 13
Date:
Thu, 2017-11-23 11:10 - 12:00
Location:
MPIM Lecture Hall

Measuring congruences among modular forms over arithmetic rings has good applications
to number theory. In particular, Hida has shown in 2013 that the non-existence of the following
two types of congruences is almost equivalent to the vanishing of the $\mu$-invariants of the
Kubota-Leopoldt $p$-adic $L$-function and the Katz anti-cyclotomic $p$-adic $L$-function:
(1) a congruence mod $p$ between a $p$-adic family of Eisenstein series and a non-CM
cuspidal Hida family; (2) a congruence mod $p$ between a non-CM and a CM cuspidal
Hida family. In this talk, I will explain my attempt to describe congruence modules that
classify such types of congruences, in the case where the Hilbert modular forms are defined
over a real quadratic field of narrow ideal class number one.

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