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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