$\Lambda$-adic Euler characteristics of elliptic curves

Abstract

Let $E_{/\mathbb{Q}}$ be a modular elliptic curve, and $p>3$ a good ordinary or semistable prime. Under mild hypotheses, we prove an exact formula for the $\mu$-invariant associated to the weight-deformation of the Tate module of $E$. For example, at ordinary primes in the range $3<p<100$, the result implies the triviality of the $\mu$-invariant of $X_0(11)$.