## $\Lambda$-adic Euler characteristics of elliptic curves

### Summary

Let $E_{/\Bbb Q}$ 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 $X_0(11)$.

### Mathematics Subject Classification

11R23, 11F80, 11G05

### Keywords/Phrases

Euler characteristics, Selmer groups, Tamagawa numbers