Anticyclotomic $\mu$-invariants of residually reducible Galois Representations

Abstract

Let $E$ be an elliptic curve over an imaginary quadratic field $K$, and $p$ be an odd prime such that the residual representation $E[p]$ is reducible. The $\mu$-invariant of the fine Selmer group of $E$ over the anticyclotomic ${\mathbb{Z}}_p$-extension of $K$ is studied. We do not impose the Heegner hypothesis on $E$, thus allowing certain primes of bad reduction to decompose infinitely in the anticyclotomic ${\mathbb{Z}}_p$-extension. It is shown that the fine $\mu$-invariant vanishes if certain explicit conditions are satisfied. Further, a partial converse is proven.

16 pages, comments welcome

References in corpus (1)

• On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes

Cited by in corpus (1)

• Iwasawa Invariants for elliptic curves over ${\mathbb{Z}}_p$-extensions and Kida's Formula