Anticyclotomic $\largeμ$-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 µ-invariant of the fine Selmer group of E over the anticyclotomic 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 Z p -extension. It is shown that the fine µ-invariant vanishes if certain explicit conditions are satisfied. Further, a partial converse is proven.