The Iwasawa -invariant of certain elliptic curves of analytic rank zero

This paper is about the Iwasawa theory of elliptic curves over the cyclotomic Zₚ-extension Q^cyc of Q. We discuss a deep conjecture of Greenberg that if E/Q is an elliptic curve with good ordinary reduction at p, and Ep is irreducible as a Galois module, then the Selmer group of E over Q^cyc has -invariant zero. We prove new cases of Greenberg's conjecture for some elliptic curves of analytic rank 0. The proof involves studying the p-adic L-function of E. The crucial input is a new technique using the Rankin-Selberg method.