Appears in collection : Mathematics is a long conversation: a celebration of Barry Mazur

Let Π (resp. Σ) be a cohomological (for the trivial coefficient) cuspidal automorphic representation of GL(3) (resp. GL(2)) over a CM number field, and assume that they are base change from unitary groups. We prove the following theorem: if the Rankin-Selberg L-function L(Π×Σ, s) does not vanish at its center, then the associated ℓ-adic Bloch-Kato Selmer group vanishes (for primes ℓ where the mod ℓ Galois representations satisfy certain mild conditions). The conditions on ℓ come from Euler system type argument. We will discuss some examples from elliptic curves. This is a joint work with Yifeng Liu, Yichao Tian, Liang Xiao, and Xinwen Zhu.