From the Birch and Swinnerton-Dyer conjecture to Nagao's conjecture

Abstract: Let E be an elliptic curve over Q, and let a_p be the Frobenius trace for each prime p. In 1965, Birch and Swinnerton-Dyer formulated a conjecture which implies the convergence of the Nagao-Mestre sum $lim_{x->infty} (1/log x) \sum_{p < x}(a_p log p)/p=-r+1/2,$ where r is the order of the zero of the L-function of E at s=1, which is predicted to be the Mordell-Weil rank of E(Q). We show that if the above limit exists, then the limit equals -r+1/2, and study the connections to the Riemann hypothesis for E. We also relate this to Nagao's conjecture for elliptic curves. Furthermore, we discuss a generalization of the above results for the Selberg classes and hence (conjecturally) for the L-function of abelian varieties, and their relations to the generalized Nagao's conjecture. This is a joint work with M. Ram Murty.

number theory

Audience: researchers in the topic

Around Frobenius distributions and related topics II