# Number Theory Seminar

Speaker: Seoyoung Kim (Queen's University)

Title: From the Birch and Swinnerton-Dyer conjecture to Nagao's conjecture
Abstract: Let E be an elliptic curve over Q, and let ap 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 limx->infty (1/log x) Σ p<x (ap 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 graphs.

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