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

Linfoot Number Theory Seminar

26th January 2021, 4:00 pm – 5:00 pm

Virtual Seminar, https://zoom.us/j/95992599567

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: 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 Riemann hypothesis for E. We also relate this to Nagao's conjecture. This is a recent joint work with M. Ram Murty.

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