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

Seoyoung Kim (Queen's University)

10-Aug-2020, 16:30-17:00 (4 years ago)

Abstract: Let $E$ be an elliptic curve over $\mathbb{Q}$ with discriminant, and let $a_p$ be the Frobenius trace for each prime p. In 1965, Birch and Swinnerton-Dyer formulated a conjecture which implies

$\lim\limits_{x \rightarrow \infty} \frac{1}{\log x} \sum_{p< x} \frac{a_p\log p}{p}=-r+\frac{1}{2},$

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 $E(\mathbb{Q})$. We show that if the above limit exits, then the limit equals $-r+\frac{1}{2}$, 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.

number theory

Audience: researchers in the discipline

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POINT: New Developments in Number Theory

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Organizers: Jessica Fintzen*, Karol Koziol*, Joshua Males*, Aaron Pollack, Manami Roy*, Soumya Sankar*, Ananth Shankar*, Vaidehee Thatte*, Charlotte Ure*
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