分野別セミナー

[講演概要]
For elliptic curves over rationals, there are a well-known conjecture of Sato-Tate and a new computational guided
murmuration phenomenon, for which the abelian Hasse-Weil zeta functions are used. In this talk, we show that both the murmurations and the Sato-Tate conjecture stand equally well for non-abelian high rank zeta functions of elliptic curves over rationals. We establish our results by carefully examining the asymptotic behaviors of the $p$-reduction
invariants $a_{E/\mathbb{F}_q}$ ($n\geq1$), the rank $n$ analogous of the rank one $a$-invariant $a_{E/\mathbb{F}_q}=1+q-N_{E/\mathbb{F}_q}$ of elliptic curve $E/\mathbb{F}_q$. Such asymptotic results are based on the counting miracle of the so-called $\alpha_{E/\mathbb{F}_q,n}$- and $\beta_{E/\mathbb{F}_q,n}$-invariants
of $E/\mathbb{F}_q$ in rank $n$, and a remarkable recursive relation on the $\beta_{E/\mathbb{F}_q,n}$-invariants, established by Weng and Zagier,
previously.
This is a joint work with Zhan Shi.