Chebyshev's Bias for Elliptic Curves

Ikuya Kaneko, Shin-ya Koyama

Published 2022-06-11Version 1

This article studies the prime number races for non-constant elliptic curves $E$ over function fields. We prove that if $\mathrm{rank}(E) > 0$, then there exist Chebyshev biases towards being negative, and otherwise it is shown under the Birch-Swinnerton-Dyer conjecture that there exist Chebyshev biases towards being positive. The proof uses the convergence of the Euler product at the centre implied by the Deep Riemann Hypothesis over function fields.