A New Aspect of Chebyshev's Bias for Elliptic Curves over Function Fields

TL;DR

This paper investigates Chebyshev biases in prime number races for elliptic curves over function fields, revealing biases depend on the rank of the curve and utilizing the Deep Riemann Hypothesis for convergence results.

Contribution

It establishes a connection between the rank of elliptic curves and Chebyshev biases, using the Deep Riemann Hypothesis to analyze prime distributions.

Findings

Bias towards negative primes if rank > 0

Bias towards positive primes if rank = 0

Convergence of Euler product at the center under Deep Riemann Hypothesis

Abstract

This work considers 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 there exist Chebyshev biases towards being positive. The main innovation entails the convergence of the partial Euler product at the centre that follows from the Deep Riemann Hypothesis over function fields.