A new aspect of Chebyshev’s bias for elliptic curves over function fields

Abstract

This work addresses the prime number races for non-constant elliptic curves E E over function fields. We prove that if r a n k ( E ) > 0 \mathrm {rank}(E) > 0 , then there exist Chebyshev biases towards being negative, and otherwise there exist Chebyshev biases towards being positive. The key input is the convergence of the partial Euler product at the centre, which follows from the Deep Riemann Hypothesis over function fields.