Elliptic curves of unbounded rank and Chebyshev's bias

TL;DR

This paper explores the deep connection between the unboundedness of elliptic curve ranks and biases in prime number races, establishing conditional equivalences based on hypotheses like the Riemann Hypothesis.

Contribution

It demonstrates a conditional equivalence between unbounded elliptic curve ranks and extreme Chebyshev biases in prime number races, linking two major conjectures in number theory.

Findings

Conditional equivalence between unbounded rank and prime bias

Large ranks imply extreme Chebyshev bias under hypotheses

Existence of biased races suggests unbounded rank and RH for elliptic curves

Abstract

We establish a conditional equivalence between quantitative unboundedness of the analytic rank of elliptic curves over $\mathbb Q$ and the existence of highly biased elliptic curve prime number races. We show that conditionally on a Riemann Hypothesis and on a hypothesis on the multiplicity of the zeros of $L(E,s)$, large analytic ranks translate into an extreme Chebyshev bias. Conversely, we show under a certain linear independence hypothesis on zeros of $L(E,s)$ that if highly biased elliptic curve prime number races do exist, then the Riemann Hypothesis holds for infinitely many elliptic curve $L$-functions and there exist elliptic curves of arbitrarily large rank.