Elliptic Curves of Unbounded Rank and Chebyshev's Bias

Abstract

We establish a conditional equivalence between quantitative unboundedness of the analytic rank of elliptic curves over 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.