Elliptic curves of unbounded rank and Chebyshev's bias

Daniel Fiorilli

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arXiv · 1304.8011

Elliptic curves of unbounded rank and Chebyshev's bias

Daniel Fiorilli

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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.

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Daniel Fiorilli. 2013-04-30. Elliptic curves of unbounded rank and Chebyshev's bias. https://arxiv.org/abs/1304.8011

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