Elliptic Curves of Unbounded Rank and Chebyshev's Bias

被引:12
作者
Fiorilli, Daniel [1 ]
机构
[1] Univ Michigan, Dept Math, Ann Arbor, MI 48109 USA
基金
加拿大自然科学与工程研究理事会;
关键词
NUMBER;
D O I
10.1093/imrn/rnt103
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We establish a conditional equivalence between quantitative unboundedness of the analytic rank of elliptic curves over 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.
引用
收藏
页码:4997 / 5024
页数:28
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