Applications of Moments of Dirichlet Coefficients in Elliptic Curve Families
Zo\"e Batterman, Aditya Jambhale, Steven J. Miller, Akash L., Narayanan, Kishan Sharma, Andrew Yang, and Chris Yao

TL;DR
This paper explores the moments of elliptic curve L-function coefficients, their relation to arithmetic properties, and investigates the Bias Conjecture through numerical analysis of specific families, linking number theory with machine learning approaches.
Contribution
It introduces a numerical approach to study the Bias Conjecture in elliptic curve families, inspired by recent machine learning investigations.
Findings
Confirmed the Bias Conjecture for certain families with known Legendre sums.
Numerical evidence suggests mixed bias behavior in the studied family.
Highlights the complexity of bias effects in elliptic curve coefficient moments.
Abstract
The moments of the coefficients of elliptic curve L-functions are related to numerous arithmetic problems. Rosen and Silverman proved a conjecture of Nagao relating the first moment of one-parameter families satisfying Tate's conjecture to the rank of the corresponding elliptic surface over Q(T); one can also construct families of moderate rank by finding families with large first moments. Michel proved that if j(T) is not constant, then the second moment of the family is of size p^2 + O(p^(3/2)); these two moments show that for suitably small support the behavior of zeros near the central point agree with that of eigenvalues from random matrix ensembles, with the higher moments impacting the rate of convergence. In his thesis, Miller noticed a negative bias in the second moment of every one-parameter family of elliptic curves over the rationals whose second moment had a calculable…
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Taxonomy
TopicsAnalytic Number Theory Research
