Applications of Moments of Dirichlet Coefficients in Elliptic Curve Families
Zoë Batterman, Aditya Jambhale, Steven J. Miller, Akash L. Narayanan, Kishan Sharma, Andrew Yang, Chris Yao
TL;DR
This work investigates how the Dirichlet-coefficient moments of elliptic curve L-functions in one-parameter families relate to arithmetic invariants such as rank and zero distributions. Building on Nagao–Tate and Michel results, it analyzes the Bias Conjecture for the second moment and tests whether non-generic families can exhibit a persistent lower-order bias; a concrete family $y^2 = x^3 + x + T^3$ provides a nuanced view, showing a positive bias on primes $p \equiv 2 \pmod{3}$ and suggesting, via numerical experiments, that the main negative bias may not always dominate. The study employs both analytic reductions (e.g., translating sextic Legendre sums to quadratics) and murmurations-inspired numerical techniques to explore bias behavior beyond closed-form second moments, highlighting the potential for refined probabilistic methods to detect subtle biases. Overall, the results reinforce the nuanced landscape of moment biases, offer partial evidence against the Strong Bias Conjecture in some non-generic cases, and point to the Weak Bias Conjecture as an open, tractable target for further exploration with enhanced computational tools.
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 closed-form expression, specifically the first lower order term which does not average to zero is on average negative. This Bias Conjecture is confirmed for many families; however, these are highly non-generic families whose resulting Legendre sums can be determined. Inspired by the recent successes by Yang-Hui He, Kyu-Hwan Lee, Thomas Oliver, Alexey Pozdnyakov and others in investigations of murmurations of elliptic curve coefficients with machine learning techniques, we pose a similar problem for trying to understand the Bias Conjecture. As a start to this program, we numerically investigate the Bias Conjecture for a family whose bias is positive for half the primes. Since the numerics do not offer conclusive evidence that negative bias for the other half is enough to overwhelm the positive bias, the Bias Conjecture cannot be verified for the family.