Lower Order Biases in Moment Expansions of One Parameter Families of Elliptic Curves

TL;DR

This paper investigates biases in the second moment of one-parameter families of elliptic curves, creating a database and framework to analyze lower order terms, revealing potential violations of existing conjectures for large primes.

Contribution

The authors develop a systematic framework and database to analyze biases in the second moment of elliptic curve families, extending investigations beyond current theoretical limits.

Findings

Potential violations of Miller's conjecture for primes up to 250,000.

Identification of biases in lower order terms of second moments.

New conjectures inspired by empirical data.

Abstract

For a fixed elliptic curve $E$ without complex multiplication, $a_p := p+1 - \#E(\mathbb{F}_p)$ is $O(\sqrt{p})$ and $a_p/2\sqrt{p}$ converges to a semicircular distribution. Michel proved that for a one-parameter family of elliptic curves $y^2 = x^3 + A(T)x + B(T)$ with $A(T), B(T) \in \mathbb{Z}[T]$ and non-constant $j$-invariant, the second moment of $a_p(t)$ is $p^2 + O(p^{{3}/{2}})$. The size and sign of the lower order terms has applications to the distribution of zeros near the central point of Hasse-Weil $L$-functions and the Birch and Swinnerton-Dyer conjecture. S. J. Miller conjectured that the highest order term of the lower order terms of the second moment that does not average to zero is on average negative. Previous work on the conjecture has been restricted to a small set of highly nongeneric families. We create a database and a framework to quickly and systematically investigate biases in the second moment of any one-parameter family. When looking at families which have so far been beyond current theory, we find several potential violations of the conjecture for $p \leq 250,000$ and discuss new conjectures motivated by the data.