Negative bias in moments of the Legendre family of elliptic curves

2026/08/01 by Ben Kane, Mikuláš Zindulka
Mathematics · #math.NT #msc:11E41 #msc:11F27 #msc:11F37 #msc:11G05

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36 pages, comments are welcome! Additional details contained in the source file

arxiv created 2026/08/01 · arxiv updated 2026/08/04

Abstract

We determine the bias in higher moments of the Legendre family of elliptic curves in view of the Negative Bias Conjecture of S. J. Miller. We show that every lower order term is either zero or negative on average for both even and odd moments, and explicitly compute the third and fourth moment. The results about Hurwitz class numbers in arithmetic progressions which we prove in the process may be useful for other applications.

Citations

• The bias conjecture for elliptic curves over finite fields and Hurwitz class numbers in arithmetic progressions
• Odd moments for the trace of Frobenius and the Sato--Tate conjecture in arithmetic progressions
• Distribution of values of Gaussian hypergeometric functions
• Second moments and the bias conjecture for the family of cubic pencils
• Distribution of moments of Hurwitz class numbers in arithmetic progressions and holomorphic projection
• Class numbers and representations by ternary quadratic forms with congruence conditions
• Legendre elliptic curves over finite fields
• La conjecture de Weil : I