Negative bias in moments of the Legendre family of elliptic curves

Ben Kane; Mikuláš Zindulka

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arXiv · 2608.00435

Negative bias in moments of the Legendre family of elliptic curves

Ben Kane·Mikuláš Zindulka

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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.

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Ben Kane, Mikuláš Zindulka. 2026-08-01. Negative bias in moments of the Legendre family of elliptic curves. https://arxiv.org/abs/2608.00435

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