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arXiv · 2303.17825

Refinements of Katz-Sarnak theory for the number of points on curves over finite fields

Jonas Bergström·Everett W. Howe·Elisa Lorenzo García·Christophe Ritzenthaler
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Abstract

This paper goes beyond Katz-Sarnak theory on the distribution of curves over finite fields according to their number of rational points, theoretically, experimentally and conjecturally. In particular, we give a formula for the limits of the moments measuring the asymmetry of this distribution for (non-hyperelliptic) curves of genus $g \geq 3$. The experiments point to a stronger notion of convergence than the one provided by the Katz-Sarnak framework for all curves of genus $\geq 3$. However, for elliptic curves and for hyperelliptic curves of every genus we prove that this stronger convergence cannot occur.

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Jonas Bergström, Everett W. Howe, Elisa Lorenzo García, Christophe Ritzenthaler. 2023-12-14. Refinements of Katz-Sarnak theory for the number of points on curves over finite fields. https://arxiv.org/abs/2303.17825

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