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Pré-Publication, Document De Travail Année : 2023

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

Résumé

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.

Dates et versions

hal-04062150 , version 1 (07-04-2023)

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Jonas Bergström, Everett W. Howe, Elisa Lorenzo García, Christophe Ritzenthaler. Refinements of Katz-Sarnak theory for the number of points on curves over finite fields. 2023. ⟨hal-04062150⟩
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