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

Part of: Algebraic number theory: global fields Arithmetic algebraic geometry Curves

Published online by Cambridge University Press:  09 January 2024

Jonas Bergström Open the ORCID record for Jonas Bergström [Opens in a new window] ,
Everett W. Howe Open the ORCID record for Everett W. Howe [Opens in a new window] ,
Elisa Lorenzo García Open the ORCID record for Elisa Lorenzo García [Opens in a new window]  and
Christophe Ritzenthaler
Jonas Bergström
Affiliation:
Department of Mathematics, Stockholms Universitet, Stockholm, Sweden e-mail: jonasb@math.su.se
Everett W. Howe
Affiliation:
Independent mathematician, San Diego, CA, United States e-mail: however@alumni.caltech.edu
Elisa Lorenzo García*
Affiliation:
Faculté des sciences, Institut de Mathématiques, Université de Neuchâtel, Neuchâtel, Switzerland
Christophe Ritzenthaler
Affiliation:
Laboratoire J.A. Dieudonné, Université Côte d’Azur, Nice, France e-mail: christophe.ritzenthaler@univ-rennes1.fr
*
e-mail: elisa.lorenzo@unine.ch
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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.

Keywords

Katz–Sarnak theorydistributionmomentsSerre’s obstruction

MSC classification

Primary: 11G20: Curves over finite and local fields 11R45: Density theorems 14H10: Families, moduli (algebraic) 14H25: Arithmetic ground fields

Information

Type
Article
Information
Canadian Journal of Mathematics , Volume 77 , Issue 2 , April 2025 , pp. 400 - 425
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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