Geodesic
Refinements of Katz–Sarnak theory for the number of points on curves over finite fields
Canadian journal of mathematics, Tome 77 (2025) no. 2, pp. 400-425

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DOI

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.
DOI : 10.4153/S0008414X2400004X
Mots-clés : Katz–Sarnak theory, distribution, moments, Serre’s obstruction 
Bergström, Jonas; Howe, Everett W.; García, Elisa Lorenzo; Ritzenthaler, Christophe. Refinements of Katz–Sarnak theory for the number of points on curves over finite fields. Canadian journal of mathematics, Tome 77 (2025) no. 2, pp. 400-425. doi: 10.4153/S0008414X2400004X
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     title = {Refinements of {Katz{\textendash}Sarnak} theory for the number of points on curves over finite fields},
     journal = {Canadian journal of mathematics},
     pages = {400--425},
     year = {2025},
     volume = {77},
     number = {2},
     doi = {10.4153/S0008414X2400004X},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X2400004X/}
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