On the L-polynomials of curves over finite fields
Ballini F, Lombardo D, Verzobio M. 2025. On the L-polynomials of curves over finite fields. Proceedings of the Royal Society of Edinburgh Section A: Mathematics.
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Author
Ballini, Francesco;
Lombardo, Davide;
Verzobio, MatteoISTA 
Corresponding author has ISTA affiliation
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Abstract
We discuss, in a non-Archimedean setting, the distribution of the coefficients of L-polynomials of curves of genus g over Fq . Among other results, this allows us to prove that the Q-vector space spanned by such characteristic polynomials has dimension g + 1. We also state a conjecture about the Archimedean distribution of the number of rational points of curves over finite fields.
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Date Published
2025-02-06
Journal Title
Proceedings of the Royal Society of Edinburgh Section A: Mathematics
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Cambridge University Press
Acknowledgement
We thank Umberto Zannier for bringing the problem to our attention, for many useful suggestions, and especially for pointing out the relevance of the equidistribution results of Katz–Sarnak, noting that they imply the case q≫g0 of theorem 1.4. In addition, the first author would like to thank Umberto Zannier for his guidance during his undergraduate studies, on a topic that ultimately inspired much of the work in this article. We are grateful to J. Kaczorowski and A. Perelli for sharing their work [Reference Kaczorowski and Perelli28] before publication. We thank Christophe Ritzenthaler and Elisa Lorenzo García for their interesting comments on the first version of this article, Zhao Yu Ma for a comment about remark 3.12, and the anonymous referees for their helpful suggestions.
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Cite this
Ballini F, Lombardo D, Verzobio M. On the L-polynomials of curves over finite fields. Proceedings of the Royal Society of Edinburgh Section A: Mathematics. 2025. doi:10.1017/prm.2025.7
Ballini, F., Lombardo, D., & Verzobio, M. (2025). On the L-polynomials of curves over finite fields. Proceedings of the Royal Society of Edinburgh Section A: Mathematics. Cambridge University Press. https://doi.org/10.1017/prm.2025.7
Ballini, Francesco, Davide Lombardo, and Matteo Verzobio. “On the L-Polynomials of Curves over Finite Fields.” Proceedings of the Royal Society of Edinburgh Section A: Mathematics. Cambridge University Press, 2025. https://doi.org/10.1017/prm.2025.7.
F. Ballini, D. Lombardo, and M. Verzobio, “On the L-polynomials of curves over finite fields,” Proceedings of the Royal Society of Edinburgh Section A: Mathematics. Cambridge University Press, 2025.
Ballini F, Lombardo D, Verzobio M. 2025. On the L-polynomials of curves over finite fields. Proceedings of the Royal Society of Edinburgh Section A: Mathematics.
Ballini, Francesco, et al. “On the L-Polynomials of Curves over Finite Fields.” Proceedings of the Royal Society of Edinburgh Section A: Mathematics, Cambridge University Press, 2025, doi:10.1017/prm.2025.7.
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