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Refinements of Katz–Sarnak theory for the number of points on curves over finite fields
Published online by Cambridge University Press: 09 January 2024
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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- © The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society