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Session I.2 - Computational Number Theory

Wednesday, June 14, 15:30 ~ 16:00

Distribution of the number of points on curves over finite fields: new results and conjectures

Christophe Ritzenthaler

Université Rennes 1 and Université Côte d'Azur, France   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, we will see how to go beyond Katz-Sarnak theory on the distribution of curves over finite fields according to their number of rational points. In particular, we present a formula for the limits of the moments measuring the asymmetry of this distribution for curves of genus $g \geq 3$. Then, with some experimental data, we will try to convince the audience that there may be a stronger notion of convergence than the one provided by the Katz–Sarnak framework for all curves of genus $\geq 3$. This was not observed before because for elliptic curves and for hyperelliptic curves of every genus, this stronger convergence cannot occur.

Joint work with JONAS BERGSTRÖM (Stockholms Universitet), EVERETT W. HOWE and ELISA LORENZO GARCÍA (université de Neuchâtel).

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