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Session I.2 - Computational Number Theory

Tuesday, June 13, 16:30 ~ 17:00

Frobenius distributions on K3 surfaces

David Kohel

Institut de Mathématiques de Marseille, France   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $S$ be a K3 surface over $\mathbb{Q}$. The zeta function of the reduction of $S$ over $\mathbb{F}_p$ takes the form $$ \frac{1}{(1-T)(1-pT)(1-p^2T)p^{-1}L(pT)} $$ where $L(T) \in \mathbb{Z}[T]$ is of degree $21$, with $L(0) = p$ and all roots on the unit circle. The polynomial $L(T)$ factors as $$ L(T) = L_{\mathrm{alg}}(T)L_{\mathrm{trc}}(T) $$ into the algebraic and transcendental parts, where the roots of former are roots of unity. The polynomial $L(T)$ carries the information of the image of Frobenius in the Sato-Tate group associated to $S$. We describe the characterization of this Galois representation associated to $S$, in terms of character theory of orthogonal groups.

Joint work with Kiran Kedlaya.

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