Picard ranks of reductions of K3 surfaces over global fields

Abstract: For a K3 surface $X$ over a number field with potentially good reduction everywhere, we prove that there are infinitely many primes modulo which the reduction of $X$ has larger geometric Picard rank than that of the generic fiber $X$. A similar statement still holds true for ordinary K3 surfaces with potentially good reduction everywhere over global function fields. In this talk, I will present the proofs via the (arithmetic) intersection theory on good integral models (and its special fibers) of $\mathrm{GSpin}$ Shimura varieties. These results are generalizations of the work of Charles on exceptional isogenies between reductions of a pair of elliptic curves. This talk is based on joint work with Ananth Shankar, Arul Shankar, and Salim Tayou and with Davesh Maulik and Ananth Shankar.

number theory

Audience: researchers in the topic

London number theory seminar

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For a record of talks predating this website see: wwwf.imperial.ac.uk/~buzzard/LNTS/numbtheo_past.html