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Sato-Tate type distributions for families of hypergeometric varieties

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NC2W02 - Crossing the bridge: New connections in number theory and physics

Studying the statistical behavior of number theoretic quantities is presently in vogue. Here we discuss point counts over finite fields for “hypergeometric families” of elliptic curves and K3 surfaces. We obtain Sato-Tate distributions for these families, which turn out to be of SU(2) type (a.k.a. semicircular) and of O3 type (a.k.a. Batman type). To prove these results, we obtain moments for the corresponding traces of Frobenius, which we analyze using the theory of differential operators on spaces of harmonic Maass forms.

This talk is part of the Isaac Newton Institute Seminar Series series.

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