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arXiv · 1712.07105

Sato-Tate distributions of twists of the Fermat and the Klein quartics

Francesc Fité·Elisa Lorenzo García·Andrew V. Sutherland
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Abstract

We determine the limiting distribution of the normalized Euler factors of an abelian threefold A defined over a number field k when A is geometrically isogenous to the cube of a CM elliptic curve defined over k. As an application, we classify the Sato-Tate distributions of the Jacobians of twists of the Fermat and Klein quartics, obtaining 54 and 23, respectively, and 60 in total. We encounter a new phenomenon not visible in dimensions 1 or 2: the limiting distribution of the normalized Euler factors is not determined by the limiting distributions of their coefficients.

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Francesc Fité, Elisa Lorenzo García, Andrew V. Sutherland. 2017-12-19. Sato-Tate distributions of twists of the Fermat and the Klein quartics. https://doi.org/10.1007/s40687-018-0162-0

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