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SatoTate distributions and Galois endomorphism modules in genus 2
 Compos. Math
"... For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the Lfunction of A. This distribution is expected to correspond to taking characteristic polynomials of a uniform random matrix in some closed subgroup of USp(4); this SatoTate g ..."
Abstract

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For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the Lfunction of A. This distribution is expected to correspond to taking characteristic polynomials of a uniform random matrix in some closed subgroup of USp(4); this SatoTate group may be obtained from the Galois action on any Tate module of A. We show that the SatoTate group is limited to a particular list of 55 groups up to conjugacy. We then classify A according to the Galois module structure on the Ralgebra generated by endomorphisms of AQ (the Galois type), and establish a matching with the classification of SatoTate groups; this shows that there are at most 52 groups up to conjugacy which occur as SatoTate groups for suitable A and k, of which 34 can occur for k = Q. Finally, we exhibit examples of Jacobians of hyperelliptic curves exhibiting each Galois type (over Q whenever possible), and observe numerical agreement with the expected SatoTate distribution by comparing moment statistics. 1
ALGEBRAIC TWISTS OF MODULAR FORMS AND HECKE ORBITS
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