Sato-Tate distributions and Galois endomorphism modules in genus 2
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For an abelian surface A over a number eld k, we study the limit- ing distribution of the normalized Euler factors of the L-function of A. This distribution is expected to correspond to taking characteristic poly- nomials of a uniform random matrix in some closed subgroup of USp(4); this Sato-Tate group may be obtained from the Galois action on any Tate module of A. We show that the Sato-Tate 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 R-algebra generated by endomorphisms of AQ (the Galois type), and establish a matching with the classi cation of Sato-Tate groups; this shows that there are at most 52 groups up to con- jugacy which occur as Sato-Tate groups for suitable A and k, of which 34 can occur for k = Q. Finally, we exhibit examples of Jacobians of hyperel- liptic curves exhibiting each Galois type (over Q whenever possible), and observe numerical agreement with the expected Sato-Tate distribution by comparing moment statistics.
CitationFité, F. [et al.]. Sato-Tate distributions and Galois endomorphism modules in genus 2. "Compositio mathematica", 2012, vol. 148, núm. 5, p. 1390-1442.
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