Sato–Tate distributions and Galois endomorphism modules in genus 2
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Kedlaya_Sato-Tate distributions.pdf
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Author(s)
Kedlaya, Kiran S.
Sutherland, Andrew Victor
Fite, Francesc
Rotger, Victor
Date Issued
July 2012
Journal
Compositio Mathematica
Publisher
Cambridge University Press
Citation
Fite, Francesc, Kiran S. Kedlaya, Victor Rotger, and Andrew V. Sutherland. “Sato–Tate distributions and Galois endomorphism modules in genus 2.” Compositio Mathematica 148, no. 05 (September 25, 2012): 1390-1442. © Foundation Compositio Mathematica 2012
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Final published version
Abstract
For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function 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 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 [superscript A][line over Q] (the Galois type), and establish a matching with the classification of Sato–Tate groups; this shows that there are at most 52 groups up to conjugacy which occur as Sato–Tate groups for suitable A and k, of which 34 can occur for k=ℚ. Finally, we present examples of Jacobians of hyperelliptic curves exhibiting each Galois type (over ℚ whenever possible), and observe numerical agreement with the expected Sato–Tate distribution by comparing moment statistics.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1112/s0010437x12000279
