• DOI:10.1112/S0010437X12000279
• Corpus ID: 256029

Sato–Tate distributions and Galois endomorphism modules in genus 2

@article{Fite2012SatoTateDA,
  title={Sato–Tate distributions and Galois endomorphism modules in genus 2},
  author={Francesc Fit'e and Kiran S. Kedlaya and V{\'i}ctor Rotger and Andrew V. Sutherland},
  journal={Compositio Mathematica},
  year={2012},
  volume={148},
  pages={1390 - 1442}
}

• Francesc Fit'e, K. Kedlaya, +1 author Andrew V. Sutherland
• Published 30 October 2011
• Mathematics
• Compositio Mathematica

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 $A_{{\overline {\mathbb Q}}}$ (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.