Sato–Tate distributions and Galois endomorphism modules in genus 2
By Francesc Fit'e, Kiran S. Kedlaya, Victor Rotger, Andrew V. Sutherland. Compositio Mathematica, 2011
114 citations recorded by the source corpus.
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.
Subfields: Artificial Intelligence, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/f53159a6a766af477f9b3d4911ba0d2bf4992e9d
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Citations in the corpus
- Counting points on hyperelliptic curves in average polynomial time (2012, methodology): “Such investigations have recently been carried out by Fit´e, Kedlaya, Rotger and Sutherland for curves of genus up to three [KS08], [KS09], [FKRS12], with particularly detailed information being obtained for genus two curves.”
- On the Sato-Tate conjecture for non-generic abelian surfaces (2013, methodology): “ecture). If one believes that all motives are automorphic, then the Sato-Tate conjecture becomes a statement about the distribution of the Satake parameters of unitary automorphic representations. In [FKRS], Fite, Kedlaya, Rotger and Sutherland described the generalization of the Sato-Tate conjectur”