2011/10/31 by Francesc Fité, Kiran S. Kedlaya, Víctor Rotger +2 · 1 citation
Computer Science · Mathematics · #Abelian group #Absolute Galois group #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Coding theory and cryptography #Combinatorics #Conjugacy class #Discrete mathematics #Distribution (mathematics) #Endomorphism #Galois group #Galois module #Group (periodic table) #Mathematical analysis #Mathematics #Pure mathematics #Type (biology) #math.AG #math.NT #msc:11G10 #msc:11G20 #msc:11M50 #msc:14G10 #msc:14K15
paper · pdf · doi:10.1112/s0010437x12000279
published as Compositio Math. 148 (2012) 1390-1442 · 59 pages, 2 figures, minor edits, to appear in Compositio Mathematica
arxiv created 2012/01/27 · openalex publication_date 2012/07/25 · arxiv updated 2019/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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_ \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.