For an abelian surface A over a number eld k, we study the limit-\uding distribution of the normalized Euler factors of the L-function of A.\udThis distribution is expected to correspond to taking characteristic poly-\udnomials of a uniform random matrix in some closed subgroup of USp(4);\udthis Sato-Tate group may be obtained from the Galois action on any Tate\udmodule of A. We show that the Sato-Tate group is limited to a particular\udlist of 55 groups up to conjugacy. We then classify A according to the\udGalois module structure on the R-algebra generated by endomorphisms of\udAQ (the Galois type), and establish a matching with the classi cation of\udSato-Tate groups; this shows that there are at most 52 groups up to con-\udjugacy which occur as Sato-Tate groups for suitable A and k, of which 34\udcan occur for k = Q. Finally, we exhibit examples of Jacobians of hyperel-\udliptic curves exhibiting each Galois type (over Q whenever possible), and\udobserve numerical agreement with the expected Sato-Tate distribution by\udcomparing moment statistics.
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