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 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.

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Let S be a smooth cubic surface over a finite field F q . It is known that #S(F q ) = 1 + aq + q 2 for some a ∈ {−2, −1, 0, 1, 2, 3, 4, 5, 7}. Serre has asked which values of a can arise for a given q. Building on special cases treated by Swinnerton-Dyer, we give a complete answer to this question. We also answer the analogous question for other del Pezzo surfaces, and consider the inverse Galois problem for del Pezzo surfaces over finite fields. Finally we give a corrected version of Manin's and Swinnerton-Dyer's tables on cubic surfaces over finite fields.Note that a smooth cubic surface S with a(S) = 7 is split, i.e. all its lines are defined over F q . It has been known for a long time, prior to the work of Swinnerton-Dyer [37], that such a surface exists over F q if and only if q = 2, 3, 5; this result appears to be first due to Hirschfeld [16, Thm. 20.1.7].We briefly explain the proof of Theorem 1.1. Any smooth cubic surface over an algebraically closed field is the blow-up of P 2 in 6 rational points in general position. Whilst this does not hold over other fields in general, the trace values 1, 2, 3, 4, 5 can be obtained from cubic surfaces which are blow-ups of P 2 in collections of closed points in general position. We therefore show that such collections exist over every finite field F q , via combinatorial arguments. Trace 0 can be obtained by blowing-up certain collections of closed points of total degree 7, and contracting a line. The existence of the remaining traces −1, −2 can be deduced from work of Rybakov [29] and Swinnerton-Dyer [37], respectively. 1.2. Corrections to Manin's and Swinnerton-Dyer's tables. Let S be a smooth cubic surface over a finite field F q . Building on work of Frame [12] and Swinnerton-Dyer [36], Manin constucted a table (Table 1 of [27, p. 176]) of the conjugacy classes of W (E 6 ) and their properties, such as the trace a(S).Urabe [39,40] was the first to notice that Manin's table contains mistakes regarding the calculation of H 1 (F q , PicS) (the issue being that H 1 (F q , PicS) must have square order). In our investigation we found some new mistakes. These concern the Galois orbit on the lines, where the mistake can be traced back to , and the index [27, §28.2] of the surface. The index is the size of the largest Galois invariant collection of pairwise skew lines overF q .Manin's table has a surface of index 2, which led him to state [27, Thm. 28.5(i)] that the index can only take one of the values 0, 1, 2, 3, 6. Our investigations reveal, however, that index 2 does not occur, and that index 5 can occur, hence the correct statement is the following. Theorem 1.2. Let S be a smooth cubic surface over a finite field. Then the index of S can only take one of the values 0, 1, 3, 5, 6.

Abstract. Let C denote the Fermat curve over Q of prime exponent ℓ. e Jacobian Jac(C) of C splits over Q as the product of Jacobians Jac(C k ), ≤ k ≤ ℓ − , where C k are curves obtained as quotients of C by certain subgroups of automorphisms of C. It is well known that Jac(C k ) is the power of an absolutely simple abelian variety B k with complex multiplication. We call degenerate those pairs (ℓ, k) for which B k has degenerate CM type. For a non-degenerate pair (ℓ, k), we compute the Sato-Tate group of Jac(C k ), prove the generalized Sato-Tate Conjecture for it, and give an explicit method to compute the moments and measures of the involved distributions. Regardless of (ℓ, k) being degenerate or not, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the ℓ-th cyclotomic eld. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.

Abstract. Let A/Q be an abelian variety of dimension g ≥ 1 that is isogenous over Q to E g , where E is an elliptic curve. If E does not have complex multiplication (CM), by results of Ribet and Elkies concerning fields of definition of elliptic Q-curves E is isogenous to a curve defined over a polyquadratic extension of Q. We show that one can adapt Ribet's methods to study the field of definition of E up to isogeny also in the CM case. We find two applications of this analysis to the theory of Sato-Tate groups: First, we show that 18 of the 34 possible Sato-Tate groups of abelian surfaces over Q occur among at most 51 Q-isogeny classes of abelian surfaces over Q; Second, we give a positive answer to a question of Serre concerning the existence of a number field over which abelian surfaces can be found realizing each of the 52 possible Sato-Tate groups of abelian surfaces.