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.

Abstract. We present a new algorithm to compute the classical modular polynomial Φ l in the rings Z[X, Y ] and (Z/mZ)[X, Y ], for a prime l and any positive integer m. Our approach uses the graph of l-isogenies to efficiently compute Φ l mod p for many primes p of a suitable form, and then applies the Chinese Remainder Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an expected running time of O(l 3 (log l) 3 log log l), and compute Φ l mod m using O(l 2 (log l) 2 + l 2 log m) space. We have used the new algorithm to compute Φ l with l over 5000, and Φ l mod m with l over 20000. We also consider several modular functions g for which Φ g l is smaller than Φ l , allowing us to handle l over 60000.

International audienceWe present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field F_q. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of log q, while our bound for the second algorithm depends primarily on log |D_E|, where D_E is the discriminant of the order isomorphic to End(E). As a byproduct, our method yields a short certificate that may be used to verify that the endomorphism ring is as claimed

Abstract.For each open subgroup G of GL2(Ẑ) containing −I with full determinant, let XG/Q denote the modular curve that loosely parametrizes elliptic curves whose Galois representation, which arises from the Galois action on its torsion points, has image contained in G. Up to conjugacy, we determine a complete list of the 248 such groups G of prime power level for which XG(Q) is infinite. For each G, we also construct explicit maps from each XG to the j-line. This list consists of 220 modular curves of genus 0 and 28 modular curves of genus 1. For each prime ℓ, these results provide an explicit classification of the possible images of ℓ-adic Galois representations arising from elliptic curves over Q that is complete except for a finite set of exceptional j-invariants.

Abstract. We present a space-efficient algorithm to compute the Hilbert class polynomial H D (X) modulo a positive integer P , based on an explicit form of the Chinese Remainder Theorem. Under the Generalized Riemann Hypothesis, the algorithm uses O(|D| 1/2+ log P ) space and has an expected running time of O(|D| 1+ ). We describe practical optimizations that allow us to handle larger discriminants than other methods, with |D| as large as 10 13 and h(D) up to 10 6 . We apply these results to construct pairing-friendly elliptic curves of prime order, using the CM method.