By Henri Darmon (auth.), Guillaume Hanrot, François Morain, Emmanuel Thomé (eds.)

This e-book constitutes the refereed lawsuits of the ninth overseas Algorithmic quantity idea Symposium, ANTS 2010, held in Nancy, France, in July 2010. The 25 revised complete papers awarded including five invited papers have been rigorously reviewed and chosen for inclusion within the booklet. The papers are dedicated to algorithmic facets of quantity thought, together with straightforward quantity idea, algebraic quantity conception, analytic quantity idea, geometry of numbers, algebraic geometry, finite fields, and cryptography.

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The curve X has good reduction at 7, and X(F7 ) = {(0, 0), (1, 0), (2, 0), (5, 0), (6, 0), (3, 6), (3, −6), ∞}. 3(2)], we know |X(Q)| ≤ 10. However, we can find 10 rational points on X: the six rational Weierstrass points, and the points (3, ±6), (10, ±120). Hence |X(Q)| = 10. Since the Chabauty condition holds, there must exist a holomorphic differQ ential ω for which ∞ ω = 0 for all Q ∈ X(Q). We can find such a differential by taking Q to be one of the rational non-Weierstrass points, then computing Q Q a := ∞ ω0 , b := ∞ ω1 and setting ω = bω0 − aω1 .

T ÔhÕ Ôa, b 2ah, f ÔhÕÕ. E are the inverse of the roots of f , and that T ÔhÕ subtracts h to each roots of f . Norms of matrices and forms. Let M Ôα, β; γ, δ Õ be a matrix in M2 ÔZÕ. The Euclidean norm is M 2 α2 β 2 γ 2 δ 2 , and the maximum norm max Ô α , β , γ , δ Õ. v 2 Õ is the inis M duced Euclidean norm, which is also the square root of the largest eigenvalue of Mt M. All the norms are equivalent: M M M 2 2 M . Smallest Reduction Matrix of Binary Quadratic Forms 35 Additionally, the induced norm is sub-multiplicative: if N È M2 ÔZÕ then MN M ¤ N and Id 1, and it is lower-bounded by the spectral radius ρÔMÕ, which is the supremum among the absolute values of the eigenvalues of M.

2g − 1) is crystalline (see the erratum to [17]), so Frobenius will act via a matrix with p-adically integral entries. 2 Tiny Integrals Q We refer to any Coleman integral of the form P ω in which P, Q lie in the same residue disc (Weierstrass or not) as a tiny integral. As an easy first case, we give an algorithm to compute tiny integrals of basis differentials. Algorithm 8 (Tiny Coleman integrals). Input: Points P, Q ∈ X(Cp ) in the same residue disc (neither equal to the point at infinity) and a basis differential ωi .

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