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30 documents, page 1 of 3

Computing arithmetic Kleinian groups

Page , Aurel (2015)
Projects: EC | ANTICS (278537)
International audience; Arithmetic Kleinian groups are arithmetic lattices in PSL_2(C). We present an algorithm which, given such a group Gamma, returns a fundamental domain and a finite presentation for Gamma with a computable isomorphism.

Computing isogenies between Abelian Varieties

Lubicz , David; Robert , Damien (2012)
Projects: EC | ANTICS (278537)
We describe an efficient algorithm for the computation of separable isogenies between abelian varieties represented in the coordinate system given by algebraic theta functions. Let $A$ be an abelian variety of dimension $g$ defined over a field of odd characteristic. Our algorithm decomposes in two principal steps. First, given a theta null point for $A$ and a subgroup $K$ isotropic for the Weil pairing, we explain how to compute the theta null point corresponding to the quotient abelian vari...

An algorithm for the principal ideal problem in indefinite quaternion algebras

Page, Aurel (2014)
Projects: EC | ANTICS (278537)
International audience; Deciding whether an ideal of a number field is principal and finding a generator is a fundamental problem with many applications in computational number theory. For indefinite quaternion algebras, the decision problem reduces to that in the underlying number field. Finding a generator is hard, and we present a heuristically subexponential algorithm.

Generalised Weber Functions

Enge , Andreas; Morain , François (2014)
Projects: EC | ANTICS (278537)
International audience; A generalised Weber function is given by $\w_N(z) = \eta(z/N)/\eta(z)$, where $\eta(z)$ is the Dedekind function and $N$ is any integer; the original function corresponds to $N=2$. We classify the cases where some power $\w_N^e$ evaluated at some quadratic integer generates the ring class field associated to an order of an imaginary quadratic field. We compare the heights of our invariants by giving a general formula for the degree of the modular equation relating $\w_...

Dirichlet series associated to cubic fields with given quadratic resolvent

Cohen, Henri; Thorne, Frank (2014)
Projects: EC | ANTICS (278537)
International audience; Let k be a quadratic field. We give an explicit formula for the Dirichlet series enumerating cubic fields whose quadratic resolvent field is isomorphic to k. Our work is a sequel to previous work of Cohen and Morra, where such formulas are proved in a more general setting, in terms of sums over characters of certain groups related to ray class groups. In the present paper we carry the analysis further and prove explicit formulas for these Dirichlet series over Q. In a ...

Short addition sequences for theta functions

Enge , Andreas; Hart , William; Johansson , Fredrik (2018)
Projects: EC | ANTICS (278537)
International audience; The main step in numerical evaluation of classical Sl2 (Z) modular forms and elliptic functions is to compute the sum of the first N nonzero terms in the sparse q-series belonging to the Dedekind eta function or the Jacobi theta constants. We construct short addition sequences to perform this task using N + o(N) multiplications. Our constructions rely on the representability of specific quadratic progressions of integers as sums of smaller numbers of the same kind. For...

Vanishing and non-vanishing theta values

Cohen, Henri; Zagier, Don (2013)
Projects: EC | ANTICS (278537)
International audience

Arithmetic on Abelian and Kummer Varieties

Lubicz , David; Robert , Damien (2016)
Projects: EC | ANTICS (278537)
International audience; A Kummer variety is the quotient of an abelian variety by the automorphism $(-1)$ acting on it. Kummer varieties can be seen as a higher dimensional generalisation of the $x$-coordinate representation of a point of an elliptic curve given by its Weierstrass model. Although there is no group law on the set of points of a Kummer variety, there remains enough arithmetic to enable the computation of exponentiations via a Montgomery ladder based on differential additions. I...

Computing Class Polynomials for Abelian Surfaces

Enge, Andreas; Thomé, Emmanuel (2014)
Projects: EC | ANTICS (278537)
International audience; We describe a quasi-linear algorithm for computing Igusa class polynomials of Jacobians of genus 2 curves via complex floating-point approximations of their roots. After providing an explicit treatment of the computations in quartic CM fields and their Galois closures, we pursue an approach due to Dupont for evaluating ϑ- constants in quasi-linear time using Newton iterations on the Borchardt mean. We report on experiments with our implementation and present an example...
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