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30 documents, page 1 of 3

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...

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_...

Computing arithmetic Kleinian groups

Page, Aurel (2014)
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.

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.

Dirichlet series associated to cubic fields with given quadratic resolvent

Cohen, Henri; Thorne, Frank (2013)
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...

A generalisation of Miller's algorithm and applications to pairing computations on abelian varieties

Lubicz , David; Robert , Damien (2015)
Projects: EC | ANTICS (278537)
International audience; In this paper, we use the theory of theta functions to generalize to all abelian varieties the usual Miller's algorithm to compute a function associated to a principal divisor. We also explain how to use the Frobenius morphism on abelian varieties defined over a finite field in order to shorten the loop of the Weil and Tate pairings algorithms. This extend preceding results about ate and twisted ate pairings to all abelian varieties. Then building upon the two precedin...