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ANTICS

Title
Algorithmic Number Theory in Computer Science
Funding
EC | FP7 | SP2 | ERC
Call
ERC-2011-StG_20101014
Contract (GA) number
278537
Start Date
2012/01/01
End Date
2016/12/31
Open Access mandate
no
Special Clause 39
no
Organizations
INRIA
More information
Detailed project information (CORDIS)

 

  • Computing isogenies between abelian varieties

    Lubicz, David; Robert, Damien (2010)
    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...

    Computing arithmetic Kleinian groups

    Page, Aurel (2012)
    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.

    Short addition sequences for theta functions

    Enge, Andreas; Hart, William; Johansson, Fredrik (2016)
    Projects: EC | ANTICS (278537)
    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 example, we show that e...

    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 , Fran├žois (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 (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 ...

    Schertz style class invariants for quartic CM fields

    Enge , Andreas; Streng , Marco (2016)
    Projects: EC | ANTICS (278537)
    A class invariant is a CM value of a modular function that lies in a certain unramified class field. We show that Siegel modular functions over $\mathbb Q$ for $\Gamma^0(N)\subseteq \operatorname {Sp}_4(\mathbb Z)$ yield class invariants under some splitting conditions on $N$. Small class invariants speed up onstructions in explicit class field theory and public-key cryptography. Our results generalise results of Schertz's from elliptic curves to abelian varieties and from classical modular f...

    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...
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