arXiv subjects

Alexandre Gélin

Publications and source records attributed to Alexandre Gélin.

3 recordsLinked to original sources

arXiv2018

Reducing the complexity for class group computations using small defining polynomials

In this paper, we describe an algorithm that efficiently collect relations in class groups of number fields defined by a small defining polynomial. This conditional improvement consists in testing directly the smoothness of principal ideals generated by small algebraic integers. This strategy leads to an algorithm for computing the class group whose complexity is possibly as low as $L_{|Δ_{\mathbf K}|}\left(\frac{1}{3}\right)$.

math.NT↗

arXiv2018

On the complexity of class group computations for large degree number fields

In this paper, we examine the general algorithm for class group computations, when we do not have a small defining polynomial for the number field. Based on a result of Biasse and Fieker, we simplify their algorithm, improve the complexity analysis and identify the optimal parameters to reduce the runtime. We make use of the classes $\mathcal D$ defined in [GJ16] for classifying the fields according to the size of the extension degree and prove that they enable to describe all the number fields.

math.NT↗

arXiv2018

Principally polarized squares of elliptic curves with field of moduli equal to Q

We give equations for 13 genus-2 curves over $\overline{\mathbb{Q}}$, with models over $\mathbb{Q}$, whose unpolarized Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal order. If the Generalized Riemann Hypothesis is true, there are no further examples of such curves. More generally, we prove under the Generalized Riemann Hypothesis that there exist exactly 46 genus-2 curves over $\overline{\mathbb{Q}}$ with field of moduli $\mathbb{Q}$ whose Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal order.

math.NT↗