Nous décrivons dans cet article les algorithmes nécessaires à une implantation efficace de la méthode de Schoof pour le calcul du nombre de points sur une courbe elliptique dans un corps fini. Nous tentons d’unifier pour cela les idées d’Atkin et d’Elkies. En particulier, nous décrivons le calcul d’équations pour $X_0\left(\ell \right)$, $\ell$ premier, ainsi que le calcul efficace de facteurs des polynômes de division d’une courbe elliptique.

For a smooth and proper curve $X_K$ over the fraction field $K$ of a discrete valuation ring $R$, we explain (under very mild hypotheses) how to equip the de Rham cohomology $H_{\mathrm{dR}}^1(X_K/K)$ with a canonical integral structure: i.e., an $R$-lattice which is functorial in finite (generically étale) $K$-morphisms of $X_K$ and which is preserved by the cup-product auto-duality on $H_{\mathrm{dR}}^1(X_K/K)$. Our construction of this lattice uses a certain class of normal proper models of $X_K$ and relative dualizing sheaves. We show that our lattice naturally...

Consider the families of curves $C^{n,A}:y²=xⁿ+Ax$ and $C_{n,A}:y²=xⁿ+A$ where A is a nonzero rational. Let $J^{n,A}$ and $J_{n,A}$ denote their respective Jacobian varieties. The torsion points of $C^{3,A}\left(\mathbb{Q}\right)$ and $C_{3,A}\left(\mathbb{Q}\right)$ are well known. We show that for any nonzero rational A the torsion subgroup of $J^{7,A}\left(\mathbb{Q}\right)$ is a 2-group, and for A ≠ 4a⁴,-1728,-1259712 this subgroup is equal to $J^{7,A}\left(\mathbb{Q}\right)\left[2\right]$ (for a excluded values of A, with the possible exception of A = -1728, this group has a point of order 4). This is a variant of the corresponding results for $J^{3,A}$ (A ≠ 4) and $J^{5,A}$. We also almost...

In this article we obtain class invariants and cyclotomic unit groups by considering specializations of modular units. We construct these modular units from functional solutions to higher order $q$-recurrence equations given by Selberg in his work generalizing the Rogers-Ramanujan identities. As a corollary, we provide a new proof of a result of Zagier and Gupta, originally considered by Gauss, regarding the Gauss periods. These results comprise part of the author’s 2006 Ph.D. thesis [6] in which...

One can define class invariants for a quartic primitive CM field $K$ as special values of certain Siegel (or Hilbert) modular functions at CM points corresponding to $K$. Such constructions were given by de Shalit-Goren and Lauter. We provide explicit bounds on the primes appearing in the denominators of these algebraic numbers. This allows us, in particular, to construct $S$-units in certain abelian extensions of a reflex field of $K$, where $S$ is effectively determined by $K$, and to bound the primes appearing...

Soient $X$ une variété abélienne sur un corps de nombres $E$ et $G$ son groupe de Mumford–Tate. Soit $v$ une valuation de $E$ et pour tout nombre premier $\ell$ tel que $v\left(\ell \right)=0$, soit $F_{\ell }\in G({\mathbf{Q}}_{\ell })$ l’automorphisme de Frobenius (géométrique) de la cohomologie étale $\ell$-adique de $X$. On montre que si $X$ a une bonne réduction ordinaire en $v$, alors il existe $F\in G\left(\mathbf{Q}\right)$ tel que, pour tout $\ell$, $F_{\ell }$ soit conjugué à $F$ dans $G\left({\mathbf{Q}}_{\ell }\right)$. On montre un résultat analogue pour le frobenius de la cohomologie cristalline de la réduction de $X$ modulo $v$.

Assuming GRH, we present an algorithm which inputs a prime $p$ and outputs the set of fundamental discriminants $D\<0$ such that the reduction map modulo a prime above $p$ from elliptic curves with CM by ${𝒪}_D$ to supersingular elliptic curves in characteristic $p$ is surjective. In the algorithm we first determine an explicit constant $D_p$ so that $\left|D\right|\>D_p$ implies that the map is necessarily surjective and then we compute explicitly the cases $\left|D\right|\<D_p$.

The aim of these notes is to generalize Laumon’s construction [20] of automorphic sheaves corresponding to local systems on a smooth, projective curve $C$ to the case of local systems with indecomposable unipotent ramification at a finite set of points. To this end we need an extension of the notion of parabolic structure on vector bundles to coherent sheaves. Once we have defined this, a lot of arguments from the article “ On the geometric Langlands conjecture” by Frenkel, Gaitsgory and Vilonen [11]...