We give upper and lower bounds for the number of points on abelian varieties over finite fields, and lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian surfaces.

Consider two families of hyperelliptic curves (over ℚ), $C^{n,a}:y²=xⁿ+a$ and $C_{n,a}:y²=x(xⁿ+a)$, and their respective Jacobians $J^{n,a}$, $J_{n,a}$. We give a partial characterization of the torsion part of $J^{n,a}\left(\mathbb{Q}\right)$ and $J_{n,a}\left(\mathbb{Q}\right)$. More precisely, we show that the only prime factors of the orders of such groups are 2 and prime divisors of n (we also give upper bounds for the exponents). Moreover, we give a complete description of the torsion part of $J_{8,a}\left(\mathbb{Q}\right)$. Namely, we show that $J_{8,a}{\left(\mathbb{Q}\right)}_{tors}=J_{8,a}\left(\mathbb{Q}\right)\left[2\right]$. In addition, we characterize the torsion parts of $J_{p,a}\left(\mathbb{Q}\right)$, where p is an odd prime, and...

The goal of this paper is to study certain $p$-adic differential operators on automorphic forms on $U(n,n)$. These operators are a generalization to the higher-dimensional, vector-valued situation of the $p$-adic differential operators constructed for Hilbert modular forms by N. Katz. They are a generalization to the $p$-adic case of the $C^{\infty }$-differential operators first studied by H. Maass and later studied extensively by M. Harris and G. Shimura. The operators should be useful in the construction of certain $p$-adic...

On décrit dans cet article une version effective d’un théorème de Rumely : on peut trouver beaucoup de points entiers sur des ouverts (assez grands) de variétés arithmétiques, tout en contrôlant la hauteur de ces points. On applique ensuite ce résultat :- aux modèles de variétés abéliennes;- à la démonstration d’un analogue arithmétique des théorèmes de Bertini.

Let $k$ be a field of characteristic $p\>0$. Let $D_m$ be a ${BT}_m$ over $k$ (i.e., an $m$-truncated Barsotti–Tate group over $k$). Let $S$ be a $k$-scheme and let $X$ be a ${BT}_m$ over $S$. Let $S_{D_m}\left(X\right)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_m$. If $p\ge 5$, we show that $S_{D_m}\left(X\right)$ is pure in $S$, i.e. the immersion $S_{D_m}\left(X\right)\hookrightarrow S$ is affine. For $p\in \{2,3\}$, we prove purity if $D_m$ satisfies a certain technical property depending only on its $p$-torsion $D_m\left[p\right]$. For $p\ge 5$, we apply the developed techniques to show that all level $m$...

Let p be a prime greater than 3. Consider the modular curve X₀(3p) over ℚ and its Jacobian variety J₀(3p) over ℚ. Let (3p) and (3p) be the group of rational torsion points on J₀(3p) and the cuspidal group of J₀(3p), respectively. We prove that the 3-primary subgroups of (3p) and (3p) coincide unless p ≡ 1 (mod 9) and $3^{(p-1)/3}\equiv 1\left(modp\right)$.

In [3] we introduced the concept of strongly modular abelian variety. This note contains some remarks and examples of this kind of varieties, especially for the case of Jacobian surfaces, that complement the results of [3].

Le but de cet article est d’étudier une conjecture de Lang énoncée sur les courbes elliptiques dans un livre de Serge Lang, puis généralisée aux variétés abéliennes de dimension supérieure dans un article de Joseph Silverman. On donne un résultat asymptotique sur la hauteur des points de Heegner sur $J_0\left(N\right)$, lequel permet de déduire que la conjecture est optimale dans sa formulation.

The main result of this paper implies that if an abelian variety over a field $F$ has a maximal isotropic subgroup of $n$-torsion points all of which are defined over $F$, and $n\ge 5$, then the abelian variety has semistable reduction away from $n$. This result can be viewed as an extension of Raynaud’s theorem that if an abelian variety and all its $n$-torsion points are defined over a field $F$ and $n\ge 3$, then the abelian variety has semistable reduction away from $n$. We also give information about the Néron models...