Soient $p$ et $q$ deux nombres premiers distincts et $X^{pq}/w_q$ le quotient de la courbe de Shimura de discriminant $pq$ par l’involution d’Atkin-Lehner $w_q$. Nous décrivons un moyen permettant de vérifier un critère de Parent et Yafaev en grande généralité pour prouver que si $p$ et $q$ satisfont des conditions de congruence explicites, connues comme les conditions du cas non ramifié de Ogg, et si $p$ est assez grand par rapport à $q$, alors le quotient $X^{pq}/w_q$ n’a pas de point rationnel non spécial.

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 $m$...

The rational points on $X_0\left(N\right)/W_N$ in the case where $N$ is a composite number are considered. A computational study of some of the cases not covered by the results of Momose is given. Exceptional rational points are found in the cases $N=91$ and $N=125$ and the $j$-invariants of the corresponding quadratic $\mathbb{Q}$-curves are exhibited.

We derive a relation between induced representations on the group ${GL}_2(\mathbb{Z}/p^2\mathbb{Z})$ which implies a relation between the jacobians of certain modular curves of level $p^2$. The motivation for the construction of this relation is the determination of the applicability of Mazur’s method to the modular curve associated to the normalizer of a non-split Cartan subgroup of ${GL}_2(\mathbb{Z}/p^2\mathbb{Z})$.

Let $N$ be an odd and squarefree positive integer divisible by at least two relative prime integers bigger or equal than $4$. Our main theorem is an asymptotic formula solely in terms of $N$ for the stable arithmetic self-intersection number of the relative dualizing sheaf for modular curves $X_1\left(N\right)/\mathbb{Q}$. From our main theorem we obtain an asymptotic formula for the stable Faltings height of the Jacobian $J_1\left(N\right)/\mathbb{Q}$ of $X_1\left(N\right)/\mathbb{Q}$, and, for sufficiently large $N$, an effective version of Bogomolov’s conjecture for $X_1\left(N\right)/\mathbb{Q}$.

We study the local factor at $p$ of the semi-simple zeta function of a Shimura variety of Drinfeld type for a level structure given at $p$ by the pro-unipotent radical of an Iwahori subgroup. Our method is an adaptation to this case of the Langlands-Kottwitz counting method. We explicitly determine the corresponding test functions in suitable Hecke algebras, and show their centrality by determining their images under the Hecke algebra isomorphisms of Goldstein, Morris, and Roche.

La correspondance de Shimizu et Jacquet-Langlands donne des relations entre les quotients de la partie nouvelle de la jacobienne $J_0\left(N\right)$ de $X_0\left(N\right)$ et ceux de la partie nouvelle de la jacobienne de certaines courbes de Shimura associées. Nous comparons dans ce texte les congruences entre formes modulaires pour des quotients qui sont associés dans cette correspondance.

We define Picard cycles on each smooth three-sheeted Galois cover C of the Riemann sphere. The moduli space of all these algebraic curves is a nice Shimura surface, namely a symmetric quotient of the projective plane uniformized by the complex two-dimensional unit ball. We show that all Picard cycles on C form a simple orbit of the Picard modular group of Eisenstein numbers. The proof uses a special surface classification in connection with the uniformization of a classical Picard-Fuchs system....

For any prime number p > 3 we compute the formal completion of the Néron model of J0(p) in terms of the action of the Hecke algebra on the Z-module of all cusp forms (of weight 2 with respect to Γ0(p)) with integral Fourier development at infinity.