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.

For a commutative algebraic group $G$ over a perfect field $k$, Ribet defined the set of almost rational torsion points $G_{tors,k}^{ar}$ of $G$ over $k$. For positive integers $d$, $g,$ we show there is an integer $U_{d,g}$ such that for all tori $T$ of dimension at most $d$ over number fields of degree at most $g$, $T_{tors,k}^{ar}\subseteq T\left[U_{d,g}\right]$. We show the corresponding result for abelian varieties with complex multiplication, and under an additional hypothesis, for elliptic curves without complex multiplication. Finally, we show that except for an explicit finite...

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.

Nous construisons des familles ordinaires $p$-adiques de formes modulaires pour le groupe ${\mathrm{GSp}}_{2g}$. Notre travail généralise et précise des travaux antérieurs de Hida.

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....

Let $J_1$ be the Jacobian of the modular curve associated with ${\Gamma }_1\left(Np\right),(p,N)=1$ and $J_0$ the one associated with ${\Gamma }_1\left(N\right)\cap {\Gamma }_0\left(p\right)$. We study $J_1[p-1]$ as a Hecke and Galois-module. We relate a certain matrix of $p$-adic periods to the infinitesimal deformation of the $U_p$-operator.

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.

Let $X$ be a curve over a field $k$ with a rational point $e$. We define a canonical cycle ${\Delta }_e\in Z^2(X^3{)}_{\mathrm{hom}}$. Suppose that $k$ is a number field and that $X$ has semi-stable reduction over the integers of $k$ with fiber components non-singular. We construct a regular model of $X^3$ and show that the height pairing $⟨{\tau }_{*}\left({\Delta }_e\right),{\tau }_{*}^{\text{'}}\left({\Delta }_e\right)⟩$ is well defined where $\tau$ and ${\tau }^{\text{'}}$ are correspondences. The paper ends with a brief discussion of heights and $L$-functions in the case that $X$ is a modular curve.