For $n\in {𝔽}_q\left[T\right]$ let $G$ be a subgroup of the Atkin-Lehner involutions of the Drinfeld modular curve $X_0\left(𝔫\right)$. We determine all $𝔫$ and $G$ for which the quotient curve $G\setminus X_0\left(𝔫\right)$ is rational or elliptic.

Let $J_0\left(𝔫\right)$ be the Jacobian variety of the Drinfeld modular curve $X_0\left(𝔫\right)$ over ${𝔽}_q\left(t\right)$, where $𝔫$ is an ideal in ${𝔽}_q\left[t\right]$. Let $0\to B\to J_0\left(𝔫\right)\to A\to 0$ be an exact sequence of abelian varieties. Assume $B$, as a subvariety of $J_0\left(𝔫\right)$ , is stable under the action of the Hecke algebra $𝕋\subset$ End $\left(J_0\left(𝔫\right)\right)$. We give a criterion which is sufficient for the exactness of the induced sequence of component groups $0\to {\Phi }_{B,\infty }\to {\Phi }_{J,\infty }\to {\Phi }_{A,\infty }\to 0$ of the Néron models of these abelian varieties over ${𝔽}_q\left[\left[\frac{1}{t}\right]\right]$. This criterion is always satisfied when either $A$ or $B$ is one-dimensional. Moreover, we prove that the sequence...

Let $\Lambda$ be a maximal $𝔽{}_q\left[T\right]$-order in a division quaternion algebra over $𝔽{}_q\left(T\right)$ which is split at the place $\infty$. The present article gives an algorithm to compute a fundamental domain for the action of the group of units ${\Lambda }^{*}$ on the Bruhat-Tits tree $𝒯$ associated to ${\mathrm{PGL}}_2\left(𝔽{}_q\left((1/T)\right)\right)$. This action is a function field analog of the action of a co-compact Fuchsian group on the upper half plane. The algorithm also yields an explicit presentation of the group ${\Lambda }^{*}$ in terms of generators and relations. Moreover we determine an upper bound...

In this article we give an introduction to mixed motives and sketch a couple of ways to construct examples.

In this paper we study certain moduli spaces of Barsotti-Tate groups constructed by Rapoport and Zink as local analogues of Shimura varieties. More precisely, given an isogeny class of Barsotti-Tate groups with unramified additional structures, we investigate how the associated (non-basic) moduli spaces compare to the (basic) moduli spaces associated with its isoclinic constituents. This aspect of the geometry of the Rapoport-Zink spaces is closely related to Kottwitz’s prediction that their $l$-adic...