Graph Laplacians, component groups and Drinfeld modular curves

Let $\frak{p}$ be a prime ideal of $\mathbb{F}_q[T]$. Let $J_0(\frak{p})$ be the Jacobian variety of the Drinfeld modular curve $X_0(\frak{p})$. Let $\Phi$ be the component group of $J_0(\frak{p})$ at the place $1/T$. We use graph Laplacians to estimate the order of $\Phi$ as $\mathrm{deg}(\frak{p})$ goes to infinity. This estimate implies that $\Phi$ is not annihilated by the Eisenstein ideal of the Hecke algebra $\mathbb{T}(\frak{p})$ acting on $J_0(\frak{p})$ once the degree of $\frak{p}$ is large enough. We also obtain an asymptotic formula for the size of the discriminant of $\mathbb{T}(\frak{p})$ by relating this discriminant to the order of $\Phi$; in this problem the order of $\Phi$ plays a role similar to the Faltings height of classical modular Jacobians. Finally, we bound the spectrum of the adjacency operator of a finite subgraph of an infinite diagram in terms of the spectrum of the adjacency operator of the diagram itself; this result has applications to the gonality of Drinfeld modular curves

On component groups of Jacobians of quaternionic modular curves

We use a combinatorial result relating the discriminant of the cycle pairing on a weighted finite graph to the eigenvalues of its Laplacian to deduce a formula for the orders of component groups of Jacobians of modular curves arising from quaternion algebras over $\mathbb{F}_q(T)$ or $\mathbb{Q}$. Our formula over $\mathbb{Q}$ recovers a result of Jordan and Livn\'e

Local diophantine properties of modular curves of D\cal{D}-elliptic sheaves

We study the existence of rational points on modular curves of $\cal{D}$-elliptic sheaves over local fields and the structure of special fibres of these curves. We discuss some applications which include finding presentations for arithmetic groups arising from quaternion algebras, finding the equations of modular curves of $\cal{D}$-elliptic sheaves, and constructing curves violating the Hasse principle.Comment: 24 page

On Jacquet-Langlands isogeny over function fields

We propose a conjectural explicit isogeny from the Jacobians of hyperelliptic Drinfeld modular curves to the Jacobians of hyperelliptic modular curves of $\mathcal{D}$-elliptic sheaves. The kernel of the isogeny is a subgroup of the cuspidal divisor group constructed by examining the canonical maps from the cuspidal divisor group into the component groups.Comment: 29 page