Graph Laplacians, component groups and Drinfeld modular curves

TL;DR

This paper investigates the structure and size of the component group of Jacobians of Drinfeld modular curves using graph Laplacians, providing asymptotic estimates and bounds with applications to modular curve properties.

Contribution

It introduces graph Laplacian techniques to estimate the order of component groups and relates these to the discriminant of the Hecke algebra, advancing understanding of Drinfeld modular curves.

Findings

Estimated the order of the component group as the degree of the prime increases.

Showed the component group is not annihilated by the Eisenstein ideal for large primes.

Derived bounds on the spectrum of adjacency operators related to modular curve gonality.

Abstract

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.