On component groups of Jacobians of quaternionic modular curves

TL;DR

This paper derives a formula for the orders of component groups of Jacobians of quaternionic modular curves using combinatorial graph theory and spectral analysis, extending known results over the rational numbers.

Contribution

It introduces a new combinatorial approach linking graph discriminants and Laplacian eigenvalues to compute component group orders for quaternionic modular curves.

Findings

Derived a formula for component group orders over function fields and rationals.

Recovers and generalizes a known result of Jordan and Livne9.

Establishes a connection between graph discriminants and Jacobian component groups.

Abstract

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.