The Langlands-Kottwitz approach for the modular curve

TL;DR

This paper applies the Langlands-Kottwitz method to compute local factors of the Hasse-Weil zeta-function for modular curves at bad reduction places, also proving a related conjecture.

Contribution

It demonstrates how the Langlands-Kottwitz approach can be used for modular curves and proves a specific conjecture of Haines and Kottwitz.

Findings

Computed local factors of the Hasse-Weil zeta-function at bad reduction places.

Proved a conjecture of Haines and Kottwitz in this context.

Validated the effectiveness of the Langlands-Kottwitz method for modular curves.

Abstract

We show how the Langlands-Kottwitz method can be used to determine the local factors of the Hasse-Weil zeta-function of the modular curve at places of bad reduction. On the way, we prove a conjecture of Haines and Kottwitz in this special case.