Regular models of modular curves in prime level over ${\mathbb Z}_p^{\mathrm{ur}}$

TL;DR

This paper constructs regular models for certain modular curves over unramified extensions of p-adic integers and computes the connected components of the fibers of their Jacobians' Néron models, advancing understanding of their arithmetic structure.

Contribution

It provides explicit regular models for modular curves associated with split and non-split Cartan subgroups over ${ m Z}_p^{ m ur}$, and calculates the connected components of the fibers of their Jacobians' Néron models.

Findings

Explicit regular models for modular curves at prime level.

Determination of the group of connected components of the fibers.

Enhanced understanding of the arithmetic of modular Jacobians.

Abstract

We give regular models for modular curves associated with (normalizer of) split and non-split Cartan subgroups of ${\mathrm{GL}}_2 ({\mathbb F}_p )$ (for $p$ any prime, $p\ge 5$). We then compute the group of connected components of the fiber at $p$ of the N\'eron model of their Jacobians.