Coarse base change fails for some modular curves

TL;DR

This paper demonstrates that the property of coarse base change for certain modular curves fails in infinitely many cases, providing explicit examples that fill gaps in existing mathematical literature.

Contribution

It identifies specific modular curves where coarse base change fails, expanding understanding of the limitations of base change properties in algebraic geometry.

Findings

Failure of coarse base change for infinitely many H

Examples include H = Γ₁(4) over F₂ and Z_{(2)}

Completes open entries in Katz and Mazur's table

Abstract

For a congruence level $H \subset \mathrm{GL}_2(\hat{\mathbb{Z}})$, the formation of the modular curve $X_H$, i.e., of the coarse moduli space of the level $H$ modular stack $\mathscr{X}_H$, is known to commute with arbitrary base change in a wide range of cases. We exhibit infinitely many $H$, for instance, $H = \Gamma_1(4)$, for which this coarse base change property fails. In our examples failure is witnessed for base change to $\mathbb{F}_2$ and for any $\mathbb{Z}_{(2)}$-fiberwise dense open substack of $\mathscr{X}_H$. These examples fill in several open entries in a table in the book of Katz and Mazur.