Coarse base change fails for some modular curves

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 ${\mathcal{X}}_H$, 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 ${\mathcal{X}}_H$. These examples fill in several open entries in a table in the book of Katz and Mazur.

arXiv comment7 pages; final version, to appear in Algebraic Geometry