某些模曲线的粗基线变更失败
Coarse base change fails for some modular curves
摘要：
对于一致性水平$H \subset \mathrm{GL}_2(\hat{\mathbb{Z}})$，模曲线$X_H$的形成，即，的粗模空间的水平$H$模块堆栈$\mathscr{X}_H$，是已知的任意基变化在广泛的情况下交换。我们展示了无穷多个$H$，例如，$H = Γ_1(4)$，对于这些$H$，这种粗略的基变化属性失败。在我们的实施例中，对于$\mathbb{F}_2$的碱基变化和对于任何$\mathbb{Z}_{(2)}$ -$\mathscr{X}_H$的纤维状致密开放子叠层，都观察到失败。这些例子填补了卡茨和马祖尔书中表格中的几个开放条目。
英文摘要：
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, $H = Γ_1(4)$, 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.
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