Wild Stacky Curves and Rings of Mod p Modular Forms
Andrew Kobin, David Zureick-Brown
TL;DR
The paper develops a framework to compute and bound the log canonical rings of wild stacky curves and applies it to rings of mod $p$ modular forms, connecting stack geometry with ethereal forms. It extends the tame theory of Voight–ZB to wild ramification, providing generator/relation bounds tied to refined signatures and establishing a geometric mechanism for ethereal phenomena, including infinite families in characteristics $2$ and $3$. The authors prove a wild-version refinement of Rustom's conjecture, describe global tame-by-wild root stack structures for modular curves, and supply Magma-based computational tools to identify ethereal generators and their $q$-expansions. The results yield precise conditions under which ethereal weight-$2$ forms arise, reveal how ramification affects modular-form rings, and open avenues toward nonstandard level structures and higher-dimensional moduli problems. Overall, the work blends stack-theoretic methods with modular-forms arithmetic to illuminate nonliftable phenomena and provide practical computational approaches for mod $p$ modular forms.
Abstract
We extend work of Voight and the second author to compute the log canonical ring of a wild stacky curve over a field of characteristic $p > 0$, which allows us to compute rings of mod $p$ modular forms of level $Γ_{0}(N)$. Our approach also reveals that in characteristics $2$ and $3$, there are infinitely many levels $N$ for which there are weight $2$ modular forms of level $Γ_{0}(N)$ that do not lift to characteristic $0$.