2020/03/24 by Sean Howe, Howe, Sean · 1 citation
Mathematics · #11F33 #11F77 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2003.11129
openalex publication_date 2020/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Following a suggestion of Peter Scholze, we construct an action of \mathbbGm on the Katz moduli problem, a profinite-étale cover of the ordinary locus of the p-adic modular curve whose ring of functions is Serre's space of p-adic modular functions. This action is a local, p-adic analog of a global, archimedean action of the circle group S1 on the lattice-unstable locus of the modular curve over ℂ. To construct the \mathbbGm-action, we descend a moduli-theoretic action of a larger group on the (big) ordinary Igusa variety of Caraiani-Scholze. We compute the action explicitly on local expansions and find it is given by a simple multiplication of the cuspidal and Serre-Tate coordinates q; along the way we also prove a natural generalization of Dwork's equation τ=log q for extensions of ℚp/ℤp by μp^∞ valid over a non-Artinian base. Finally, we give a direct argument (without appealing to local expansions) to show that the action of \mathbbGm integrates the differential operator θ coming from the Gauss-Manin connection and unit root splitting, and explain an application to Eisenstein measures and p-adic L-functions.