nLab p-adic modular form




A p-adic modular form is a limit of modular forms under a topology that encodes congruences between its Fourier coefficients.


The original definition is due to Serre. Consider the space of modular forms over \mathbb{Q}, and recall that they are fully determined by their q-expansions, therefore we may write them as power series f= n=0 a nq n[[q]]f=\sum_{n=0}^{\infty}{a_{n}}q^{n}\in\mathbb{Q}[[q]]. Since \mathbb{Q} embeds into the p-adic numbers p\mathbb{Q}_{p}, we may consider these modular forms as elements of p[[q]]\mathbb{Q}_{p}[[q]] as well. We define a valuation v pv_{p} on this space by

v p(f):=inf n(v p(a n))v_{p}(f):=\inf_{n}( v_{p}(a_{n}))

where the v pv_{p} on the right-hand side is the p-adic valuation on p\mathbb{Q}_{p} with v p(p)=1v_{p}(p)=1.

A p-adic modular form is a power series f= n=0 a nq n p[[q]]f=\sum_{n=0}^{\infty}{a_{n}}q^{n}\in\mathbb{Q}_{p}[[q]] such that there exists a sequence of modular forms f 1,f 2,f_{1},f_{2},\ldots with f iff_{i}\to f.

Another different definition was later formulated by Katz.

Over rigid analytic spaces


(Calegari13, Definition 2.1.13)

Let XX be a modular curve and let X rigX^{\rig} be its associated rigid analytic space. The p-adic modular forms of weight kk are the global sections H 0(X rig[0],ω k)H^{0}(X^{\rig}[0],\omega^{k}), where X rig[0]X^{\rig}[0] denotes the ordinary locus of X rigX^{\rig}.


  • Frank Calegari, Congruences Between Modular Forms, 2013 (pdf)

  • Ellen Eischen, An Introduction to Eisenstein Measures, 2021 (arxiv:2101.01879)

Last revised on November 26, 2022 at 20:17:41. See the history of this page for a list of all contributions to it.