P-adic modular form
title: "P-adic modular form" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["modular-forms", "p-adic-numbers"] topic_path: "general/modular-forms" source: "https://en.wikipedia.org/wiki/P-adic_modular_form" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
In mathematics, a p-adic modular form is a p-adic analog of a modular form, with coefficients that are p-adic numbers rather than complex numbers. introduced p-adic modular forms as limits of ordinary modular forms, and shortly afterwards gave a geometric and more general definition. Katz's p-adic modular forms include as special cases classical p-adic modular forms, which are more or less p-adic linear combinations of the usual "classical" modular forms, and overconvergent p-adic modular forms, which in turn include Hida's ordinary modular forms as special cases.
Serre's definition
Serre defined a p-adic modular form to be a formal power series with p-adic coefficients that is a p-adic limit of classical modular forms with integer coefficients. The weights of these classical modular forms need not be the same; in fact, if they are then the p-adic modular form is nothing more than a linear combination of classical modular forms. In general the weight of a p-adic modular form is a p-adic number, given by the limit of the weights of the classical modular forms (in fact a slight refinement gives a weight in Zp×Z/(p–1)Z).
The p-adic modular forms defined by Serre are special cases of those defined by Katz.
Katz's definition
A classical modular form of weight k can be thought of roughly as a function f from pairs (E,ω) of a complex elliptic curve with a holomorphic 1-form ω to complex numbers, such that f(E,λω) = λ−k**f(E,ω), and satisfying some additional conditions such as being holomorphic in some sense.
Katz's definition of a p-adic modular form is similar, except that E is now an elliptic curve over some algebra R (with p nilpotent) over the ring of integers R0 of a finite extension of the p-adic numbers, such that E is not supersingular, in the sense that the Eisenstein series E**p–1 is invertible at (E,ω). The p-adic modular form f now takes values in R rather than in the complex numbers. The p-adic modular form also has to satisfy some other conditions analogous to the condition that a classical modular form should be holomorphic.
Overconvergent forms
Main article: Overconvergent form
Overconvergent p-adic modular forms are similar to the modular forms defined by Katz, except that the form has to be defined on a larger collection of elliptic curves. Roughly speaking, the value of the Eisenstein series E**k–1 on the form is no longer required to be invertible, but can be a smaller element of R. Informally the series for the modular form converges on this larger collection of elliptic curves, hence the name "overconvergent".
References
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