2013/06/18 by David Goss, Goss, David
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1306.4344
openalex publication_date 2013/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The classical theory of p-adic (elliptic) modular forms arose in the 1970's from the work of J.-P. Serre \citese1 who took p-adic limits of the q-expansions of these forms. It was soon expanded by N. Katz \citeka1 with a more functorial approach. Since then the theory has grown in a variety of directions. In the late 1970's, the theory of modular forms associated to Drinfeld modules was born in analogy with elliptic modular forms \citego1, \citego2. The associated expansions at ∞ are quite complicated and no obvious limits at finite primes \mathfrak v were apparent. Recently, however, there has been progress in the \mathfrak v-adic theory, \citevi1. Also recently, A. Petrov \citepe1, building on previous work of \citelo1, showed that there is an intermediate expansion at ∞ called the "A-expansion," and he constructed families of cusp forms with such expansions. It is our purpose in this note to show that Petrov's results also lead to interesting \mathfrak v-adic cusp forms à la Serre. Moreover the existence of these forms allows us to readily conclude a mysterious decomposition of the associated Hecke action.