A construction of b-adic modular forms

Text. The classical theory of p-adic (elliptic) modular forms arose in the 1970's in the work of J.-P. Serre [Se1] who took p-adic limits of the q-expansions of these forms. It was soon expanded by N. Katz [Ka1] with a more functorial approach. In the late 1970's, the theory of modular forms associated to. Drinfeld modules was born in analogy with elliptic modular forms [Go1,Go2]. The associated expansions at infinity are quite complicated with no obvious limits at finite primes b. Recently, A. Petrov [Pei] showed that there is an intermediate expansion at infinity called the "A-expansion," and 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 b-adic cusp forms a la Serre. Moreover the existence of these forms allows us to readily conclude a mysterious decomposition of the associated Hecke action. Video. For a video summary of this paper, please click here or visit http://youtu.be/xzezUI7-3yc. (C) 2013 Elsevier Inc. All rights reserved.