A construction of vv-adic modular forms

TextThe classical theory of p-adic (elliptic) modular forms arose in the 1970ʼs 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 1970ʼs, the theory of modular forms associated to Drinfeld modules was born in analogy with elliptic modular forms [Go1] and [Go2]. The associated expansions at ∞ are quite complicated with no obvious limits at finite primes vv. Recently, A. Petrov [Pe1] showed that there is an intermediate expansion at ∞ called the “A  -expansion,” and constructed families of cusp forms with such expansions. It is our purpose in this note to show that Petrovʼs results also lead to interesting vv-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.VideoFor a video summary of this paper, please click here or visit http://youtu.be/xzezUI7-3yc.