A theta operator on Picard modular forms modulo an inert prime

(an 2 1 Fp) of such a form, μ is given by qd=dq: It lifts, by the same formula, to the space of p-adic modular forms. This suggests a relation with the Tate twist of the mod p Galois representation attached to f; if the latter is a Hecke eigenform. Over C; this operator has been considered already by Ramanujan, where it fails to preserve modularity “by a multiple of E2": Maass modi...ed it so that modularity is preserved, sacri...cing holomorphicity. Shimura studied Maass’ di¤erential operators on more general symmetric domains, as well as their iterations. They have become known as Maass-Shimura operators, and play an important role in the theory of automorphic forms [Sh3, chapter III]. At the same time, Serre’s p-adic operator has been studied in relation to mod p Galois representations, congruences between modular forms, p-adic families of modular forms and p-adic L-functions. As an example we cite Coleman’s celebrated classicality theorem, asserting that “overconvergent modular forms of small slope are classical” [Col]. A key step in Coleman’s original proof of that theorem was the observation that, although the p-adic theta operator did not preserve the space of overconvergent modular forms, for any k ̧ 0; μ mapped overconvergent forms of weight ¡k to overconvergent forms of weight k + 2: