Entire theta operators at unramified primes

Abstract

Starting with work of Serre, Katz, and Swinnerton-Dyer, theta operators have played a key role in the study of $p$-adic and $\mathrm{mod}p$ modular forms and Galois representations. This paper achieves two main results for theta operators on automorphic forms on PEL-type Shimura varieties: 1) the analytic continuation at unramified primes $p$ to the whole Shimura variety of the $\mathrm{mod}p$ reduction of $p$-adic Maass--Shimura operators {\it a priori} defined only over the $\mu$-ordinary locus, and 2) the construction of new $\mathrm{mod}p$ theta operators that do not arise as the $\mathrm{mod}p$ reduction of Maass--Shimura operators. While the main accomplishments of this paper concern the geometry of Shimura varieties and consequences for differential operators, we conclude with applications to Galois representations. Our approach involves a careful analysis of the behavior of Shimura varieties and enables us to obtain more general results than allowed by prior techniques, including for arbitrary signature, vector weights, and unramified primes in CM fields of arbitrary degree.

arXiv commentAccepted for publication in IMRN. 42 pages

References in corpus (9)

• p-adic Differential Operators on Automorphic Forms on Unitary Groups
• The μ-ordinary Hasse invariant of unitary Shimura varieties
• Theta operators on unitary Shimura varieties
• Hecke operators and the coherent cohomology of Shimura varieties
• p-adic q-expansion principles on unitary Shimura varieties
• The weight reduction of mod $p$ Siegel modular forms for $GS{p}_4$
• $p$-adic families of automorphic forms in the $\mu$-ordinary setting
• Differential operators and families of automorphic forms on unitary groups of arbitrary signature
• Differential operators mod $p$: analytic continuation and consequences

Cited by in corpus (3)

• Differential operators mod $p$: analytic continuation and consequences
• Geometric weight-shifting operators on Hilbert modular forms in characteristic p
• Automorphic Forms on Unitary Groups