We study certain mod
differential operators that act on automorphic forms over Shimura varieties of type A
or C. We show that, over the ordinary locus, these operators agree with the mod
reduction
of the
-adic
theta operators previously studied by some of the authors. In the characteristic
,
-adic
case, there is an obstruction that makes it impossible to extend the theta
operators to the whole Shimura variety. On the other hand, our mod
operators extend (“analytically continue”, in the language of de Shalit and Goren) to
the whole Shimura variety. As a consequence, motivated by their use by Edixhoven
and Jochnowitz in the case of modular forms for proving the weight part
of Serre’s conjecture, we discuss some effects of these operators on Galois
representations.
Our focus and techniques differ from those in the literature. Our intrinsic,
coordinate-free approach removes difficulties that arise from working with
-expansions
and works in settings where earlier techniques, which rely on explicit calculations, are
not applicable. In contrast with previous constructions and analytic continuation
results, these techniques work for any totally real base field, any weight, and all
signatures and ranks of groups at once, recovering prior results on analytic
continuation as special cases.
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