Phantom holomorphic projections arising from Sturm’s formula

Abstract

We show the analytic continuation of certain Siegel Poincaré series to their critical point for weight three in genus two. We proof that this continuation posesses a nonhomomorphic part and describe it. We show that Sturm's operator also produces a nonhomorphic share for weight three, we call it a phantom term. Weight three is the distinguished weight for genus two where this phenomenon arises.

Sturm’s operator for scalar weight in arbitrary genus

“…We prove its failure for arbitrary Siegel genus m ≥ 2 and scalar weight κ = m + 1. This generalizes a result for genus two in [4]. …”

“…Thus the image Sh is not the holomorphic projection ofh, and the following main result is a direct consequence of Theorem 2.1. In case m = 2 it was shown in [4] that the components given by ∆ [2] + [Γ, 1] are indeed the only ones on which Sturm's operator for weight 3 fails to be the holomorphic projection. This was done by giving a system of Poincaré series.…”

“…By analytic continuation (which is indeed holomorphic in these cases) of the Poincaré series it is also seen to be true for classical m = 1 and κ = 2 ([2]), and for m = 2 and κ = 4 ( [3]), that is in the case of low weight κ = 2m. But in [4] it was shown that this method fails in case of genus m = 2 and weight κ = m + 1 = 3. First, the analytic continuations of the Poincaré series fail to be holomorphic, but have nonzero parts in the spectral component generated by a holomorphic cuspform of weight κ − 2.…”

Sturm's operator acting on vector valued K‐types

“…The same result holds true for weight in case ([5, 6]). However, in case of weight and rank we showed jointly with R. Weissauer ([8]) that Sturm's operator produces, along with the holomorphic projection, a second term This phantom term arises as the non‐holomorphic Maass shift of a holomorphic form of weight one (see section 4 for the exact definition of . Later ([7]) we generalized this result to general rank and .…”

“…Our results obtained so far are limited by the explicit computability of phantom terms. Nevertheless, by [7, 8], and the above, the following interpretation is at hand. A holomorphic cusp form of weight ρ generates a holomorphic representation of the symplectic group G of minimal K ‐type ρ.…”