Phantom holomorphic projections arising from Sturm's formula

TL;DR

This paper investigates the analytic continuation of Siegel Poincaré series at weight three in genus two, revealing a nonhomomorphic 'phantom' component produced by Sturm's operator, a phenomenon unique to this weight.

Contribution

It demonstrates the existence of a nonhomomorphic 'phantom' term in the analytic continuation of Siegel Poincaré series and Sturm's operator at weight three in genus two.

Findings

Identification of a nonhomomorphic 'phantom' term in the continuation.

Description of the 'phantom' term's origin and properties.

Unique occurrence of this phenomenon at weight three for genus two.

Abstract

We show the analytic continuation of certain Siegel Poincar\'e 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.