On recursive properties of certain p-adic Whittaker functions

TL;DR

This paper explores recursive relations of p-adic Whittaker functions, enabling explicit computation and linking to derivatives of Euler products, with implications for Eisenstein series and Kudla's conjectures.

Contribution

It establishes recursive properties of p-adic Whittaker functions and connects these to derivatives of Euler products, supporting Kudla's conjectures in higher dimensions.

Findings

Recursive relations for p-adic Whittaker functions derived.

Explicit computation of these functions in arbitrary dimensions possible.

Implications for derivatives of Euler products and Kudla's conjectures.

Abstract

We investigate recursive properties of certain p-adic Whittaker functions (of which representation densities of quadratic forms are special values). The proven relations can be used to compute them explicitly in arbitrary dimensions, provided that enough information about the orbits under the orthogonal group acting on the representations is available. These relations have implications for the first and second special derivatives of the Euler product over all p of these Whittaker functions. These Euler products appear as the main part of the Fourier coefficients of Eisenstein series associated with the Weil representation. In case of signature (m-2,2), we interpret these implications in terms of the theory of Borcherds' products on orthogonal Shimura varieties. This gives some evidence for Kudla's conjectures in higher dimensions.