On the calculation of the ramified Siegel series

TL;DR

This paper derives a general explicit formula for the ramified Siegel series over nonarchimedean local fields, including cases where the additive character is primitive, extending previous known results.

Contribution

It provides the first explicit formula for the ramified Siegel series in the general case for arbitrary dimension n, covering nonarchimedean, non-dyadic fields.

Findings

Explicit formulas for degrees n=1, 2, 3

General formula for arbitrary n

Results applicable to nonarchimedean, non-dyadic fields

Abstract

The ramified Siegel series is an important factor that appears in the Fourier coefficient of the Siegel Eisenstein series.Many formulas for the ramified Siegel series under various conditions are already known.However, an explicit formula for the general case has not yet been obtained.We derive a formula for the Siegel series with arbitrary dimension $n$, assuming that the additive character $\psi$ is primitive.Our results cover nonarchimedean, non-dyadic local fields $F$, including the case $F=\mathbb{Q}_p$.We also give explicit values of the ramified Siegel series for degrees $n=1, 2,$ and $3$.