On the rationality of a paramodular Siegel Eisenstein series

TL;DR

The paper proves that the Fourier coefficients of the paramodular Siegel Eisenstein series $E_{k,\u03b7}$ of level $N^2$ and even weight $k\ge4$ are algebraic, lying in a number field. It combines the explicit Fourier expansion from Pierce--Schmidt (2025) with a rank-based decomposition of coefficients, analyzing rank 1 and rank 2 terms via Gauss sums and Dirichlet $L$-values. Rank 1 coefficients lie in ${\mathbb Q}(\u03b7, G(\u03b7))$, while rank 2 coefficients lie in ${\mathbb Q}(\u03b7, \sqrt{|D|} G(\alpha), G(\beta), i)$, implying all coefficients lie in ${\mathbb Q}(i, \u03b7, \zeta_N)$. A key technical input is the rationality of local integrals $K(k,T,\chi_p)$ and the result that $G(\alpha)\sqrt{|D|}$ belongs to ${\mathbb Q}(\u03b7, \zeta_N, i)$. These findings sharpen automorphic rationality phenomena for paramodular level Eisenstein series and connect coefficient algebraicity to explicit Dirichlet-character data and adelic local factors.

Abstract

We consider the rationality of the Fourier coefficients of a particular paramodular Siegel Eisenstein series of level $N^2$ with weight $k\geq 4$. We show that the coefficients lie in a number field.

Theorems & Definitions (1)

• Theorem 1.1