Siegel Eisenstein Series with Paramodular Level

TL;DR

The paper constructs a paramodular Siegel Eisenstein series of level $N^2$ from a primitive Dirichlet character of conductor $N$ and computes its Fourier expansion for weight $k\ge4$, proving that the form is a paramodular newform and that its adelization yields an irreducible automorphic representation. The authors develop an adelic framework using the global induced representation $J_\chi(s)$, choose local sections with paramodular level data, and apply a descent to the Siegel upper half-space to obtain a holomorphic Siegel modular form on $\mathbb{H}_2$. Fourier coefficients $c_T$ are analyzed via a product of archimedean and non-archimedean local integrals, with explicit formulas derived for both unramified and ramified places, including rank-1 and rank-2 cases and, in particular, a detailed rank-2 computation that involves L-functions, Gauss sums, and elliptic-curve point counts. The results yield explicit expressions for the Fourier expansion, confirm the newform property, and illuminate the paramodular representation-theoretic structure of the associated automorphic representation, contributing to the understanding of paramodular forms and their L-functions.

Abstract

Starting with a primitive Dirichlet character of conductor $N$, we construct a paramodular Siegel Eisenstein series of level $N^2$ and weight $k\geq4$. We calculate the Fourier expansion of the holomorphic Siegel modular form thus constructed. The function is a paramodular newform, and its adelization generates an irreducible automorphic representation.