Recursion formulas for the Fourier coefficients of Siegel Eisenstein series of an odd prime level
Keiichi Gunji
TL;DR
This work analyzes the ramified $p$-Euler factors of Siegel Eisenstein series of level $p$ with trivial or quadratic character, developing a framework that diagonalizes the ramified components via $U(p)$-eigenfunctions and provides explicit recursion formulas. By combining functional equations with inductive relations, the author derives a closed recursion for the $U(p)$-characteristic ramified Siegel series $S_n^{(\nu)}(\psi,N,s)$ in terms of lower-rank data, with explicit base cases and the key transition term $H_n(\psi,N,s)$ (or $H_n(\chi_p,N,s)$). The results yield practical, computable expressions for Fourier coefficients of Siegel Eisenstein series at level $p$ and extend Sato–Hironaka-type analyses to the ramified setting, including both parity-dependent and character-specific intricacies. The approach refines Katsurada’s inductive framework and leverages the $U(p)$-orbit decomposition to produce an explicit, checkable recursion that applies to all cusps and both characters, enabling efficient computation of Fourier coefficients in low and high degrees. These contributions advance explicit arithmetic of Siegel modular forms at ramified primes and provide tools for further exploration of Eisenstein series with level structure.
Abstract
In this paper we treat the Fourier coefficients of Siegel Eisenstein series of level $p$ with trivial or quadratic character, for an odd prime $p$. The Euler $p$-factor of the Fourier coefficient is called the ramified Siegel series. First we show that the ramified Siegel series attached to each cusp can be decomposed to $U(p)$-eigenfunctions explicitly, next we give recursion formulas of such $U(p)$-characteristic ramified Siegel series.