A semi-ordinary $p$-stabilization of Siegel Eisenstein series for symplectic groups and its $p$-adic interpolation

TL;DR

This paper constructs a $p$-stabilization of Siegel Eisenstein series for symplectic groups, providing explicit Fourier coefficient formulas and establishing their $p$-adic interpolation, leading to the creation of a $ ext{Lambda}$-adic family that generalizes classical Eisenstein series.

Contribution

It introduces a new $p$-stabilization method for Siegel Eisenstein series and develops their $p$-adic interpolation, extending the theory beyond the classical case.

Findings

Explicit Fourier coefficient formulas for $p$-stabilized series

Existence of a $ ext{Lambda}$-adic form interpolating families of Eisenstein series

Generalization of ordinary $ ext{Lambda}$-adic Eisenstein series for ${ m GL}(2)${}

Abstract

For any rational prime $p$, we define a certain $p$-stabilization of holomorphic Siegel Eisenstein series for the symplectic group ${\rm Sp}(2n)_{/\mathbb{Q}}$ of an arbitrary genus $n \ge 1$. In addition, we derive an explicit formula for the Fourier coefficients and conclude their $p$-adic interpolation problems. Consequently, for any odd prime $p$, we deduce the existence of a $\Lambda$-adic form (in the sense of A. Wiles, H. Hida and R.L. Taylor) such that after taking a suitable constant multiple, it interpolates $p$-adic analytic families of the above-mentioned $p$-stabilized Siegel Eisenstein series with nebentypus characters locally trivial at $p$ and Siegel Eisenstein series with nebentypus characters locally non-trivial at $p$ simultaneously. This can be viewed as a quite natural generalization of the ordinary $\Lambda$-adic Eisenstein series for ${\rm GL}(2)$.