Using Katsurada’s determination of the Eisenstein series to compute Siegel eigenforms

Abstract

Abstract. We compute Hecke eigenform bases of spaces of level one, degree three Siegel modular forms and 2-Euler factors of the eigenforms through weight 22. Our method uses the Fourier coefficients of Siegel Eisenstein series, which are fully known and computationally tractable by the work of H. Katsurada; we also use P. Garrett's decomposition of the pullback of the Eisenstein series through the Witt map. Our results support I. Miyawaki's conjectural lift, and they give examples of eigenforms that are congruence neighbors.

Computation of Gross-Keating invariants

“…There is a different computer code to obtain F (B; X), which is publicly available. King's LISP code [Kin03] and its recent extension by King, Poor, Shurman, and Yuen in [KPSY18] computes F (B; X) using the genus symbol of B. This is based on the recursive formula in [Kat99], which is more complicated than (1.5).…”

“…Although this is a wellknown result, it still requires some work to convert the steps described in the literature (for example, [Wat60,Cas78]) into an explicit algorithm. We have included a simple procedure for completeness, which we learned from the code used in [KPSY18].…”

“…Meanwhile, Katsurada obtained a recursive formula for the local Siegel series of the half-integral matrix B of any degree over Z p in [Kat99]. Katsurada's formula is of central importance in the algorithmic calculation of the Fourier coefficients of the Siegel-Eisenstein series, and therefore has made a significant impact in the computational area of Siegel modular forms and the classification of even unimodular lattices in high dimensions [Kin03,KPSY18].…”

“…To the best of our knowledge, this is the first computer implementation to calculate the Gross-Keating invariants. There is a LISP code for the Siegel series, used in [Kin03,KPSY18] based on Katsurada's formula [Kat99], but it is more specialized in computing the Fourier series of the Siegel-Eisenstein series, and does not address the Gross-Keating invariants; see Subsection 1.5 for more comments.…”

Quinary forms and paramodular forms

“…There are some tables of paramodular forms, based on Fourier series expansions, due mostly to Poor, Yuen and some coauthors (see [PY15], [PSY17], [KPSY18] and [BPP + 19]). A different approach using quinary forms, analogous to Birch's use of ternary quadratic forms, can be used to compute Hecke eigenvalues more easily.…”