[Paper Review] An explicit formula for the Siegel series of a quadratic form over a non-archimedian local field

Tamotsu Ikeda, Hidenori Katsurada|arXiv (Cornell University)|Feb 22, 2016

Advanced Algebra and Geometry27 references3 citations

TL;DR

This paper provides an explicit formula for the Siegel series of a half-integral matrix over a non-archimedean local field of characteristic zero, expressing it universally in terms of the extended Gross-Keating (EGK) datum of the matrix. The key contribution is a unified formula that works for all primes, including p=2, and establishes that the Siegel series is completely determined by the EGK invariant, generalizing and simplifying prior results for p-adic fields.

ABSTRACT

Let F be a non-archimedian local field of characteristic 0, and O the ring of integres in F. We give an explicit formula for the Siegel series of a half-integral matrix over O. This formula expresses the Siegel series of a half-integral matrix $B$ explicitly in terms of the Gross-Keating invariant of $B$ and its related invariants.

Motivation & Objective

• To provide a universal, explicit formula for the Siegel series of a half-integral matrix over any non-archimedean local field of characteristic zero.
• To resolve the lack of a unified treatment for the case p=2 in previous formulas, which were complicated and separate from the odd p case.
• To clarify which invariants govern the Siegel series by introducing the extended Gross-Keating (EGK) datum as the complete determining invariant.
• To generalize and reformulate earlier results—particularly those of Katsuda (2017) for p-adic fields—into a single, coherent framework applicable to all non-archimedean local fields.

Proposed method

• The authors define a Laurent polynomial $\widetilde{F}(B,X)$ in $X^{1/2}$ and $X^{-1/2}$ associated with the Siegel series $b(B,s)$ of a non-degenerate half-integral matrix $B$.
• They introduce the extended Gross-Keating (EGK) datum $\mathrm{EGK}(B)$, which consists of the block sizes $n_i$, exponents $m_i$, and signs $\zeta_i$ derived from the Jordan decomposition and invariants of the upper-left submatrices of $B$.
• An $\mathrm{EGK}$ datum $G$ is axiomatized as a tuple $(n_1,\ldots,n_r; m_1,\ldots,m_r; \zeta_1,\ldots,\zeta_r)$ satisfying certain integrality and parity conditions.
• A universal Laurent polynomial $\widetilde{\mathcal{F}}(G;Y,X)$ is constructed in terms of $Y = q^{1/2}$ and $X$, with explicit expressions given in Proposition 4.1.
• The main result establishes that $\widetilde{F}(B,X) = \widetilde{\mathcal{F}}(\mathrm{EGK}(B); q^{1/2}, X)$, proving the Siegel series is completely determined by the EGK datum.
• The method handles both dyadic and non-dyadic cases uniformly by refining the definition of invariants in the $p=2$ case.

Experimental results

Research questions

• RQ1 What invariants fully determine the Siegel series of a half-integral matrix over a non-archimedean local field of characteristic zero?
• RQ2 Can a single, unified formula for the Siegel series be constructed that works for both odd and even residual characteristics, especially for $p=2$?
• RQ3 How can the extended Gross-Keating invariant be systematically defined to capture the full arithmetic of the matrix in both dyadic and non-dyadic cases?
• RQ4 Is the Siegel series of a half-integral matrix completely determined by its extended Gross-Keating datum?
• RQ5 Can the formula be used to compute Fourier coefficients of Siegel modular forms and special values of L-functions in a uniform way?

Key findings

• The Siegel series $\widetilde{F}(B,X)$ of a half-integral matrix $B$ is completely determined by its extended Gross-Keating (EGK) datum $\mathrm{EGK}(B)$, as shown by the identity $\widetilde{F}(B,X) = \widetilde{\mathcal{F}}(\mathrm{EGK}(B); q^{1/2}, X)$.
• The formula unifies the previously separate cases for odd $p$ and $p=2$, resolving the complexity and lack of clarity in earlier work by Katsuda (2017).
• For the 10 matrices $B_1, \ldots, B_{10}$ of degree 4 with discriminant $\mathfrak{D}_B = 5$, the computed $\widetilde{F}_i(X)$ values match the known Fourier coefficients of the Duke-Imamoglu-Ikeda lift, confirming consistency with prior computations by Breulmann-Kuss.
• The explicit expressions for $\widetilde{F}_i(X)$ in terms of $S_k = X^k + X^{-k}$ and powers of $q^{-1/2}$ are derived and verified, with $\widetilde{F}_1(X) = 1$ and $\widetilde{F}_5(X) = S_2 + q^{-1/2}S_1 + 1 + q$, among others.
• The extended GK datum $\mathrm{EGK}_2(B)$ captures the $2$-adic structure of $B$, and the formula correctly reproduces the $2$-adic Siegel series for all 10 test matrices.
• The universal polynomial $\widetilde{\mathcal{F}}(G;Y,X)$ is explicitly constructed and shown to depend only on the $\mathrm{EGK}$ datum $G$, enabling systematic computation of Siegel series across all non-archimedean local fields.