[Paper Review] The Siegel variance formula for quadratic forms

Naser T. Sardari|arXiv (Cornell University)|Apr 17, 2019

Advanced Algebra and Geometry24 references4 citations

TL;DR

This paper introduces a smooth variance sum for quadratic forms using oscillator representations and Siegel modular forms, deriving a precise formula for the variance of Fourier coefficients. It establishes a sharp upper bound for the variance when n=1 and applies this to prove a cutoff phenomenon in the representation of integers by unimodular lattices and to achieve an optimal diophantine exponent bound for p-integral points on quadrics.

ABSTRACT

We introduce a smooth variance sum associated to a pair of positive definite symmetric integral matrices $A_{m imes m}$ and $B_{n imes n}$, where $m\geq n$. By using the oscillator representation, we give a formula for this variance sum in terms of a smooth sum over the square of a functional evaluated on the $B$-th Fourier coefficients of the vector valued holomorphic Siegel modular forms which are Hecke eigenforms and obtained by the theta transfer from $O_{A_{m imes m}}$. By using the Ramanujan bound on the Fourier coefficients of the holomorphic cusp forms, we give a sharp upper bound on this variance when $n=1$. As applications, we prove a cutoff phenomenon for the probability that a unimodular lattice of dimension $m$ represents a given even number. This gives an optimal upper bound on the sphere packing density of almost all even unimodular lattices. Furthermore, we generalize the result of Bourgain, Rudnick and Sarnak~\cite{Bourgain}, and also give an optimal bound on the diophantine exponent of the $p$-integral points on any positive definite $d$-dimensional quadric, where $d\geq 3$. This improves the best known bounds due to Ghosh, Gorodnik and Nevo~\cite{GGN} into an optimal bound.

Motivation & Objective

• To develop a smooth variance sum for pairs of positive definite symmetric integral matrices A and B with m ≥ n.
• To derive a formula for this variance in terms of Fourier coefficients of vector-valued holomorphic Siegel modular forms that are Hecke eigenforms via theta lift from O_A.
• To establish a sharp upper bound on the variance when n=1 using the Ramanujan bound on cusp form Fourier coefficients.
• To apply the variance formula to prove a cutoff phenomenon in the probability that unimodular lattices represent a given even integer.
• To generalize results of Bourgain-Rudnick-Sarnak and improve the best-known diophantine exponent bound for p-integral points on positive definite d-dimensional quadrics (d ≥ 3).

Proposed method

• Introduce a smooth variance sum associated with a pair of positive definite symmetric integral matrices A (m×m) and B (n×n), with m ≥ n.
• Use the oscillator representation to relate the variance sum to the square of a functional evaluated on B-th Fourier coefficients of vector-valued holomorphic Siegel modular forms.
• Apply the theta lift construction to obtain Hecke eigenforms from the orthogonal group O_A.
• Leverage the Ramanujan bound on Fourier coefficients of holomorphic cusp forms to derive a sharp upper bound for the variance when n=1.
• Use the variance formula to analyze the probability of representation of integers by unimodular lattices under the Siegel mass measure.
• Apply the variance formula to derive optimal bounds on the diophantine exponent for p-integral points on positive definite quadrics in d ≥ 3 variables.

Experimental results

Research questions

• RQ1 What is the precise formula for the smooth variance sum of quadratic forms in terms of Siegel modular forms?
• RQ2 How can the Ramanujan bound on Fourier coefficients be used to derive a sharp upper bound for the variance when n=1?
• RQ3 Does a cutoff phenomenon occur in the probability that an odd (or even) unimodular lattice of dimension m represents a given odd (or even) integer?
• RQ4 Can the diophantine exponent for p-integral points on d-dimensional positive definite quadrics be improved to an optimal bound?
• RQ5 What is the implication of the variance formula for the sphere packing density of almost all even unimodular lattices?

Key findings

• The paper establishes a complete formula for the Siegel variance sum in terms of the B-th Fourier coefficients of Hecke eigenforms arising from the theta lift of O_A.
• For n=1, a sharp upper bound of order 5113^(-t) is derived for the probability that a unimodular lattice of dimension m represents a given even integer 2q, with cutoff at q ~ m/(2πe).
• The cutoff phenomenon is confirmed: for q ≤ m/(2πe) + (0.5/πe)log(m) - t - 1, the representation probability is ≤ 5113^(-t), and for q ≥ m/(2πe) + (1.6/πe)log(m) + t, it is ≥ 1 - 5113^(-t), with t = o(m).
• The sphere packing density of almost all even unimodular lattices is shown to be o(m^{2+ε}2^{-m}) with μ₀-probability 1 + O(m^{-δ+ε}) for any δ > 0 and ε > 0.
• The diophantine exponent for p-integral points on any positive definite d-dimensional quadric (d ≥ 3) is bounded optimally, improving upon the previous best bound by Ghosh-Gorodnik-Nevo.
• Conjecture 1.2 suggests that for large m, every even integer 2q ≥ (1+ε)m/(2πe) is representable by every even unimodular lattice, implying a sphere packing density of (1+ε)^m 2^{-m} for such lattices.