Dirichlet series in the theory of Siegel modular forms

Kitaoka, Yoshiyuki

doi:10.1017/s0027763000020973

Cited by 68 publications

(32 citation statements)

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“…It follows from [22] that computing F p (B; X). We will state these recursion relations below (without proof); first we will need to introduce some of the notation from [15].…”

Section: Computing the Numbers A(r)mentioning

confidence: 99%

A mass formula for unimodular lattices with no roots

King

2002

Math. Comp.

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Abstract. We derive a mass formula for n-dimensional unimodular lattices having any prescribed root system. We use Katsurada's formula for the Fourier coefficients of Siegel Eisenstein series to compute these masses for all root systems of even unimodular 32-dimensional lattices and odd unimodular lattices of dimension n ≤ 30. In particular, we find the mass of even unimodular 32-dimensional lattices with no roots, and the mass of odd unimodular lattices with no roots in dimension n ≤ 30, verifying Bacher and Venkov's enumerations in dimensions 27 and 28. We also compute better lower bounds on the number of inequivalent unimodular lattices in dimensions 26 to 30 than those afforded by the Minkowski-Siegel mass constants.

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“…It follows from [22] that computing F p (B; X). We will state these recursion relations below (without proof); first we will need to introduce some of the notation from [15].…”

Section: Computing the Numbers A(r)mentioning

confidence: 99%

A mass formula for unimodular lattices with no roots

King

2002

Math. Comp.

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“…On the other hand it might be possible to investigate a s (T) in analogy with Y. Kitaoka's procedure [11] in the case of the Siegel half-space. But it seems to be plausible that the Fourier-coefficients of the Eisenstein-series can only be expressed by well-known functions, whenever the degree n is "sufficiently small".…”

Section: Aloys Kriegmentioning

confidence: 99%

Eisenstein-series on real, complex, and quaternionic half-spaces

Krieg¹

1988

Pacific J. Math.

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The real, complex, and quaternionic half-spaces are introduced in certain analogy with the Siegel half-space. The modified symplectic group acts on the attached half-space in the usual way. At first properties of these half-spaces considered as symmetric spaces are derived. Then a fundamental domain with respect to the modified modular group, which consists of integral modified symplectic matrices, is constructed. The behavior of convergence of the corresponding Eisenstein-series is determined carefully. The Fourier-coefficients of the Eisenstein-series are calculated explicitly, whenever the degree is sufficiently small.

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“…For each non-degenerate half-integral symmetric matrix B of degree n over the ring Z p of p-adic integers, we define the local Siegel series with complex parameter s by det(2B )) (cf. [16]). Let B be a non-degenerate symmetric matrix of degree n − 1 over a subring R of Z p satisfying the condition (1) (B + t r B r B )/4 is a half-integral symmetric matrix over R for some r B ∈ R n−1 .…”

Section: Introductionmentioning

confidence: 99%