THETA CONSTANTS AND DIFFERENTIAL EQUATIONS

W. Zudilin

April 13, 2000

For any genus g consider the theta constants (thetanulls, or Thetanullwerte)

ϑ_{a}(τ) = ϑ_{(a′,a′′)}(τ) = ∑_{n∈ℤ^{g}} exp(πi^{t}(n + \frac{1}{2}a′)τ (n + \frac{1}{2}a′) + πi^{t}(n + \frac{1}{2}a′)a′′)

attached to the even 2-characteristics a = (a′, a′′) from 𝔎_{+} ⊂ (ℤ/2ℤ)^{2g} = {a : ta′a′′ ≡ 0 mod 2}, where τ = (τ_{jl})_{j,l=1,...,g} is a symmetric g × g-matrix from Siegel half space ℌ_{g} = {τ : Im τ > 0} of degree g. We generalize well-known results of M. Halphen, R. A. Rankin, K. Mahler for theta constants of genus 1 and their derivatives to the case of arbitrary genus.

Consider partial derivations δ,

δ_{jj} = \frac{1}{πi} \frac{∂}{∂τ_{jj}} , j = 1, …, g, δ_{jl} = \frac{1}{2πi} \frac{∂}{∂τ_{jl}} = δ_{lj}, j, l = 1, …, g, j ≠ l,

and logarithmic derivatives of the theta constants

ψ_{a,jl} = ψ_{a,jl}(τ) := \frac{δ_{jl}ϑ_{a}}{ϑ_{a}} = ψ_{a,lj}, a ∈ 𝔎_{+}, j, l = 1, …, g.

To the ψ-functions, we assign the quadratic forms

ψ_{a} = ψ_{a}[z] := ∑_{j,l=1}^{g} ψ_{a,jl} · z_{j}z_{l};

then we define quartic forms

δψ_{a} = δψ_{a}[z] := ∑_{j,l,m,p=1}^{g} δ_{jl}ψ_{a,mp} · z_{j}z_{l}z_{m}z_{p}.

Typeset by AMS-TEX

Key words and phrases. Theta constants, nonlinear partial differential equations, algebraic independence.

Theorem 1 [Z]. The logarithmic derivatives of theta constants satisfy the system of partial differential equations

ϑ_{a}^{4} · δψ_{a} = \frac{1}{2^{g−2}} ∑_{b∈𝔎_{+}}(−1)^{<a,b>}ϑ_{b}^{4} · ψ_{b}^{2} − 2ϑ_{a}^{4} · ψ_{a}^{2}, a ∈ 𝔎_{+},

where <a, b> = ^{t}a′b′′ −^{t}b′a′′ mod 2.

Consider the rings

Q_{g} = ℚ[ϑ_{a}, ψ_{a,jl}]_{a∈𝔎_{+};j,l=1,...,g} and Q′_{g} = ℚ[ψ_{a,jl}]_{a∈𝔎_{+};j,l=1,...,g}.

Theorem 1 then implies that the fraction field of Q_{g} is stable under δ_{jl}-derivations.

Theorem 2 [BZ]. The ring Q_{g} has transcendence degree 2g^{2} + g over ℚ.

For small genus g = 1, 2, 3 the results above can be sharpened as follows (see [BZ], [Z]):

(i) theta constants in genus ≤ 3 are algebraic over the fraction field of Q′_{g}; (ii) the rings Q′_{1} and Q′_{2} (and the fraction field of Q′_{3}) are δ-stable.

In particular, the rings Q′_{1}, Q′_{2}, and Q′_{3} have transcendance degree 3, 10, and 21 over ℚ respectively.

The proof of Theorem 1 lies on Riemann’s relations for abelian theta functions and the heat equation, while the proof of Theorem 2 combines results from differential Galois theory with a modular interpretation of the period matrix of abelian varieties. The constant 2g^{2} + g is dim Sp_{2g}(ℂ), and the sympectic group Sp_{2g}(ℂ) occurs as a differential Galois group of a Picard–Vessiot extension for a system of linear partial differential equations.

This talk was partly supported by Russian Foundation for Basic Research, grant no. 00-01-10527.

References

[BZ] D. Bertrand and W. Zudilin, On the transcendence degree of the differential field generated by Siegel modular forms, Prépublication de l’Institut de Mathématiques de Jussieu, no. 248,

     Mars 2000, 20 pp.; http://xxx.lanl.gov/abs/math/0006176.

[Z] W. Zudilin, Theta constants and differential equations, Mat. Sb. 191 (2000), no. 12, 46 pp.

Moscow Lomonosov State University