THETA CONSTANTS AND DIFFERENTIAL EQUATIONS

W. Zudilin

September 30, 1999

The proof of well-known (since 1996) Nesterenko’s Theorem [1] leans on three important points:

(i) functions

\begin{matrix}P (q) = 1 − 24 ∑_{n=1}^{∞} σ_{1}(n)q^{n}, \\ Q(q) = 1 + 240 ∑_{n=1}^{∞} σ_{3}(n)q^{n},  R(q) = 1 − 504 ∑_{n=1}^{∞} σ_{5}(n)q^{n}, \end{matrix}  (1)

where σ_{k}(n) = ∑_{d|n} d^{k}, are algebraically independent over ℂ(q) (corollary from Mahler’s Theorem, see [2]);

(ii) functions (1) satisfy the system of differential equations

δP = \frac{1}{12}(P^{2} − Q), δQ = \frac{1}{3}(P Q − R), δR = \frac{1}{2}(P R − Q^{2}),

where δ = q \frac{d}{dq} (Ramanujan’s Theorem, see [3]); (iii) functions (1) are holomorphic in the unit circle |q| < 1 and have integral

coefficients of polynomial growth in their Taylor expansions. Functions (1) are algebraic over the field K which is the algebraic extension

of ℂ(q) generated by (one-dimensional) theta constants

\begin{aligned}ϑ_{2} = ϑ_{2}(q) & = 2 ∑_{n=0}^{∞} q^{(n+1/2)^{2}} = 2q^{1/4} + 2q^{9/4} + 2q^{25/4} + ⋯, \\ ϑ_{3} = ϑ_{3}(q) & = 1 + 2 ∑_{n=1}^{∞} q^{n^{2}} = 1 + 2q + 2q^{4} + 2q^{9} + ⋯, \\ ϑ_{4} = ϑ_{4}(q) & = 1 + 2 ∑_{n=1}^{∞}(−1)^{n}q^{n^{2}} = 1 − 2q + 2q^{4} − 2q^{9} + ⋯ \end{aligned}  (2)

and all their δ-derivatives (see, e.g., [4], Chapter 6). Note that the variable q is connected with a modular parameter τ by the rule q = e^{πiτ}, Im τ > 0 (see [5],

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

Chapter 21) and this assignment transforms Taylor expansions to Fourier ones. Jacobi’s third order differential equation for functions (2) is too complicated for applications, but there is another “basis” of the field K, namely logarithmic δ-de-rivatives of theta constants

ψ_{2} = \frac{δϑ_{2}}{ϑ_{2}} ,  ψ_{3} = \frac{δϑ_{3}}{ϑ_{3}} ,  ψ_{4} = \frac{δϑ_{4}}{ϑ_{4}} ,  (3)

and another form of “three important points”:

(i) functions (3) are algebraically independent over ℂ(q); (ii) functions (3) satisfy the system of differential equations

δ(ψ_{2} + ψ_{3}) = 4ψ_{2}ψ_{3}, δ(ψ_{3} + ψ_{4}) = 4ψ_{3}ψ_{4}, δ(ψ_{4} + ψ_{2}) = 4ψ_{4}ψ_{2}  (4)

(Halpen’s Theorem proved before Ramanujan, see [6]); (iii) functions (3) are holomorphic in the unit circle |q| < 1 and have integral

coefficients (except ψ_{2}(0) = 1/4) of polynomial growth in their Taylor expansions.

Moreover, functions (3) are connected with theta constants by formulas

ϑ_{2}^{4} = \frac{1}{4}(ψ_{3} − ψ_{4}), ϑ_{3}^{4} = \frac{1}{4}(ψ_{2} − ψ_{4}), ϑ_{4}^{4} = \frac{1}{4}(ψ_{2} − ψ_{3});  (5)

(see, in slightly different notation, [7], Chapter 11). Recently Yu. Nesterenko [8] proved an equivalent to his theorem statement in terms of functions (3) and discovered this way easier than previous one because of known information on the structure of solutions of Halphen’s system (4).

My own interest to theta constants and their differential properties is initiated by Nesterenko’s Theorem. In this talk I state some multi-dimensional analogues of (ii)–(iii) for generalized Halphen’s functions.

We fix a dimension (or a genus) g and define g-dimensional theta function (with characteristic a = (a′, a′′) ∈ ℤ^{2g}) by the series

ϑ_{a}(z, T) = ∑_{n∈ℤ^{g}} exp(πi(n + \frac{1}{2}a′)^{t}T(n + \frac{1}{2}a′) + 2i(n + \frac{1}{2}a′)^{t}(z + \frac{π}{2} a′′)) (6)

where z ∈ ℂ^{g} and T is a symmetric complex matrix with a positive imaginary part (the space ℌ_{g} of dimension g(g + 1)/2 of such matrices is an open set in ℂ^{g×g} and is called Siegel upper half-space). We write vectors as columns and denote the transposition by index t; this means that n^{t}z is the scalar product of vectors n and z. The series (6) converges absolutely and uniformly with respect to z and T on any compact subset of ℂ^{g} ×ℌ_{g}; therefore it determines the holomorphic function on ℂ^{g} × ℌ_{g} (see [9]).

The quasi-periodicity of (6) with respect to the lattice πℤ^{g}+πTℤ^{g} ⊂ ℂ^{g} allows us to consider the only functions with reduced characteristics a ∈ ℤ^{2g}/2ℤ^{2g}. Usually we omit the parameter T writing theta functions and define theta constants as the values of even theta functions (6) at the point z = 0. Since

ϑ_{a}(−z) = (−1)^{|a|}ϑ_{a}(z), |a| = a′^{t}a′′, a = (a′, a′′) ∈ ℤ^{2g}

(see, e.g., [9]), we see that the function ϑ_{a}(z) is even if and only if the number |a| is even. Hence all non-trivial theta constants are assigned to even characteristics from the set 𝔎_{0} = {a ∈ ℤ^{2g}/2ℤ^{2g} : |a| ≡ 0 (mod 2)}, #𝔎_{0} = 2^{g−1}(2^{g} + 1).

Consider differential operators

δ_{jk} = \frac{1}{πi} \frac{∂}{∂τ_{jk}} , j, k = 1, …, g,

and all possible “logarithmic” derivatives of theta constants:

ψ_{a,jk} = ψ_{a,jk}(T) = \frac{δ_{jk}ϑ_{a}}{ϑ_{a}} ,  a ∈ 𝔎_{0}, j, k = 1, …, g.  (7)

From the definition given above, we get the almost obvious result on arithmetical properties of functions (7).

Theorem 1. Functions (7) have the following Fourier expansions:

ψ_{a,jk}(T) = ψ_{a,jk}^{(0)} + ∑_{N} ψ_{a,jk}^{(N)} · exp(πi · tr(NT)),  (8)

where tr is a matrix trace and the summation in (8) is over non-negative symmetric matrices with integral elements,

ψ_{a,jk}^{(0)} = \{ \begin{matrix}0 & if a′_{j}a′_{k} = 0, \\ \frac{1}{4} & if j = k and a′_{j} = 1, \\ \frac{1}{2} \frac{e^{πiτ_{jk}/2} − e^{−πiτ_{jk}/2}}{e^{πiτ_{jk}/2} + e^{−πiτ_{jk}/2}} & if j ≠ k and a′_{j} = a′_{k} = 1, \end{matrix}

and ψ_{a,jk}^{(N)} ∈ ℤ.

Theorem 1 is the point (iii) in our program. To claim results on differential equations, I need a certain compact notation for functions (7). Put

ψ_{a} = z^{t}Ψ_{a}z, Ψ_{a} = (ψ_{a,jk})_{j,k=1,...,g},  a ∈ 𝔎_{0},  (9)
δ = z^{t}Δz,  Δ = (δ_{jk})_{j,k=1,...,g}.  (10)

Theorem 2. Functions (9) satisfy the system of differential equations

ϑ_{a}^{4} · δψ_{a} = \frac{1}{2^{g−2}} ∑_{b∈𝔎_{0}}(−1)^{|a,b|}ϑ_{b}^{4} · ψ_{b}^{2} − 2ϑ_{a}^{4} · ψ_{a}^{2},  a ∈ 𝔎_{0},  (11)

where |a, b| = a′^{t}b′′ − b′^{t}a′′ ≡ |b, a| (mod 2).

Notice that non-compact form of the system (11) is

ϑ_{a}^{4} · δ_{ls}ψ_{a,jk} = \frac{1}{2^{g−2}N_{jkls}} ∑_{b∈𝔎_{0}}(−1)^{|a,b|}ϑ_{b}^{4} ·\begin{matrix}∑ & ∗ \\ j_{1},k_{1},l_{1},s_{1} \end{matrix} ψ_{b,j_{1}k_{1}}ψ_{b,l_{1}s_{1}} − ϑ_{a}^{4} · ψ_{a,ls}ψ_{a,jk}

− \frac{1}{N_{jkls}} ϑ_{a}^{4} ·\begin{matrix}∑ & ∗ \\ j_{1},k_{1},l_{1},s_{1} \end{matrix} ψ_{a,j_{1}k_{1}}ψ_{a,l_{1}s_{1}},  a ∈ 𝔎_{0},  j, k, l, s = 1, …, g,

where denote by ^{∑∗} the summation over all different perturbations of the set {j, k, l, s} and by N_{jkls} the number of such perturbations.

It follows from Theorem 2 that the algebraic extension of ℂ(τ_{jk}, e^{πiτ_{jk}})_{j,k=1,...,m} generated by theta constants and all their partial derivatives has a finite degree of algebraic independence over ℂ.

Theorem 3. For any (even or not) given characteristic c ∈ ℤ^{g}, c ≠ 0, functions (9) satisfy the system of differential equations

ϑ_{a+c}^{2}ϑ_{a}^{2}δ(ψ_{a+c} + ψ_{a}) = \frac{1}{2^{g−1}} ∑_{b∈𝔎}(−1)^{|a,b|}(−1)^{c′^{t}(a′′+b′′)}ϑ_{b+c}^{2}ϑ_{b}^{2}(ψ_{b+c} + ψ_{b})^{2}

+ 4ϑ_{a+c}^{2}ϑ_{a}^{2}ψ_{a+c}ψ_{a} − 2ϑ_{a+c}^{2}ϑ_{a}^{2}(ψ_{a+c} + ψ_{a})^{2},  a ∈ 𝔎,  (12)

where 𝔎 = 𝔎_{c} = {a ∈ ℤ^{2g}/2ℤ^{2g} : a, a+c ∈ 𝔎_{0}}. Moreover, theta constants and (9) have relations

\begin{aligned}ϑ_{a+c}^{2}ϑ_{a}^{2} & = \frac{1}{2^{g}} ∑_{b∈𝔎}(−1)^{|a,b|}(−1)^{c′^{t}(a′′+b′′)}ϑ_{b+c}^{2}ϑ_{b}^{2}, a ∈ 𝔎, & (13) \\ ϑ_{a+c}^{2}ϑ_{a}^{2}(ψ_{a+c} + ψ_{a}) & = \frac{1}{2^{g}} ∑_{b∈𝔎}(−1)^{|a,b|}(−1)^{c′^{t}(a′′+b′′)}ϑ_{b+c}^{2}ϑ_{b}^{2}(ψ_{b+c} + ψ_{b}), & (14) \end{aligned}

a ∈ 𝔎.

In spite of a symmetry of (11) and (12), these systems have a weak likeness with Halphen’s one (4). First there are a lot of relations between participants of (11) or (12); secondly the sets of irregular points of these systems contain the variety of zeros of theta constants.

However, as was shown in the papers [10] and [11], in the case g = 2 it is possible to state a system similar to (4).

Theorem 4 (Ohyama’s Theorem). Suppose even characteristics a_{1}, a_{2}, a_{3}, a_{4} ∈ ℤ^{4}/2ℤ^{4} are “vertices of non-trivial parallelogram” (in other words, a_{1}−a_{2} ≡ a_{3}−a_{4} (mod 2ℤ^{4})). Then corresponding functions ψ_{a_{1}}, ψ_{a_{2}}, ψ_{a_{3}}, ψ_{a_{4}} satisfy the system of differential equations

δ(ψ_{a_{1}} +ψ_{a_{2}} +ψ_{a_{3}} +ψ_{a_{4}}) = (ψ_{a_{1}} +ψ_{a_{2}} +ψ_{a_{3}} +ψ_{a_{4}})^{2}−2(ψ_{a_{1}}^{2} +ψ_{a_{2}}^{2} +ψ_{a_{3}}^{2} +ψ_{a_{4}}^{2}). (15)

Unfortunately, in the case g > 2, a system of Halphen’s type for (9) is not still obtained since nowadays. In the case g = 3, I can reduce from Theorem 3 that δ-derivatives of functions (9) can be expressed as rational functions in terms of themselves with degree of denominators at most 9.

I skip the proof of Theorem 2 and give hints for proving Theorem 3 and reducing Theorem 4 from it.

By O(z^{n}), n = 0, 1, 2, …, denote the class of holomorphic near the point z = 0 functions f(z) that have Taylor expansions starting from degree n, that is

\frac{∂^{m_{1}+···+m_{g}}f}{∂z_{1}^{m_{1}} ⋯ ∂z_{g}^{m_{g}}} (0) = 0, m_{1} + ⋯ + m_{g} < n.

Lemma 1. In the neighborhood of the point z = 0 following expansions are valid:

ϑ_{a}(z) = ϑ_{a} · (1 − 2ψ_{a} + \frac{2}{3}ψ_{a}^{2} + \frac{2}{3}δψ_{a}) + O(z^{6}), a ∈ 𝔎_{0},

where functions ψ_{a}, a ∈ 𝔎_{0}, and differential operator δ are defined by (9) and (10).

Lemma 1 follows from the multi-dimensional analogue of the heat equation:

δ_{jk}ϑ_{a}(z, T) = −\frac{1}{4} \frac{∂^{2}}{∂z_{j} ∂z_{k}} ϑ_{a}(z, T), j, k = 1, …, g, a ∈ ℤ^{2g}

(see, e.g., [12], Teil 1, Kapitel 1).

Fixing c ∈ ℤ^{2g}, we introduce the notation

θ_{a}(z_{1}, z_{2}, z_{3}, z_{4}) = ϑ_{a+c}(z_{1})ϑ_{a+c}(z_{2})ϑ_{a}(z_{3})ϑ_{a}(z_{4}),

useful from the point of view of Riemann’s addition formulas (see [12], Teil 2, Kapitel VII). Two specializations of these formulas give us following assertion.

Lemma 2. For each a ∈ 𝔎_{c} we have

(−1)^{c′^{t}a′′}θ_{a}(z, z, z, z) = \frac{1}{2^{g}} ∑_{b∈𝔎_{c}}(−1)^{|a,b|}(−1)^{c′^{t}b′′}θ_{b}(2z, 0, 0, 0),  (16)

(−1)^{c′^{t}a′′}θ_{a}(z, 0, z, 0) = \frac{1}{2^{g}} ∑_{b∈𝔎_{c}}(−1)^{|a,b|}(−1)^{c′^{t}b′′}θ_{b}(z, 0, z, 0).  (17)

Actually the summation in right-hand sides of identities (16) and (17) can be twice decreased, e.g., we can group summands θ_{b}(2z, 0, 0, 0), θ_{b+c}(2z, 0, 0, 0) in (16), and θ_{b}(z, 0, z, 0), θ_{b+c}(z, 0, z, 0) in (17) into pairs. Such a division into pairs is well defined because the map b ↦ b + c is a (non-trivial) bijection of the set 𝔎. Build a factor-set 𝔎^{∗} = 𝔎_{c}^{∗} choosing one element from each pair. Then we rewrite identities (16) and (17) in a form

θ_{a}(z, z, z, z) = \frac{1}{2^{g−1}} ∑_{b∈𝔎_{c}^{∗}}(−1)^{|a,b|}(−1)^{c′^{t}(a′′+b′′)} \frac{θ_{b}(2z, 0, 0, 0) + θ_{b}(0, 0, 2z, 0)}{2}, (18)

θ_{a}(z, 0, z, 0) = \frac{1}{2^{g−1}} ∑_{b∈𝔎_{c}^{∗}}(−1)^{|a,b|}(−1)^{c′^{t}(a′′+b′′)}θ_{b}(z, 0, z, 0)  (19)

for each a ∈ 𝔎^{∗}.

Lemma 3. The symmetric matrix

M = M_{c} = \frac{1}{2^{g−1}} ((−1)^{|a,b|}(−1)^{c′^{t}(a′′+b′′)})_{a,b∈𝔎^{∗}}

of size 2^{g−2}(2^{g−1}+1) corresponding to identities (18) and (19), satisfies the equation

M^{2} = \frac{1}{2}(M + E),

where E is an identity matrix of the same size.

Proof of Theorem 3. Using Lemma 1, we have

ϑ_{1}^{2}(z)ϑ_{2}^{2}(z) =ϑ_{1}^{2}ϑ_{2}^{2} · (1 − 2ψ_{1} + \frac{2}{3}ψ_{1}^{2} + \frac{2}{3}δψ_{1})^{2}(1 − 2ψ_{2} + \frac{2}{3}ψ_{2}^{2} + \frac{2}{3}δψ_{2})^{2} + O(z^{6})

= ϑ_{1}^{2}ϑ_{2}^{2} · (1 − 4(ψ_{1} + ψ_{2}) + \frac{16}{3}ψ_{1}ψ_{2} + \frac{16}{3}(ψ_{1} + ψ_{2})^{2} + \frac{4}{3}δ(ψ_{1} + ψ_{2})) + O(z^{6}),

\frac{ϑ_{1}(2z)ϑ_{2} + ϑ_{2}(2z)ϑ_{1}}{2}

= ϑ_{1}ϑ_{2} · (1 − 4(ψ_{1} + ψ_{2}) − \frac{32}{3}ψ_{1}ψ_{2} + \frac{16}{3}(ψ_{1} + ψ_{2})^{2} + \frac{16}{3}δ(ψ_{1} + ψ_{2})) + O(z^{6}),

ϑ_{1}(z)ϑ_{2}(z)

= ϑ_{1}ϑ_{2} · (1 − 2(ψ_{1} + ψ_{2}) + \frac{8}{3}ψ_{1}ψ_{2} + \frac{2}{3}(ψ_{1} + ψ_{2})^{2} + \frac{2}{3}δ(ψ_{1} + ψ_{2})) + O(z^{6}),

where I omit the numeration of theta constants and their logarithmic derivatives for simplicity. Substituting last expansions in (18) and (19) and comparing the homogeneous components of degree 0, 2, and 4, we obtain (13), (14), and

4W + 4X + Y = 4M(−2W + X + Y ),  (20)
4W + X + Y = M(4W + X + Y ),  (21)

where

\begin{matrix}W = (ϑ_{a+c}^{2}ϑ_{a}^{2}ψ_{a+c}ψ_{a})_{a∈𝔎^{∗}}, X = (ϑ_{a+c}^{2}ϑ_{a}^{2}(ψ_{a+c} + ψ_{a})^{2})_{a∈𝔎^{∗}}, \\ Y = (ϑ_{a+c}^{2}ϑ_{a}^{2}δ(ψ_{a+c} + ψ_{a}))_{a∈𝔎^{∗}} \end{matrix}

are corresponding vector-columns. Since

δ(ϑ_{a+c}^{2}ϑ_{a}^{2}(ψ_{a+c} + ψ_{a})) = 2ϑ_{a+c}^{2}ϑ_{a}^{2}(ψ_{a+c} + ψ_{a})^{2} + ϑ_{a+c}^{2}ϑ_{a}^{2}δ(ψ_{a+c} + ψ_{a}), a ∈ 𝔎^{∗},

δ-differentiation of (14) gives us

2X + Y = M(2X + Y ).  (22)

Using (20)–(22) and Lemma 3, we obtain

Y = 4W + 2(M − E)X,  (23)

that is, exactly, the matrix form of the system (12).

This completes the proof of Theorem 3.

Proof of Theorem 4. Now I show how to reduce Theorem 4 from Theorem 3 if g = 2. Choose any characteristic c ∈ ℤ^{4}/2ℤ^{4}, c ≠ 0. The corresponding set 𝔎^{∗} = 𝔎_{c}^{∗} contains three elements, say a_{1}, a_{2}, a_{3}. It can be assumed that the order of these elements is chosen such that the matrix M = M_{c} has a form

M = \frac{1}{2} ( \begin{matrix}1 & 1 & 1 \\ 1 & 1 & −1 \\ 1 & −1 & 1 \end{matrix} ) .

Putting

θ_{j} = ϑ_{a_{j}+c}^{2}ϑ_{a_{j}}^{2}, S_{j} = ψ_{a_{j}+c} + ψ_{a_{j}}, P_{j} = ψ_{a_{j}+c}ψ_{a_{j}}, j = 1, 2, 3,

we rewrite the system of differential equations (12) (equivalently, (23)) as

\begin{aligned}θ_{1}δS_{1} & = 4θ_{1}P_{1} − (θ_{1}S_{1}^{2} − θ_{2}S_{2}^{2} − θ_{3}S_{3}^{2}), \\ θ_{2}δS_{2} & = 4θ_{2}P_{2} + (θ_{1}S_{1}^{2} − θ_{2}S_{2}^{2} − θ_{3}S_{3}^{2}), \\ θ_{3}δS_{3} & = 4θ_{3}P_{3} + (θ_{1}S_{1}^{2} − θ_{2}S_{2}^{2} − θ_{3}S_{3}^{2}). \end{aligned}  (24)

In our case, relations (13), (14) look like

θ_{1} − θ_{2} − θ_{3} = 0, θ_{1}S_{1} − θ_{2}S_{2} − θ_{3}S_{3} = 0,

hence,

\frac{S_{2} − S_{3}}{θ_{1}} = \frac{S_{1} − S_{3}}{θ_{2}} = \frac{S_{2} − S_{1}}{θ_{3}} .

This implies that

θ_{1}S_{1}^{2} − θ_{2}S_{2}^{2} − θ_{3}S_{3}^{2} = θ_{1}(S_{1} − S_{2})(S_{1} − S_{3})
= −θ_{2}(S_{2} − S_{1})(S_{2} − S_{3})
= −θ_{3}(S_{3} − S_{1})(S_{3} − S_{2}).  (25)

Substituting (25) in (24) and cancellating theta constants, we obtain the system

\begin{aligned}δS_{1} & = 4P_{1} − (S_{1} − S_{2})(S_{1} − S_{3}), \\ δS_{2} & = 4P_{2} − (S_{2} − S_{1})(S_{2} − S_{3}), \\ δS_{3} & = 4P_{3} − (S_{3} − S_{1})(S_{3} − S_{2}). \end{aligned}  (26)

Thus, in our notation, the system (15) follows from (26) by adding any two of equations:

\begin{aligned}δ(S_{1} + S_{2}) & = 4P_{1} + 4P_{2} − (S_{1} − S_{2})^{2}, \\ δ(S_{1} + S_{3}) & = 4P_{1} + 4P_{3} − (S_{1} − S_{3})^{2}, \\ δ(S_{2} + S_{3}) & = 4P_{2} + 4P_{3} − (S_{2} − S_{3})^{2}, \end{aligned}

since any two among three pairs {a_{1}, a_{1} + c}, {a_{2}, a_{2} + c}, and {a_{3}, a_{3} + c} form “non-trivial parallelogram”.

This completes the proof of Theorem 4.

The point (i) of our program is still an open question even in the case g = 2. I do not want to conjecture anything on this subject. To discuss this problem we need a modular point of view on functions (7).

In the case g > 1 consider the symplectic group

Sp_{2g}(ℝ) = \{γ = ( \begin{matrix}A & B \\ C & D \end{matrix} ) : γ^{t} ( \begin{matrix}0 & E \\ −E & 0 \end{matrix} ) γ = ( \begin{matrix}0 & E \\ −E & 0 \end{matrix} )\},  (27)

where A, B, C, D are square matrices of size g and E is the identity matrix. One has the same definition for the Siegel modular group Sp_{2g}(ℤ), the only difference from (27) is in integer-valued matrices A, B, C, D. Both of groups, Sp_{2g}(ℝ) or Sp_{2g}(ℤ), act on Siegel’s upper half-space ℌ_{g} ⊂ ℂ^{g(g+1)/2} by the rule

T ↦ γT = (AT + B)(CT + D)^{−1}, γ = ( \begin{matrix}A & B \\ C & D \end{matrix} ) .

Following statement shows a connection between differentiating and above acting.

Lemma 4. Suppose a function F (T): ℌ_{g} → ℂ is holomorphic, the matrix differential operator Δ is defined by (10) (so, ΔF is a symmetric matrix of size g which elements are corresponding partial derivatives of F). Then

(ΔF)(γT) = (CT + D) · ΔF (γT) · (CT + D)^{t} for all γ = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Sp_{2g}(ℤ).

Using the functional equation for theta constants

ϑ_{a}(γT) = ξ · det^{1/2}(CT + D) · ϑ_{a}(T), γ = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Γ_{1,2}, ξ^{8} = 1

(see [9], Chapter II) and Lemma 4, we have the following result on modular properties of functions (7).

Theorem 5. For each characteristic a ∈ 𝔎_{0} matrix-valued function Ψ_{a} consisting of functions (7) satisfies the functional equation

Ψ_{a}(γT) = \frac{1}{2}(CT + D) · C^{t} + (CT + D) · Ψ_{a}(T) · (CT + D)^{t}, γ ∈ Γ_{1,2}.

In particular, for a, b ∈ 𝔎_{0} the function det(Ψ_{a} − Ψ_{b})(T) is a modular form of weight 2 with respect to Γ_{1,2}.

Theorem 5 can help to get the multi-dimensional analogues of formulas (5), at least in cases g = 2 and g = 3. This problem seems to be more computational than mathematical.

References

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2. K. Mahler, On algebraic differential equations satisfied by automorphic functions, J. Austral. Math. Soc. 10 (1969), 445–450.

3. S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philosoph. Soc. 22 (1916), no. 9, 159–184; G. H. Hardy, P. V. Sechu Aiyar, B. M. Wilson (eds.), Collected Papers of Srinivasa Ramanujan, Chelsea Publ., New York, 1962, pp. 136–162.

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9. D. Mumford, Tata Lectures on Theta, II, Progress in Math., vol. 28, Birkhäuser, Boston– Basel–Stuttgart, 1984.

10. Y. Ohyama, Differential equations of theta constants of genus two, Algebraic Analysis of Singular Perturbations (Kyoto, 1996), Sūrikaisekikenkyūsho Kōkyūroku, vol. 968, Kyoto Univ., Kyoto, 1996, pp. 96–103 (Japanese); English transl., Preprint, Osaka Univ., Osaka, 1996.

11. Y. Ohyama, Nonlinear equations on theta constants of genus two, Preprint, Osaka Univ., Osaka, 1996.

12. A. Krazer, Lehrbuch der Thetafunktionen, Leipzig, 1903; Reprint, Chelsea Publ., New York, 1970.

Moscow State University