A study of elliptic gamma function and allies

We study analytic and arithmetic properties of the elliptic gamma function $$ \prod_{m,n=0}^\infty\frac{1-x^{-1}q^{m+1}p^{n+1}}{1-xq^mp^n}, \qquad |q|,|p|<1, $$ in the regime $p=q$; in particular, its connection with the elliptic dilogarithm and a formula of S. Bloch. We further extend the results to more general products by linking them to non-holomorphic Eistenstein series and, via some formulae of D. Zagier, to elliptic polylogarithms.

A known functional equation of the elliptic gamma function [3,Theorem 4.1] represents an SL 3 (Z) symmetry of (z; τ , σ ). The problem of determining its behaviour in the regime σ = τ under SL 2 (Z) transformations is specifically addressed in [2], where the (logarithm of the) infinite product is related to the elliptic dilogarithm via a formula of S. Bloch [1]. Our principal aim in this note is recasting analytic and arithmetic (modular) properties of the function θ 1 (z; τ ) and its relatives, in particular, linking them to non-holomorphic Eisenstein series and the elliptic dilogarithm. This programme is carried out in Sects. 2-4; it gives a new proof of Bloch's formula and related results from [2]. In Sect. 5 we go further to discuss similar features of products that generalise ones for θ 0 and θ 1 ; their relationship with non-holomorphic Eisenstein series and formulae from [7] allow us to link them to elliptic polylogarithms.

Period functions
A natural way of measuring failure of weight k modular behaviour under the transformation (z, τ ) → (ẑ,τ ) for a function f (z, τ ) is through the period function

Lemma 1 We have
Observe that the expression in the parentheses on the right-hand side measures the failure of k-parity of f (z, τ ).

Lemma 3
The function (4) admits the following representation: Proof As shown in the proof of Theorem 5.2 in [3], and the assumptions |x|, |x −1q | < 1 are made to ensure convergence. (The latter can be dropped in the final result by appealing to the analytic continuation in z.) Recalling the transformation (2), using interchanging summation and summing over k, we obtain (This formula can be alternatively derived from logarithmically differentiating identity (2) with respect to τ and further integrating the result with respect to z.) Substituting (z/τ , −1/τ ) for (z, τ ) translates the result into the desired relation.

Non-holomorphic modularity
where It follows then from Lemma 1 and the parity relations (3) that We summarise our finding in the following claim.

Lemma 4 We have
Lemma 3 leads to the following expansions of the functions F + and F − .
Proof For F + substitute the expression of T (z; τ ) from Lemma 3 into the computation This leads to the formula and the latter simplifies to the expression given in the statement of Theorem 1 by elementary manipulation.
For F − we proceed as follows. We have Multiply this expression by τ − τ = 2i Im τ and use A(τ − τ ) = 2i Im z to get to deduce the expression for F − as in the theorem.
A consequence of this expansion is the invariance of under translation τ → τ + 1.

Lemma 5 We have
Proof The functions L(z, τ ) and U (z, τ ) (hence their complex conjugates) are clearly invariant under translation τ → τ + 1. The result follows from noticing that is also invariant under the transformation.
We summarise the results in this section as follows.
In other words, it behaves like a Jacobi form of weight 1 on SL 2 (Z).

Elliptic dilogarithm
Theorem 2 provides a natural link between the period function F (z; τ ) and the elliptic dilogarithm its companion. Namely, the expansion in the theorem can be stated as This is essentially the result discussed in [2, Section 1]. Viewing now z as an element of the lattice R + Rτ , so that A andÂ in the representation z = −Â + Aτ are fixed, we find out that the τ -derivative 1 2πi is the Eisenstein series i 4π 3 m,n∈Z e 2π i(mÂ+nA) (mτ + n) 3 of weight 3, where the notation indicates omitting the term m = n = 0 from the summation. Integrating we obtain This is equation (7) in [2]. Sinceẑ = z/τ = A −Â/τ = A +Âτ , it follows that The latter is a (non-holomorphic) modular form of weight 1, and combined with equation (6) is the formula of Bloch mentioned previously.

Lemma 6
We have, for k ≥ 1, Proof Apply Lemma 1 and the relation We further use that the τ -derivative of ln Q k (z; τ ) is an Eisenstein series. of weight k + 2. This is true for k = 1 (see Sect. 4), while for k ≥ 1 we observe the functional equation The equality G k+2 (A,Â; τ ) = E k+2 (A,Â; τ ) then follows by induction on k using the fact that the constant terms of both functions at τ = ∞ (or q = 0) agree.
Integrating we obtain Since both sides continuously depend on A andÂ, the formula remains valid also for ln Q k (z; τ ).
As in our computation in Sect. 4 we obtain Thus, for positive integers a and b. Finally, observe that the non-holomorphic Eisenstein series (8) can be identified with the elliptic polylogarithms using a formula of Zagier [7,Proposition 2]. This leads to the following general result. Theorem 4 For k ≥ 1 and z = Aτ −Â, we have

Conclusion
This final (and very short!) part is devoted to highlighting some directions for further research.
This consideration does not exclude, however, a possibility for modified products (7) and related functions F k to exist such that the latter ones have true modular behaviour for each k ≥ 1. It sounds to us a nice problem to determine such modular objects. Several arithmetic problems related to the case k = 1 (originating from the elliptic gamma function) are still open. Our personal favourites include connection of (5) with the Mahler measure and mirror symmetry; see, for example, observation in [6].