Finiteness of a cohomology associated with certain Jackson integrals (Q1177402)

Let \(E\) be an elliptic curve of modulus \(q=e^{2\pi\sqrt{-1}\tau}\) for \(\text{Im }\tau>0\). Let \(\{{\mathfrak z}_ j\), \(j=1,dots,2n\}\) denote a basis of \(H_ 1(E^ n,\mathbb{Z})\) such that \(\{{\mathfrak z}_ j,{\mathfrak z}_{n+j}\}\) represents a pair of canonical loops in \(E\). Let \(X\) be the factor space of \(H_ 1(E^ n,\mathbb{Z})\) by the abelian subgroup \(A=\langle {\mathfrak z}_ 1,\dots,{\mathfrak z}_ n\rangle\) and \(\overline{X}\) the factor space of the dual \(H^{1,0}(E^ n)^*\) of the first component of the Hodge decomposition of \(H^ 1(E^ n,\mathbb{C})\) by \(A\). Let \(R^ \times (\overline {X})\) be the group of non -zero rational functions on \(\overline{X}\). This article gives the structure theorem for \(H^ 1(X,R^ \times(\overline {X}))\) by means of basic cocycles \(b_ \chi(\omega)\) with factorization into powers of \(q\) raised to linear forms in \(\Hom(X,\mathbb{Z})\). This is a \(q\)-analogue of M. Sato's theory of \(b\)-functions for prehomogeneous vector spaces. \(b_ \chi(\omega)\) serves as character of the \(q\)-analogue of relative invariants called quasi-meromorphic functions, which include the \(q\)-analogue of the theta function. Several interesting related problems are proposed, including difference system for the Jackson integrals of quasi-meromorphic functions. Partial answers are discussed. Examples are given in basic hypergeometric functions.