THE COHOMOLOGY GROUP ASSOCIATED WITH JACOBI CUSP FORMS OVER CAYLEY NUMBERS

Abstract. We consider the cohomology group associated with Jacobi cusp forms over Cayley num-bers as an example of the general theory of the cohomology groups associated with cusp forms developed by Eichler and Shimura. The theory establishes isomorphisms between the cohomology groups and the vector spaces of vector-valued cusp forms and then expresses their common di-mensions in terms of certain geometric invariants in the corresponding quotient spaces. This is an analogue of the Riemann-Roch theorem applied to the cases of vector-valued modular forms. The theory of Jacobi forms over Cayley numbers was initiated by the author and Krieg [4, 5, 6] in 1991 in order to investigate the Maa space on the 10-dimensional Hermitian upper half-plane over Cayley numbers of degree two. Jacobi forms over Cayley numbers appeared as Fourier-Jacobi coefficients of modular forms on the upper half-plane there. They are of special interest for two main reasons. First, they are related to modular forms on the 27-dimensional exceptional domain considered by Baily [1]. Second, the arithmetic of integral Cayley numbers leads to special results, which cannot be obtained in the general case