N ov 2 00 7 Jacobi Forms of Degree One and Weil Representations Nils

We discuss the notion of Jacobi forms of degree one with matrix index, we state dimension formulas, give explicit examples, and indicate how closely their theory is connected to the theory of invariants of Weil representations associated to finite quadratic modules. 1 Jacobi forms of degree one Jacobi forms of degree one with matrix index F gained recent interest, mainly due to applications in the theory of Siegel and orthogonal modular forms, and in the geometry of moduli spaces. Of particular interest among these Jacobi forms are those of critical weight, i.e. those whose weight equals (rank(F ) + 1)/2. There are no Jacobi forms of of index F and weight strictly less than rank(F )/2, and for weights strictly greater than (rank(F ) + 3)/2 we have at least an explicit and easily computable dimension formula (see Theorem 1 below). From the point of view of algebraic geometry Jacobi forms of degree one are holomorphic functions φ(τ, z) of a variable τ in the Poincaré upper half plane H and of complex variables z ∈ C such that, for fixed τ , the function φ(τ, ·) is a theta function on the algebraic variety C/Λτ , where Λτ denotes the lattice Zτ + Z, and such that, for any τ and all A in a subgroup of finite index in SL(2,Z), the theta function φ(τ, ·) on C/Λτ and φ(Aτ, ·) on the isomorphic torus C/ΛAτ are related by a simple transformation formula. Thus, for fixed τ , the function φ(τ, ·) corresponds to a holomorphic section of a positive line bundle on C/Λτ . The positive line bundles on Xτ = C /Λτ are (up to translation and isomorphism) parameterized by their Chern classes inH(Xτ ,Z). It is not difficult to show that the cone of positive Chern classes in H(Xτ ,Z) is in one to one correspondence with the set of symmetric,