Title:

On Jacobi forms of lattice index

Jacobi forms arise naturally in number theory in several ways: theta series arise as functions of lattices, Siegel modular forms give rise to Jacobi forms through their FourierJacobi expansion and the largest Mathieu group gives rise to semiholomorphic MaassJacobi forms, for example. Jacobi forms of lattice index have applications in the theory of reflective modular forms and that of vertex operator algebras, among other areas. Poincaré and Eisenstein series are building blocks for every type of automorphic forms. We define Poincaré series for Jacobi forms of lattice index and show that they reproduce Fourier coefficients of cusp forms under the Petersson scalar product. We compute the Fourier expansions of Poincaré and Eisenstein series and give an explicit formula for the Fourier coefficients of the trivial Eisenstein series in terms of values of Dirichlet Lfunctions at negative integers. For even weight and fixed index, we obtain nontrivial linear relations between the Fourier coefficients of nontrivial Eisenstein series and those of the trivial one. This result is used to obtain formulas for the Fourier coefficients of Eisenstein series associated with isotropic elements of small order. A more efficient way of breaking down a given space of automorphic forms is into its oldspace and its newspace. We study the linear operators leading to a theory of newforms for Jacobi forms of lattice index, namely Hecke operators, operators arising from the action of the orthogonal group of the discriminant module associated with the lattice in the index and level raising operators. We show that these operators commute with one another and are therefore suitable to define a newform theory. We define the level raising operators of type U(I) (for every isotropic subgroup I of the discriminant module associated with the lattice in the index) and show that they preserve cusp forms and Eisenstein series. We give a formula for the action of the level raising operators U(I) and V(l) and operators W(s) arising from the action of the orthogonal group on cusp forms and Eisenstein series. We obtain a description of some of the oldforms in a given space of Jacobi forms using these operators and the relation between Jacobi forms and vectorvalued modular forms for the dual of the Weil representation.
