We construct some families of automorphic forms on Grassmannians which have singularities along smaller sub Grassmannians, using Harvey and Moore's extension of the Howe (or theta) correspondence to modular forms with poles at cusps. Some of the applications are as follows. We construct families of holomorphic automorphic forms which can be written as infinite products, which give many new examples of generalized Kac-Moody superalgebras. We extend the Shimura and Maass-Gritsenko correspondences to modular forms with singularities. We prove some congruences satisfied by the theta functions of positive definite lattices, and find a sufficient condition for a Lorentzian lattice to have a reflection group with a finite volume fundamental domain. We give some examples suggesting that these automorphic forms with singularities are related to Donaldson polynomials and to mirror symmetry for K3 surfaces.

Automorphic forms with singularities on Grassmannians

Automorphic forms on Os + 2,2(R) and infinte products.

Values of L -series of modular forms at the center of the critical strip

Traces of singular moduli

The object of this paper is to generalize these results to higher level. Thus we want to study the space S£+1/2(4N) of cusp forms of weight fc + 1/2 on Γ0(4ΛΟ (Ν e N) which have a Fourier expansion s in (*). Because this will be our basic object of study, we n w break with tradition and redefine the level of a modular form of half-integral weight s N if the form satisfies a transformation law under Γ0(4Ν) and has a Fourier expansion s in (*); we also drop the "4" and the "+" and write Sk+i/2(N) instead * ^) i* °f T* (n). More generally, if χ is a DirichJet character

Winfried Kohnen

Papers

Modular Forms of Half-Integral Weight on ...0(4).