FORMS, HECKE OPERATORS, AND THETA SERIES

Abstract. We analyse the behavior of Siegel theta series attached to arbitrary rank lattices under the symplectic group, and define half-integral weight Siegel modular forms. Then we introduce Hecke operators for half-integral weight Siegel forms, explicitly describing the action on Fourier coefficients (and giving an explicit choice for the matrices giving the action of each Hecke operator). We introduce generators of the Hecke algebra whose action on Fourier coefficients is more transparant. Applying these operators to theta series, we show that the average Siegel theta series of half-integral weight are eigenforms for the Hecke operators attached to primes not dividing the level; we explicitly compute the eigenvalues. Quadratic forms abound in mathematics, as they capture the geometric notions of distance and orthogonality. Siegel asked: Given a lattice L = Zx1 ⊕ · · · ⊕ Zxm with geometry given by a positive definite quadratic form Q, and given another quadratic form T, how many sublattices of L have their geometry given by T? T