LATTICES AND THETA-FUNCTION IDENTITIES .2. THETA-SERIES

The gluing construction of lattices is used to generate and study a number of theta function densities. It is shown, for example, that Riemann's formula, a fundamental degree 4 identity, can be derived from a degree 2 identity. It is proved that all in a large class of identities can be derived from these lattice techniques. A list of 24 independent quadratic identities in the Jacobi functions curly-theta(1) and curly-theta(3), conjectured to be complete, with all but two of them seeming to be new, is given. The theta series of glue classes is also investigated, and it is shown that all of them, as well as the functions curly-theta(a,b), satisfy polynomials whose coefficients are linear combinations of theta series of lattices; these polynomials have some interesting properties.