LATTICES AND THETA-FUNCTION IDENTITIES .1. THETA-CONSTANTS

The gluing construction of lattices is used to generate and study a number of theta constant identities. It is shown, for example, that the Jacobi identity, a well-known degree 4 identity, can be derived from a degree 2 identity. A list of all quadratic identities in theta(3) derivable from this lattice method, containing over 30 algebraically independent identities, and conjectured to yield all quadratic identities in theta(3) derivable by any means, is included. In contrast, only two independent quadratic identities in theta(3) have been found elsewhere in the mathematical literature. The theta constants of lattices and their glue classes are investigated, and the fact that the theta constant of each glue class is a root of a polynomial whose coefficients are linear combinations of theta constants of lattices is shown.