Theta Relations

Source abstract

Motivated by the classical quartic and cubic theta identities, we establish a product formula for Siegel--Jacobi theta series associated with rationally isometric positive-definite quadratic spaces. Our main theorem expresses a product of theta series of arbitrary Siegel degree as a finite double character sum over quotients of commensurable lattices. For the root lattice $A_2$, it yields a finite Fourier identity for cubes of cubic theta functions, a new proof of the two-variable cubic identity, and seven independent relations in degree two. A second specialization gives a new square analogue of the two-variable cubic identity. At the origin it reduces to $a(q)a(q^2)=b(q)b(q^2)+c(q)c(q^2)$; in degree two, the same seven relation patterns recur with each cube replaced by the corresponding product at $τ$ and $2τ$. We further extend the cubic identities on $A_2$ to $\ell$-th-power identities on $A_{\ell-1}$ for every $\ell\geq3$, and to $2\ell$-th-power identities on $A_{\ell-1}$ for every $\ell\geq2$, in arbitrary Siegel degree. The doubled-order construction recovers Jacobi's quartic identity and gives its analogues in arbitrary Siegel degree.