Conjecture on theta-blocks of order 1

Borcherds Products Everywhere

“…Theta blocks are the invention of V. Gritsenko, N.-P. Skoruppa and D. Zagier, see [15] for a full treatment. We will only cite the properties we need here.…”

“…For a φ ∈ J w.h. k,m , the generalized valuation ord defined in [15] by ord(φ; x) = min (n,r)∈supp(φ)…”

“…These results will be presented in our next article. Acknowledgements: We thank the authors of [15] for explaining their work on theta blocks to the last two authors much in advance of its publication. We thank F. Cléry for helpful discussions.…”

“…As it is rather easy to search for theta blocks, we call this the Borcherds Products Everywhere Theorem. The proof uses the theory of Borcherds products for paramodular forms as given by Gritsenko and Nikulin [13], the recent theory of theta blocks due to Gritsenko, Skoruppa and Zagier [15], and a theory of generalized valuations on rings of formal series presented here in section 4. Let η be the Dedekind Eta function and ϑ be the odd Jacobi theta function and write ϑ ℓ (τ, z) = ϑ(τ, ℓz).…”

“…Let η be the Dedekind Eta function and ϑ be the odd Jacobi theta function and write ϑ ℓ (τ, z) = ϑ(τ, ℓz). The most general theta block [15] can be written η…”