Colloquium abstracts

Bernhard Heim
German University of Technology, Oman
September 27, 2012

On a hidden property of Borcherds products:  Borcherds products are special automorphic forms for the orthogonal group $O_{2,n}(\R)$ of signature $(2,n)$, lifts from weakly modular forms. To state explicit examples we identify $O_{2,2}$ with $SL_2 \times SL_2$ and $O_{2,3}$ with the Siegel modular group of degree $2$. Let $j$ be the modular invariant with Fourier coefficients $c(n)$, normalized with $c(0)=0$. Then one of the most famous examples is given by $$j(z)-j(w) = p^{-1} \prod_{m=1, n\geq -1}^{\infty} \left( 1 - p^m q^n\right)^{c(nm)} \quad \left(p:=e^{2 \pi i z}, q:=e^{2 \pi i w}\right), $$ related to the denominator formula of the monster Lie algebra. This formula had been known before by Koike, Zagier. New was the link to the Moonshine conjecture and a systematic construction of automorphic products called Borcherds lifts due to Borcherds. The case of Siegel modular forms has been first studied by Gritsenko and Nikulin. They found a deep connection between the Igusa function $\Delta_5$ and a certain Kac-Moody algebr