Eisenstein series for orthogonal groups and Kneser neighbours

Source abstract

We study Eisenstein series for orthogonal groups of even unimodular lattices of signature $(n,2)$ which arise as lifts of elliptic modular forms for $\mathrm{SL}_2(\mathbb{Z})$ on 1-dimensional cusps. When the weight is small they have to be defined by analytic continuation. In general, they will be smooth but non-holomorphic after regularization. We show exactly in which cases we obtain holomorphic modular forms by computing the intertwining factors appearing in Langlands' constant term formula in terms of standard parameters and Rankin-Selberg $L$-values. This is related to Kneser's neighbour theory of positive-definite even lattices, which also governs the eigenvalues of these Eisenstein series. In particular, we describe the space of singular weight modular forms and construct the function $Φ_{12}$, the denominator function of the Fake Monster Algebra, as an Eisenstein series. We apply these results to show that the corresponding orthogonal modular varieties are of general type.