Dimension formulas for automorphic forms on orthogonal groups
Final Report Abstract
Automorphic forms on orthogonal groups play an important role in several branches of mathematics, e.g. in the theory of moduli spaces and in the theory of vertex operator algebras. They can be interpreted as sections of line bundles over certain locally symmetric spaces. The spaces of automorphic forms of fixed weight are finite-dimensional and their dimensions can be computed by means of the Hirzebruch-Riemann-Roch theorem and the holomorphic Lefshetz formula. Thereto one has to construct a smooth toroidal compactification of the underlying locally symmetric space which allows the calculation of the intersection products of the Chern classes of the cotangent bundle and the logarithmic cotangent bundle with the boundary divisors. In this project we investigate automorphic forms related to the unique even unimodular lattice of signature (10, 2). We show that the Weyl chambers of the reflection group of the even unimodular lattice of signature (9,1) give rise to a nice toroidal compactification of the corresponding modular variety. We study the geometry of this compactification. In particular we describe the boundary divisors and certain special divisors and determine their intersection products. Then we develop a recursion procedure which reduces the intersection products of the boundary divisors with the Chern classes to well-known geometric invariants. The missing steps to an explicit dimension formula in this case are to control the recursions and the application of the holomorphic Lefshetz formula.
Publications
- Towards a dimension formula for automorphic forms on O(II2,10), Dissertation, Technical University of Darmstadt, 2021
M. Rössler
(See online at https://doi.org/10.26083/tuprints-00019022)
