Lattices with many Borcherds products
Jan Hendrik Bruinier, Stephan Ehlen, and Eberhard Freitag

TL;DR
This paper classifies a finite set of even lattices of signature (2,n) with trivial cusp form spaces, showing each has a Borcherds product divisor, and extends results to finite quadratic modules.
Contribution
It proves finiteness and provides a classification of lattices with Borcherds products, extending to finite quadratic modules.
Findings
Finite classification of lattices with trivial cusp form spaces
Explicit list of such lattices
All Heegner divisors are realized as Borcherds product divisors
Abstract
We prove that there are only finitely many isometry classes of even lattices of signature for which the space of cusp forms of weight for the Weil representation of the discriminant group of is trivial. We compute the list of these lattices. They have the property that every Heegner divisor for the orthogonal group of can be realized as the divisor of a Borcherds product. We obtain similar classification results in greater generality for finite quadratic modules.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Algebra and Geometry · Finite Group Theory Research · Algebraic structures and combinatorial models
