Finiteness of 2-reflective lattices of signature (2,n)

TL;DR

This paper proves that only finitely many even lattices of signature (2,n) with n>6 admit 2-reflective modular forms, confirming a conjecture for n>6 and identifying the unique case at n=26.

Contribution

It establishes the finiteness of 2-reflective lattices for n>6 and confirms a conjecture by Gritsenko and Nikulin, identifying the unique unimodular lattice at n=26.

Findings

Finitely many 2-reflective lattices exist for n>6.

No such lattices exist for n>25 except the (2,26) unimodular lattice.

Confirmed the Gritsenko-Nikulin conjecture for n>6.

Abstract

A modular form for an even lattice L of signature (2,n) is said to be 2-reflective if its zero divisor is set-theoretically contained in the Heegner divisor defined by the (-2)-vectors in L. We prove that there are only finitely many even lattices with n>6 which admit 2-reflective modular forms. In particular, there is no such lattice in n>25 except the even unimodular lattice of signature (2,26). This proves a conjecture of Gritsenko and Nikulin in the range n>6.