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

Shouhei Ma

Published 2014-09-10Version 1

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>29. A similar, but weaker finiteness result is also obtained for n=4, 5, 6. This gives an answer to a weak version of Gritsenko-Nikulin's conjecture.