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

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 $n\geq7$ which admit $2$-reflective modular forms. In particular, there is no such lattice in $n\geq26$ except the even unimodular lattice of signature $(2,26)$. This proves a conjecture of Gritsenko and Nikulin in the range $n\geq7$.

[1]  Shouhei Ma.  Finiteness of stable orthogonal modular varieties of non-general type , 2013, 1309.7121.

[2]  J. Bruinier.  On the converse theorem for Borcherds products , 2012, 1210.4821.

[3]  K. Hulek,et al.  Uniruledness of orthogonal modular varieties , 2012, 1202.3361.

[4]  V. Gritsenko.  Reflective modular forms in algebraic geometry , 2010, 1005.3753.

[5]  Nils R. Scheithauer.  On the classification of automorphic products and generalized Kac–Moody algebras , 2006 .

[6]  W. J. McGraw.  The rationality of vector valued modular forms associated with the Weil representation , 2003 .

[7]  J. Bruinier.  Borcherds Products on O(2,l) and Chern Classes of Heegner Divisors , 2002 .

[8]  E. Looijenga.  Compactications defined by arrangements II: locally symmetric varieties of type IV , 2002, math/0201218.

[9]  V. Nikulin,et al.  On classification of Lorentzian Kac-Moody algebras , 2002, math/0201162.

[10]  J. Bruinier,et al.  Eisenstein series attached to lattices¶and modular forms on orthogonal groups , 2001 .

[11]  R. Borcherds.  Reflection groups of Lorentzian lattices , 1999, math/9909123.

[12]  V. Gritsenko,et al.  The Arithmetic Mirror Symmetry and¶Calabi–Yau Manifolds , 1996, alg-geom/9612002.

[13]  V. Gritsenko,et al.  Automorphic Forms and Lorentzian Kac--Moody Algebras. Part I , 1996, alg-geom/9610022.

[14]  R. Borcherds.  Automorphic forms with singularities on Grassmannians , 1996, alg-geom/9609022.

[15]  R. Borcherds.  Automorphic forms onOs+2,2(R) and infinite products , 1995 .

[16]  V. Gritsenko,et al.  K3 surfaces, Lorentzian Kac--Moody algebras and Mirror Symmetry , 1995, alg-geom/9510008.

[17]  Gregory Margulis,et al.  Discrete Subgroups of Semisimple Lie Groups , 1991 .

[18]  V V Nikulin,et al.  Integral Symmetric Bilinear Forms and Some of Their Applications , 1980 .

[19]  A remark on discriminants of moduli ofK3 surfaces as sets of zeros of automorphic forms , 1996 .

[20]  F. Scattone.  On the Compactification of Moduli Spaces for Algebraic K3 Surfaces , 1988 .

[21]  志村 五郎,et al.  Introduction to the arithmetic theory of automorphic functions , 1971 .