872 research outputs found
The Modular Form of the Barth-Nieto Quintic
Barth and Nieto have found a remarkable quintic threefold which parametrizes
Heisenberg invariant Kummer surfaces which belong to abelian surfaces with a
(1,3)-polarization and a lecel 2 structure. A double cover of this quintic,
which is also a Calabi-Yau variety, is birationally equivalent to the moduli
space {\cal A}_3(2) of abelian surfaces with a (1,3)-polarization and a level 2
structure. As a consequence the corresponding paramodular group \Gamma_3(2) has
a unique cusp form of weight 3. In this paper we find this cusp form which is
\Delta_1^3. The form \Delta_1 is a remarkable weight 1 cusp form with a
character with respect to the paramodular group \Gamma_3. It has several
interesting properties. One is that it admits an infinite product
representation, the other is that it vanishes of order 1 along the diagonal in
Siegel space. In fact \Delta_1 is an element of a short series of modular forms
with this last property. Using the fact that \Delta_1 is a weight 3 cusp form
with respect to the group \Gamma_3(2) we give an independent construction of a
smooth projective Calabi-Yau model of the moduli space {\cal A}_3(2).Comment: 20 pages, Latex2e RIMS Preprint 120
The Igusa modular forms and ``the simplest'' Lorentzian Kac--Moody algebras
We find automorphic corrections for the Lorentzian Kac--Moody algebras with
the simplest generalized Cartan matrices of rank 3:
A_{1,0} =
2 0 -1
0 2 -2
-1 -2 2
and
A_{1,I} =
2 -2 -1
-2 2 -1
-1 -1 2
For A_{1,0} this correction is given by the Igusa Sp_4(Z)-modular form
\chi_{35} of weight 35, and for A_{1,I} by a Siege modular form of weight 30
with respect to a 2-congruence subgroup. We find infinite product or sum
expansions for these forms. Our method of construction of \chi_{35} leads to
the direct construction of Siegel modular forms by infinite product expansions,
whose divisors are the Humbert surfaces with fixed discriminants. Existence of
these forms was proved by van der Geer in 1982 using some geometrical
consideration. We announce a list of all hyperbolic symmetric generalized
Cartan matrices A of rank 3 such that A has elliptic or parabolic type, A has a
lattice Weyl vector, and A contains the affine submatrix \tilde{A}_1.Comment: 40 pages, no figures. AMS-Te
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