×

zbMATH — the first resource for mathematics

The lowest term of the Schottky modular form. (English) Zbl 0747.11022
The Schottky modular form \(J\) is a modular form of weight 8 on the Siegel half space \(\mathbb{H}_ 4\) of complex symmetric \(4\times 4\) matrices \(\tau=(\tau_{ij})\) with position definite imaginary part. One can expand \(J=\sum_{n\in\mathbb{N}^ 4}J_ nq^{n_ 1}_{11} \cdots q^{n_ 4}_{44}\), \(n=(n_ 1,\ldots,n_ 4)\), where \(q_{ij}:=e^{2\pi\sqrt{-1}\tau_{ij}}\) into a power series in \(q_{11},\ldots,q_{44}\). Then each \(J_ n\) is a polynomial in \(q_{ij}^{\pm 1}\). By using the period map on totally degenerate curves one can compute the lowest non-trivial term \(J_{(1,1,1,1)}\) of \(J\) up to a constant. By using the formula of Igusa one can compute one coefficient explicitly to get the constant. The result is: \[ J_{(1,1,1,1)}=(-1) 2^{16} \prod_{1\leq i<j\leq 4}q^{-1}_{ij} (\Delta H-G) \] where \[ \Delta=\prod_{1\leq i<j\leq 4}(q_{ij}-1), H=\prod_{1\leq i<j\leq 4}q_{ij}-\left(\sum_{1\leq i\leq 4}\prod_{{1\leq k<l\leq 4}\atop{k,l\neq i}}q_{kl}\right)+q_{12}q_{34}+q_{13}q_{24}+q_{14}q_{23}, \] \[ G=q_{12}q_{34} \prod_{{1\leq i<j\leq 4}\atop{(i,j)\neq(1,2),(3,4)}} (q_{ij}-1)^ 2+q_{13}q_{24} \prod_{{1<i<j\leq 4}\atop{(i,j)\neq(1,3),(2,4)}}(q_{ij}-1)^ 2+q_{14}q_{23} \prod_{{1\leq i<j\leq 4}\atop{(i,j)\neq(1,4),(2,3)}}(q_{ij}-1)^ 2. \]

MSC:
11F55 Other groups and their modular and automorphic forms (several variables)
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Bourbaki, A.: Groupes et alg?bres de Lie. Paris: Hermann 1968 · Zbl 0186.33001
[2] Gerritzen, L.: The Torelli map at the boundary of the Schottky space. J. Reine Angew. Math.405, 29-47 (1990) · Zbl 0687.14036 · doi:10.1515/crll.1990.405.29
[3] Gerritzen, L.: Equations defining the periods of totally degenerate curves. (Preprint 12/90) · Zbl 0783.14017
[4] Igusa, J.: Schottky’s invariant and quadratic forms, Christoffel Symposium. Basel Boston Stuttgart: Birkh?user 1981 · Zbl 0482.10030
[5] Igusa, J.: On the irreducibility of Schottky’s divisor. J. Fac. Sci. Univ. Tokyo, Sect. I A28, 531-545 (1981) · Zbl 0501.14026
[6] Ichikawa, T.: The universal periods of curves and the Schottky problem (to appear) · Zbl 0783.14018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.