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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 \[ \Delta=\prod_{1\leq i<j\leq 4}(q_{ij}-1), H=\prod_{1\leq i<j\leq 4}q_{ij}-\left(\sum_{1\leq i\leq 4}\prod_{{1\leq k<l\leq 4}\atop{k,l\neq i}}q_{kl}\right)+q_{12}q_{34}+q_{13}q_{24}+q_{14}q_{23}, \] \[ G=q_{12}q_{34} \prod_{{1\leq i<j\leq 4}\atop{(i,j)\neq(1,2),(3,4)}} (q_{ij}-1)^ 2+q_{13}q_{24} \prod_{{1<i<j\leq 4}\atop{(i,j)\neq(1,3),(2,4)}}(q_{ij}-1)^ 2+q_{14}q_{23} \prod_{{1\leq i<j\leq 4}\atop{(i,j)\neq(1,4),(2,3)}}(q_{ij}-1)^ 2. \]

11F55 Other groups and their modular and automorphic forms (several variables)
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