In this paper, we will introduce a new matrix and call this matrix the associate matrix. Then we will show that among three matrices which are a (conformal) period matrix, a combinatorial period matrix and an associate matrix, there exists an equation which is the main result.
and call this matrix [[LAMBDA].sub.K] the associate matrix.
Let (M, {a, b},K) be a triangulated Riemann surface with the period matrix [product], the combinatorial period matrix [[product].sub.K] and the associate matrix [[LAMBDA].sub.k].
In this section, we will introduce a new matrix which is uniquely determined by a triangulated Riemann surface as well as a period matrix and a combinatorial period matrix and call this new matrix the associate matrix of a triangulated Riemann surface.
We define the associate matrix [[LAMBDA].sub.K] of (M, {a, b}, K) by
Note that the associate matrix is well defined by choosing any Riemannian metric in the conformal class of the Riemann surface.
Let (M {a b} K) be a triangulated Riemann surface with the period matrix [product], the combinatorial period matrix [[product].sub.K] and the associate matrix [[LAMBDA].sub.K].
By Theorem 4.2, we see that an associate matrix is an element of the Siegel upper half space.