The aim of our study is to develope an arithmetic theory of abelian functions on the basis of the moduli space of principally polarized abelian surfaces with level (2, 4).
One of our subjects is to understand the above moduli space, and the other is to study of torsion points of abeliar surfaces.
Now we shall explain our results. First, we invesigate the relation between the moduli apace of hyperelliptic curves, of genus 2, with level (2, 4) structure and that of abelian surfaces. We showed that these spaces are considered as subsets of SO_3 (C), naturally. Here the nine quotients of the square of theta constants form a special orthogonal matrix.
Next we shall explain the second result.
Let tau be a point of the Siegel upper-half space of degree 2. We denote by A (tau) and K (tau), the abelian surface and the kummer surface associated to tau. Assume that the principally polarized abelian surface A (tau) is the Jacobian variety of a hyperelliptic curve of genus 2. Put jalpha (tau) =rhetaalphaO (2tau) /rheta_<oo> (2tau) (alpha*1/2ZETA^2/ZETA^2), then the kummer surface K (tau) is defined over the field F (tau) =Q (jalpha(tau)). The field generated by the ratio of p-torsion points of K (tau) will be denoted by F_p (tau) : F_p (tau) =Q (rheta_<alphao>(2tau|2(tau, 1)h) /rheta_<oo>(2tau|2(tau, 1)h) ; h1/pZETA^4/ZETA^4). Then we have the following :
Theorem Let p be an odd positive integer.
1. F_p (tau) is a Galois extension of F (tau).
2. If tau is a general point, then the Galois group of F_p/F is isomorphic to the following group :