A finiteness theorem for subgroups of Sp(4,Z).

Abstract

We prove that there are only finitely many subgroups H of finite index in Sp(4,Z) such that the corresponding quotient ${\cal H}/H$ of the Siegel upper half space of rank two is not of general type. Every subgroup of the finite index in Sp(4,Z) contains a principal congruence subgroup $\Gamma(n)$ of some level n. The variety ${\cal H}/\pm \Gamma\sb2(n)$ has a natural compactification $X\sb{n}$ that was constructed by Igusa. It is known to be of general type for n $\ge$ 4. Assume for a second that the action of $H/ \pm \Gamma(n)$ on $X\sb{n}$ has no fixed points. Then the variety $X\sb{n}/(H/ \pm \Gamma(n))$ is also of general type. Unfortunately, usually there are many fixed points, and the idea is to estimate how many fixed points of each type we have, and how they affect the dimensions of the global sections of multicanonical line bundles. By means of the rather standard tools of algebraic geometry, we show that if the variety ${\cal H}/H$ is not of general type, then $H/\Gamma(n)$ contains many elements of certain types. Then we use these elements to prove that $\vert {\bf Sp}({\bf 4, Z}) : H\vert$ is bounded.