We show that the Hecke rings act on the $l$-adic cohomology groups of suitable non-singular projective toroidal compactifications of the higher dimensional modular varieties. We extend the fixed point theory of Lefschetz to the correspondences for the Hecke rings on those compactifications. We treat here the Siegel modular case.

We deal with the class of branched surfaces $K$ such that 1) the branch set $S$ of $K$ is an embedded circle, 2) all connected components of $K\backslash S$ are orientable and their number is two or three. We show that in this class only two topological types admit expanding immersions. In the proof of the result, the Euler class of the tangent bundle of $K$ plays an important role.

Let $\theta$ be an irrational number and $A_\theta$ be the corresponding irrational rotation $C^*$-algebra. For any $k\in N\cup\{\infty\}$ let $A_\theta^k$ be the dense $*$-subalgebra of $k$-times continuously differentiable elements in $A_\theta$ with respect to the canonical action of the two dimensional torus and let $A_\theta^0=A_\theta$. In the present paper we will show that there is an approximately inner $*$-derivation of $A_\theta^\infty$ to $A_\theta^\infty$ which is not inner if and only if $\theta$ is a non-generic irrational number.

In the present paper we consider geodesics which are obtained as images of great circles of $S^3(1)$ induced by an isometric minimal immersion $f:S^3(1)\to S^{24}(r)$, $r^2=1/8$ namely, geodesics of $f(S^3(1))$. $S^{24}(r)$ being regarded as a hypersphere of $\mathbf{R}^{25}$, we can consider such geodesics as curves in $\mathbf{R}^{25}$ with curvatures $k_1,k_2,k_3$. It is found that these are constants which depend on the choice of the geodesic except the case where $f$ is a standard minimal immersion [8]. Equations satisfied by $k_1,k_2,k_3$ and the necessary and sufficient condition for an isometric minimal immersion to have a geodesic which is a circle are obtained.

Though we concentrate our topic upon the case $S^3(1)\to S^{24}(1)$, in the beginning part of the paper some properties of minimal immersions of spheres into spheres in general are recollected with some additional results.