Rational Structure of X(N) over Qand Explicit Galois Action on CM Points

This paper reviews a less known rational structure on the Siegelmodular variety $X(N) =Gamma(N) backslash H_g$ over $Q$ forintegers $g, N ge 1$. The author then describes explicitly howGalois groups act on CM points on this variety. Finally, anotherproof of the Shimura reciprocity law by using the result and the$q$-expansion principle is given.