Determination of all \(\mathbb{Q}\)-rational CM-points in the moduli space of principally polarized abelian surfaces
It is well-known that in the moduli space of elliptic curves there exist exactly 9 \(\mathbb{Q}\)-rational points corresponding to the isomorphism classes of elliptic curves whose endomorphism ring is isomorphic to the ring integers of an imaginary quadratic number field. Here an analogous question concerning the moduli space \({\mathcal A}_2\) of principally polarized abelian surfaces is studied. It is shown that there exist exactly 19 \(\mathbb{Q}\)-rational points of \({\mathcal A}_2\) corresponding to abelian surfaces whose endomorphism ring is isomorphic to the ring of integers in a quartic CM-field. In Math. Comput. 68, 307-320 (1999; Zbl 0906.14025), \textit{P. van Wamelen} found 19 curves of genus 2 whose Jacobians have these properties. So he found all such curves. The proof of the main result applies lower bounds for the class number of quartic CM-fields due to \textit{S. Louboutin} [J. Number Theory 67, 1-10 (1997; Zbl 0881.11078)].
scientific article; zbMATH DE number 1563182