Torsion points on some abelian surfaces of CM-type
Let \(A\) be a two-dimensional abelian variety over the field \(\mathbb Q\) of rational numbers, and suppose that \(A\) becomes isogenous over the algebraic closure to a product of elliptic curves \(E_1\) and \(E_2\) with complex multiplication. The author shows that \(A\) has no rational torsion points of prime order \(p > 7\). More precisely, the author proves the following result. If \(A\) as above has a rational point of prime order \(p > 3\), then the following holds. The abelian surface \(A\) is simple over \(\mathbb Q\), and the elliptic curves \(E_1\) and \(E_2\) have complex multiplication by orders in the same imaginary quadratic field \(K\). The ring of \(\mathbb Q\)-rational endomorphisms of \(A\) is an order in an imaginary quadratic field \(K'\). Let \(D\) be the discriminant of \(K\) and \(D'\) the discriminant of \(K'\). Then one of the following holds: 1) \(p = 5, D \in \{-4, -20, -40\}\), and \(D' = -4\), 2) \(p = 7, D \in \{-3, -7\}\), and \(D' = -3\). The author finishes by proving the existence of abelian varieties \(A\) as above with a point of order \(p\) for both \(p=5\) and \(p=7\).
scientific article; zbMATH DE number 5690739
scientific article