Galois representations over fields of moduli and rational points on Shimura curves (Q2922907)

Let \(B\) be an indefinite quaternion algebra of discriminant \(D >1\) and fix a maximal order \(\mathcal{O}\) in \(B\). Let \(X / \mathbb{Q}\) be the Shimura curve associated to \(B\), that is the moduli space of isomorphism classes of abelian surfaces \((A, \iota: \mathcal{O} \rightarrow \mathrm{End}(A))\) with multiplication by \(\mathcal{O}\).NEWLINENEWLINELet \(q\) be a rational prime. Let \(K\) be an imaginary quadratic fields in which \(q\) is ramified. The main theorem of the paper states that if \(B\) is not split by \(\mathbb{Q}(\sqrt{-q})\) and \(D\) is divisible by a prime \(p \geq 5\) with certain additional properties, then \(X(K)\) consists only of CM points. Moreover if \(K\not = \mathbb{Q}(\sqrt{-1}) , \mathbb{Q}(\sqrt{-3})\), then \(X(K)= \emptyset\).NEWLINENEWLINEThe method of proof consists of studying the geometry of \(X\) together with Galois representations arising from points of \(X\). The role of the auxilary prime \(p\) in the statement is to provide certain behaviour of Galois representations after reduction modulo \(p\). The strategy is based on prior works of \textit{A. Skorobogatov} [Math. Res. Lett. 12, No. 5--6, 779--788 (2005; Zbl 1131.11037)]and \textit{B. W. Jordan} [J. Reine Angew. Math. 371, 92--114 (1986; Zbl 0587.14018)].