On the simplicity of Jacobians for hyperelliptic curves of genus 2 over the field of rational numbers with torsion points of high order
In two 2012 papers of \textit{V. P. Platonov} and \textit{M. M. Petrunin} [Dokl. Math. 85, No. 2, 286--288 (2012); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 443, No. 6, 664--667 (2012; Zbl 1319.11041); Dokl. Math. 86, No. 2, 642--643 (2012); translation from Dokl. Akad. Nauk. 446, No. 3, 263--264 (2012; Zbl 1345.11045)], several families of genus \(2\) curves over \(\mathbb{Q}\) were given, whose Jacobians have \(\mathbb{Q}\)-rational torsion points of order \(m\) covering all values \(1\leq m\leq 30\). In the reviewed paper, the authors find out which of these Jacobians are absolutely simple. For certain curves they use a method of \textit{F. Leprévost} [J. Théor. Nombres Bordx. 7, No. 1, 283--306 (1995; Zbl 0864.14017)] based on the properties of the characteristic polynomial of the Frobenius endomorphism for the reduction of the curve over a certain finite field. In some cases where this method does not lead to a conclusion, the presence of zero divisors in the algebra of endomorphisms allows to conclude that the Jacobian is not absolutely simple.
scientific article; zbMATH DE number 6225257