On curves of genus 2 with Jacobian of \(\text{GL}_2\)-type
An abelian variety \(A/{\mathbb Q}\) is of \(\text{GL}_2\)-type if its algebra of \({\mathbb Q}\)-endomorphisms \(E = {\mathbb Q} \otimes \text{End}_{\mathbb Q}(A)\) is a number field of degree \([E:{\mathbb Q}] = \dim A\). It is shown by Ribet that such abelian varieties should be modular, in the sense of a conjecture of Shimura-Taniyama-Weil [see \textit{K. A. Ribet}, Abelian varieties over \({\mathbb Q}\) and modular forms, in: Algebra and topology 1992, Proc. Korea Adv. Inst. Sci. Tech., Math. Workshop, Taejon/Korea, 53-79 (1992)]. In this context all the genus 1 curves over \({\mathbb Q}\) have Jacobians of \(\text{GL}_2\)-type. Hence, it is natural to state the question which curves \(C\) of genus 2 have Jacobians of \(\text{GL}_2\)-type. In this paper the authors analyse the situation when \(C\) admits a rational Rosenhain model \(C: y^2 = f(x)\), with \(f(x) = x(x-1)(x-{\lambda})(x-{\gamma})(x-{\mu}) \in {\mathbb Q}[x]\) and the automorphism group \(\Aut(C)\) is non-abelian. Then the study of the embedding of \(\Aut(C)\) into the endomorphism algebra of the Jacobian \(J(C)\) of \(C\) makes it possible to determine, depending on the choice of \(\lambda, \gamma , \mu\), when \(J(C)\) is of \(\text{GL}_2\)-type and when it is not.
scientific article; zbMATH DE number 1251322