Shimura curve quotients with odd Jacobians (Q5931957)

In their joint paper [Ann. Math. (2) 150, 1109-1149 (1999; Zbl 1024.11040)], \textit{B. Poonen} and the reviewer introduced the concept of odd and even Jacobians. The paper under review studies quotients of Shimura curves by subgroups of the Atkin-Lehner involutions and tries to determine when they have odd Jacobians, following earlier work by \textit{B. W. Jordan} and \textit{R. Livné} [Bull. Lond. Math. Soc. 31, 681-685 (1999; Zbl 1024.11043)]. Based on the criterion developed by Poonen and Stoll and using the Cherednik-Drinfeld \(p\)-adic uniformization of Shimura curves, the author arrives at a general necessary condition for an odd Jacobian: the subgroup must have index~2 and may not contain the Fricke involution. Using more detailed results on the existence of local points, several examples (including two families) of Shimura curve quotients with odd Jacobians are given. NEWLINENEWLINENEWLINETwo points in the paper warrant a special comment. NEWLINENEWLINENEWLINE(1) The statement on p.~96 that Poonen and Stoll show that \(\langle [X_\lambda], [X_\lambda] \rangle_\lambda = 1/2\) if and only if \([X_\lambda]\) is nontrivial in the Shafarevich-Tate group is not correct. It is possible that the pairing vanishes even though \([X_\lambda] \neq 0\). However, in this case the Jacobian is even. For an example, see Poonen-Stoll, p.~1140, Prop.~28. NEWLINENEWLINENEWLINE(2) The statement in the middle of p.~97 that Jordan and Livné exhibited the first known family of non-hyperelliptic curves with odd Jacobian is also not correct. Such a family (certain plane quartics) is already given in Poonen-Stoll, p.~1140, Prop.~29.