ABC and the Hasse principle for quadratic twists of hyperelliptic curves
Let \(C\) be an irreducible smooth curve, defined over \({\mathbb Q}\), admitting an involution \(\iota:C \to C\) (that is, an automorphism of order two). For each \(d \in {\mathbb Q}^{\times}/{\mathbb Q}^{\times 2}\) there is an irreducible smooth curve \({\mathcal T}_{d}(C, \iota)\), also defined over \({\mathbb Q}\), which is isomorphic to \(C\) (the isomorphism defined over \({\mathbb Q}(\sqrt{d})\)). It is said that \({\mathcal T}_{d}(C, \iota)\) violates the Hasse principle if \({\mathcal T}_{d}(C, \iota)({\mathbb A}_{\mathbb Q}) \neq \emptyset\) and \({\mathcal T}_{d}(C, \iota)({\mathbb Q})=\emptyset\), where \({\mathbb A}_{\mathbb Q}\) is the adele ring over \({\mathbb Q}\). In the paper [\textit{P. L. Clark} and \textit{J. Stankewicz}, Proc. Am. Math. Soc. 146, No. 7, 2839--2851 (2018; Zbl 1448.11113)] there is provided a criteria on \(C\) and \(\iota\) in order to have infinitely many values of \(d \in {\mathbb Q}^{\times}/{\mathbb Q}^{\times 2}\) such that \({\mathcal T}_{d}(C, \iota)\) violates the Hasse principle. That criteria is given by four conditions. One of these conditions is to assume \((C/\iota)({\mathbb Q})\) to be finite. If \(C\) is hyperelliptic and \(\iota\) is the hyperelliptic involution, one has that \((C/\iota)({\mathbb Q})={\mathbb P}^{1}({\mathbb Q})\) is infinite, so one cannot use the above criteria in this case. In the paper under review, the authors prove, under the assumption of the ABC conjecture, that if \(C\) is a hyperelliptic curve of genus at least three, then (i) the hyperelliptic involution \(\iota\) has no \({\mathbb Q}\)-rational points if and only if (ii) there are infinitely many values of \(d \in {\mathbb Q}^{\times }/{\mathbb Q}^{\times 2}\) such that \({\mathcal T}_{d}(C, \iota)\) violates the Hasse principle if and only if (iii) there some \(d \in {\mathbb Q}^{\times }/{\mathbb Q}^{\times 2}\) such that \({\mathcal T}_{d}(C, \iota)\) violates the Hasse principle. The hard part is to prove that (i) implies (ii). In order to obtain such an implication, the author first states Theorem 3 which, under the assumption that \(C({\mathbb A}_{\mathbb Q}) \neq \emptyset\), asserts that the set of primes integers congruent to \(1\) mod \(8\) such that \({\mathcal T}_{d}(C, \iota)({\mathbb A}_{\mathbb Q}) \neq \emptyset\) has positive density. Then, they use a result in [\textit{A. Granville}, Int. Math. Res. Not. 2007, No. 8, Article ID rnm027, 25 p. (2007; Zbl 1129.11028)] which, under the assumption of the ABC conjecture, asserts that the number of \(d \in {\mathbb Q}^{\times}/{\mathbb Q}^{\times 2}\) with \(|d| \leq X\) such that \({\mathcal T}_{d}(C, \iota)({\mathbb Q})\) has points which are not fixed points of \(\iota\) is bounded above (up to constants) by \(X^{2/3}\). The paper is five page long.
scientific article; zbMATH DE number 6933366