On quadratic twists of hyperelliptic curves
Let \(K\) be a field with \(\text{char}(K)\neq 2\), \(f(x)\in K[x]\) a separable polynomial and \(C|K:\, y^2=f(x)\) a hyperelliptic curve of genus \(g\geq 2\) over \(K\) (thus \(\text{deg}(f)\in\{2g+1,2g+2\}\)). Let \(d\in K\setminus K^2\) and set \(K':=K(\sqrt{d})\), which becomes a separable quadratic extension of \(K\). Let \(C_d\) be the hyperelliptic quadratic twist obtained from \(C|K'\) and which is defined by an equation of type \(dy^2=f(x)\); in particular, \(C\) and \(C_d\) are isomorphic over \(K'\). In this context a good reference is [\textit{Q. Liu}, Algebraic geometry and arithmetic curves. Transl. by Reinie Erné. Oxford: Oxford University Press (2006; Zbl 1103.14001)].NEWLINENEWLINEFor \(K\) a complete discrete valuation field, the main result of this paper is concerning minimal regular models of \(C_d\) which mainly depends on the fact that \(C(K)\neq\emptyset\). Then a nice application is giving, namely on the existence of hyperelliptic curves \(C\) of positive genus defined over \(\mathbb Q\), the existence of certain integers \(d\) such that \(C_d(\mathbb Q_p)=C_d(\mathbb Q)=\emptyset\) for infinitely many prime numbers \(p\).
scientific article; zbMATH DE number 6349623