Finite descent obstruction for curves over function fields
Let \(N\) be a positive integer and \(Y_N/\mathbb{Q}\) the affine curve parametrizing triples \((E,P,C)\), where \(E\) is an elliptic curve, \(P\) a point on \(E\) of exact order \(N\) and \(C\) a cyclic subgroup of \(E\) of order \(N\) such that \(P\) and \(C\) generate \(E[N]\). Let \(K\) be a global function field, \(S\) a finite set of places of \(K\) and \(O_S\) the ring of \(S\)-integers of \(K\). We denote by \(\mathcal{X}\) the \(S\)-integral model of a twist of \(Y_m\), where \(m > 1\), corresponding to a representation \(\bar{\rho} : G_K \rightarrow \mathrm{GL}_2(\mathbb{Z}/m)\). We consider the set \(\mathcal{X}^{f-\mathrm{cov}}\) of elliptic curves \(E_v/K_v\) with good reduction for \(v \not \in S\) for which, for all primes \(\ell\), there exist Galois representations \(\rho_{\ell} : G_K \rightarrow \mathrm{GL}_2(\mathbb{Z}/m)\), such that \(\rho_{\ell}\) restricted to a decomposition group at \(v\) corresponds to the action of \(G_{K_v}\) on the Tate module \(T_{\ell}E_v\). In this paper it is proved the following result: If \(\mathcal{X}^{f-\mathrm{cov}}\) has a point with multiplicative reduction at some place in \(S\) then \(\mathcal{X}(O_S)\) does too. Furthermore, it is shown with an example that without the non-integrality hypothesis the proof does not work.
scientific article; zbMATH DE number 6136410
scientific article