Recent zbMATH articles in MSC 14K15https://zbmath.org/atom/cc/14K152021-06-15T18:09:00+00:00WerkzeugGeneric point and Mordell-Weil rank jump.https://zbmath.org/1460.140962021-06-15T18:09:00+00:00"Colliot-Thélène, Jean-Louis"https://zbmath.org/authors/?q=ai:colliot-thelene.jean-louisSummary: Let \(k\) be a number field and \(U\) a smooth integral \(k\)-variety. Let \(X \to U\) be an abelian scheme. We consider the set \(\mathcal{R}\) of rational points \(m \in U(k)\) such that the Mordell-Weil rank of the fibre \(U_m\) is strictly greater than the Mordell-Weil rank of the generic fibre. We prove the following results. If the \(k\)-variety \(X\) is \(k\)-unirational, then \(\mathcal{R}\) is dense for the Zariski topology on \(U\). If \(X\) is \(k\)-rational, then \(\mathcal{R}\) is not thin in \(U\).Effective bounds on the dimensions of Jacobians covering abelian varieties.https://zbmath.org/1460.140952021-06-15T18:09:00+00:00"Bruce, Juliette"https://zbmath.org/authors/?q=ai:bruce.juliette"Li, Wanlin"https://zbmath.org/authors/?q=ai:li.wanlinSummary: We show that any abelian variety over a finite field is covered by a Jacobian whose dimension is bounded by an explicit constant. We do this by first proving an effective and explicit version of \textit{B. Poonen}'s Bertini theorem [Ann. Math. (2) 160, No. 3, 1099--1127 (2005; Zbl 1084.14026)] over finite fields, which allows us to show the existence of smooth curves arising as hypersurface sections of bounded degree and genus. Additionally, for simple abelian varieties we prove a better bound. As an application, we show that for any elliptic curve \(E\) over a finite field and any \(n\in \mathbb{N} \), there exist smooth curves of bounded genus whose Jacobians have a factor isogenous to \(E^n\).