Density of positive rank fibers in elliptic fibrations
A one-parameter elliptic fibration over a field \(K\) is a familly of elliptic curves \(E_r: Y^2=X^3+A(r)X+B(r)\) where \(A\) and \(B\) are polynomials with coefficients in \(K\). A conjecture of Mazur states that, for \(K={\mathbb Q}\), the set \(\{r\in{\mathbb Q}\,| \, \text{rank}(E_r({\mathbb Q})>0\}\) is either dense in \({\mathbb R}\) or finite. The conjecture is known to be true under the parity conjecture. Also, there are unconditional results of Rohrlich and Kuwata-Wang proving the conjecture for the quadratic twists of a fixed elliptic curve by a quadratic or a cubic polynomial. In the paper under review the Mazur conjecture is settled affirmatively for all general quadratic and cubic filtrations, i.e., whenever \(\max(\deg A,\deg B)\leq 3\). By the method of proof, the same result holds when \({\mathbb Q}\) is replaced by any real number field.
scientific article; zbMATH DE number 5161947