Alvaro Lozano-Robledo

math.NT — Number Theory

An elliptic curve of rank $\geq 34$ and root number $-1$ over a quadratic field

Contributed by Dylan T. Costa, Alvaro Lozano-Robledo

In this article, we find a quadratic twist over $\mathbb{Q}$ of rank $\geq 2$ (and $\geq 3$ assuming parity) of an elliptic curve $E/\mathbb{Q}$ of rank $\geq 32$, recently found by Elkies, Klagsbrun, and Li. We use a family of quadratic twists of rank $\geq 2$ over $\mathbb{Q}(t)$ constructed by Mestre to search for twists $E^D$ with root number $-1$, computing all root numbers without factoring the (very large) twisting parameter $D$. As a result, we exhibit an explicit integer $D$, with $411$ digits (a square times an integer with $381$ digits), such that $E^D$ has root number $-1$ and two independent rational points. In particular, the rank of $E$ over $\mathbb{Q}(\sqrt{D})$ is at least $34$, unconditionally, and at least $35$ if we assume the Parity Conjecture.