| Résume | We take a section $P$ of infinite order on an elliptic surface and consider points where some multiple $nP$ is tangent to the zero section. (These are "unlikely intersections" and our consideration of them is motivated by a question in geography of surfaces. It is also analogous to the question of whether elements of an elliptic divisibility sequence are square-free.) In characteristic zero, we show finiteness and give a sharp upper bound, relying heavily on a canonical parallel transport in a family of elliptic curves (the "Betti foliation") and a certain real-analytic one-form. Although the finiteness statement looks completely reasonable in characteristic $p$, it's not clear what would replace the (non-algebraic) 1-form. Time permitting, I will explain how ongoing work with Felipe Voloch connects tangencies to the $p$-descent map and allows us to bound them in characteristic $p$ as well.
Joint work with G. Urzua and F. Voloch. |
