Algebra and Number Theory Seminar: Doug Ulmer

Title: More unlikely intersections on elliptic surfaces (w/ G. Urzua and F. Voloch)

Abstract:  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.)  In characteristic zero, Urzua and I 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.  More recently with Felipe Voloch, we connect tangencies to p-descent maps and bound them in characteristic p.  We also find a new family of unlikely intersections in characteristic zero related to a famous homomorphism of Manin, and we correct inaccuracies in the literature about this homomorphism.