Unlikely intersections for algebraic curves.

Unlikely intersections for algebraic curves.

Thu, 09/06/2011
16:00 		David Masser Logic Seminar Add to calendar
Number Theory Seminar Add to calendar 		L3
Unlikely intersections for algebraic curves.
In the last twelve years there has been much study of what happens when an algebraic curve in $ n $-space is intersected with two multiplicative relations $ x_1^{a_1} \cdots x_n^{a_n}~=~x_1^{b_1} \cdots x_n^{b_n}~=~1 \eqno(\times) $ for $ (a_1, \ldots ,a_n),(b_1,\ldots, b_n) $ linearly independent in $ {\bf Z}^n $. Usually the intersection with the union of all $ (\times) $ is at most finite, at least in zero characteristic. In Oxford nearly three years ago I could treat a special curve in positive characteristic. Since then there have been a number of advances, even for additive relations $ \alpha_1x_1+\cdots+\alpha_nx_n~=~\beta_1x_1+\cdots+\beta_nx_n~=~0 \eqno(+) $ provided some extra structure of Drinfeld type is supplied. After reviewing the zero characteristic situation, I will describe recent work, some with Dale Brownawell, for $ (\times) $ and for $ (+) $ with Frobenius Modules and Carlitz Modules.