Title: Differential equations of order one (positive characteristic)
Abstract: This is joint work with Marius van der Put. We recall the definition from Grothendieck’s [EGA4] of a certain non-commutative algebra D of differential operators in characteristic p > 0, serving as an analogue of the classical Weyl algebra generated by z and d/dz. The notion “stratified order one differential equation f(y1, y, z) = 0″ in characteristic p > 0 is defined in terms of the algebra D. In the talk, relations of this definition with more classical notions such as super-singularity and the Cartier operator are given. The definition allows one to formulate an analogue of the Grothendieck-Katz conjecture, which in the present instance asserts that an order one equation in characteristic zero admits infinitely many algebraic solutions, precisely when for almost all primes p, a reduction modulo p of the equation is stratified. We can prove this conjecture in the special case of autonomous equations.