| Résume | In 1963, Manin used a construction related to the work of Fuchs, Gauss, and Picard on linear differential operators to prove a function field version of the Mordell conjecture about rational points on algebraic curves. Over the years, Manin's construction has been reinterpreted in various ways , most notably using differential algebra. Indeed, the analytic presentation is generally regarded as a heuristic, as it was even in Manin's proofs. I will describe some work (joint with T. Dupuy and J. Freitag) using o-minimality and differential algebra exploiting the analytic presentation to explain some known properties of the Manin maps (e.g. Manin's theorem of the kernel and the one basedness of Manin kernels of non-isotrivial simple abelian varieties), to produce examples of simple abelian varieties with Manin kernels of intermediate rank, and to show why such examples cannot come from low dimensional families of abelian varieties. |
