Research in motives and number theory

Question

By what I have seen a modern approach to the study of number theory and arithmetic geometry is by motives, motivic cohomology and something like that. But, what are the applications of motives and motivic cohomology in number theory ( I've seen something related to L functions)?, there are other "topological inspired" theories which applies in this sense (maybe like K-theory or topological modular forms)? Finally which are the new trends research in number theory and arithmetic geometry? Thank you for the answers!

Answer 1

Instead of trying to give to give even an imprecise account of motivic cohomology - an impossible task - I'll try a foolhardy "philosophical" approach. One example being worth a hundred speeches, as the French say, let me focus on a problem which I think is the perfect illustration of what we're discussing about. The starting point is the famous Bloch-Kato conjectures on the "special values" of the Zeta function of a number field (or more generally, of the L-functions), which generalize Dedekind's "analytical class number formula" and reveal a deeply mysterious relationship between the transcendental world (the above functions) and the algebraic world (the ideal class group and the higher Quillen K-groups). These conjectures have been proved recently, at least for abelian number fields, thanks to an original - not to say platonist - idea of Grothendieck, the so called search for "motives".

Remember Plato's "apolog of the cave": we live in a cave, and the physical reality which we perceive consists in shadows cast on the walls by the sun in our backs; to really understand the true "reality" (whatever this word means), we must turn around and face the archetype which projects these shadows. Grothendieck applied this philosophical concept to algebraic/arithmetic geometry : around a given variety are floating a host of dissimilar cohomologies (Betti, de Rahm, étale, p-adic...) coming from dissimilar worlds (algebraic topology, differential geometry, algebraic geometry, number theory...), but for which comparison theorems are available when passing to an algebraic closure. However such a passage destroys all the arithmetical properties we are interested in. Following Plato, Grothendieck suggested to look not at the shadows but at the archetype, the conjectural motivic cohomology. The "mathematical sunrays" projecting the shadows were even suggested to be some variants of Chern classes. A harebrained idea, it may seem, but which has occupied people for decades, until the final success of Voevodsky (Fields medalist in 2002). Vv. immediately used his new weapon to bring down two other conjectures (by Bloch-Kato and Lichtenbaum-Quillen) giving precise relationship beween K-theory and Galois/étale cohomology, and that was how the conjecture on special values was settled.