topos and number thory
because i know just a few things about all this stuff, i may just ask general questions. So, in the recent talk given by Professor Lafforgue, nothing was said about number theory. What Grothendieck did in this area, and can one expect for the future ?
Grothendieck's new algebraic geometry, i.e. scheme theory, unifies algebraic geometry and algebraic number theory into one theory. As any commutative ring can be seen as an affine scheme, this is true in particular of the field of rational numbers and its finite extensions, the so-called "number fields", of the ring of integers and the integer rings of number fields, of their localisations which are the p-adic fields and their integer rings, or of the residual fields of these local rings which are the finite fields. Thus, all these objects of arithmetic nature are seen as geometric objects, and any algebraic variety defined by families of polynomial equations with coefficients in one of these fields or rings can be seen as a relative scheme above the affine scheme defined by this base field or ring (i.e. as having a structure morphism to this base affine scheme in the category of schemes).
By definition, a scheme is a topological space (with Zariski's topology) endowed with a certain structure sheaf of commutative rings. Every scheme has therefore an associated topos endowed with a certain internal ring and internal modules on this ring (such as the sheaf of differentials and its external powers and, very often, a "polarisation", i.e. a locally free module of rank 1 which determines a way of embedding it into projective spaces). This defines cohomological invariants of this scheme: these are vector spaces (or modules) on the field (or the commutative ring) of coefficients which has allowed the scheme under consideration to be defined.
In addition, Grothendieck associated to these same schemes other sites (which are no longer topological spaces) such as the étale site or, in characteristic p, the crystalline site. These sites define other toposes with internal rings which in turn have cohomological invariants (like any topos with an internal ring). Thus, Grothendieck was able to associate to any scheme étale cohomology modules (with coefficients in the finite quotient rings of Z by finite powers of a prime number l), l-adic cohomology modules (with coefficients in the ring of l-adic integers) and, for p-characteristic (projective and smooth) schemes, modules of crystalline cohomology (with coefficients in the ring of p-adic integers or more generally the ring of Witt vectors of the considered base field).
Defining these invariants would have been impossible without the general notions of site, topos, and cohomology of a ringed topos. By construction, these modules or spaces of cohomology of a scheme on a field of coefficients are endowed with an action of the Galois group of this field. The study of these spaces or modules endowed with the action of this Galois group has been at the heart of algebraic number theory for half a century.
The fact that any algebraic variety on the field Q of rational numbers (or more generally on a number field) has l-adic cohomology spaces which are linear representations of the Galois group of Q (or more generally of the number field considered) allows to deepen a lot the knowledge of both the geometry of these varieties and of this Galois group. Indeed, the action of the Galois group on the cohomology spaces of a variety is related to the geometry of this variety. In this respect, the most important result is the Grothendieck-Lefschetz fixed point formula which relates the traces of the powers of the Frobenius elements acting on the cohomology spaces to the numbers of points with coefficients in finite fields of the reductions modulo a prime number p of an integral model of this variety.
Etale topology also allows to associate to any scheme the category of covers of this scheme which are locally trivial for this topology. If the scheme under consideration is connected, the choice of a geometrical point of this scheme defines an equivalence of this category of étale covers onto the category of finite sets endowed with an action of a certain profinite group called the fundamental (étale) group of this scheme at this point.
If X is, for example, a variety on a field K which is geometrically connected (i.e. is connected and remains connected after passing to an algebraic closure K' of K), its fundamental group has as a distinguished subgroup the "geometric" fundamental group of X, i.e. the fundamental group of the variety X' on K' deduced from X by changing the base from K to K'). And the quotient group identifies with the Galois group of K. Consequently, any such variety X over K defines a natural morphism from the Galois group of K to the group of outer automorphisms of the "geometric" fundamental group of X. One can wonder whether, in the case of certain "rich enough" varieties X naturally defined on a certain field K, this homomorphism is injective and whether its image can be characterised, which would provide a geometrical description of the Galois group of K. Grothendieck conjectured that it should be so if K = Q is the field of rational numbers and X is the "tower" consisting of moduli spaces of curves of genus g with n marked points with their natural geometric relationships. This is in fact the most direct approach to the most central problem of number theory: determining and understanding the Galois group of Q.
Langlands' programme proposes a partial and more indirect approach to the same problem, which consists in looking at the irreducible linear representations of this Galois group and predicting that they are naturally parameterised by objects from another part of mathematics, harmonic analysis : "automorphic representations". In fact, the deepest results obtained in the Langlands programme are all based on the study of the l-adic cohomology spaces of certain families of particularly rich and interesting varieties : "Shimura varieties" over Q, and the moduli spaces of Drinfeld "shtukas"over "algebraic function fields of a curve on a finite field".
Thanks a lot Professor Lafforgue for your detailed reply. I would like to understand it a little more, but step by step. I apologize for my basic questions. It will be an opportunity for me to learn more on topos.
First, you said : "Every scheme has therefore an associated topos endowed with a certain internal ring and internal modules on this ring (such as the sheaf of differentials and its external powers and, very often, a "polarisation", i.e. a locally free module of rank 1 which determines a way of embedding it into projective spaces). This defines cohomological invariants of this scheme: these are vector spaces (or modules) on the field (or the commutative ring) of coefficients which has allowed the scheme under consideration to be defined."
Could you help me to understand these sentences ?