In recent decades, \p\ adic geometry and \p\ adic cohomology theories have become indispensable tools in number theory, algebraic geometry, and the theory of automorphic representations. Tensoring up to q l, this gives a 1dimensional galois representation, the socalled l adic cyclotomic character. Here ldenotes a prime number invertible in k, and z l is the ring of l adic integers, the inverse limit of the rings zlr as rgoes to in nity. For a smooth projective geometrically uniruled threefold defined over a perfect field we show that there exists a canonical abelian variety over the field, namely the second algebraic representative, whose rational tate modules model canonically the third l adic cohomology groups of the variety for all primes l. An introduction to the cohomology of groups peter j. Compactly supported cohomology and nearby cycle cohomology of. Algebraic geometry i shall assume familiarity with. Big galois representations and functions volume 151 issue 4. The techniques used are a combination of results from group theory, algebraic groups and homological algebra. Introduction rankinselberg and cohomology comparison with delignes conjecture. Choosing m lkfor lsome xed prime, and kvarying over all positive integers, and making noncanonical, but a compatible system of identi cations zmz.
We have recapitulated the definition of constructible ladic sheaves over. Then as a complex manifold, it is locally analytic and contractible. In this note i try to give a very brief introduction to etale cohomology, especially to adic cohomology. Read download introduction to etale cohomology pdf pdf download. Rigid cohomology and p adic point counting par alan g. It can be seen as an analogue of the delignebeilinson cohomology in the p adic world.
Question feed subscribe to rss question feed to subscribe to this rss feed, copy and paste this url into your rss reader. More generally, x j is the class of a linear subspace cp n. In 9, it was shown that this is also the case for syntomic cohomology of varieties over a p adic local filed k. First we can reduce to the case where xis curve by an appropriate form of weak lefschetz. On the adic cohomology of varieties over number fields and its. Etale cohomology is an important branch in arithmetic geometry. The machinery of etale cohomology and its relative, ladic cohomology, is formidable. Weil cohomology theory when chark p 0, the l adic cohomology with k ql, for. Number theory learning seminar stanford university. Overconvergent cohomology of hilbert modular varieties and p. An adic gk representation comes from geometry if it is isomorphic to a subquotient of hi et xksep,q for some x and i. Etale cohomology was invented by grothendieck and his collaborators in the 1960s. An adic representation of gq is a nite dimensional qvector space v equipped with a continuous homomorphism.
Very soon, we will have to assume k finitely generated over. In mathematics, the etale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by grothendieck in order to prove the weil conjectures. As an application we obtain that, for an algebraic variety x and associated rigid analytic variety x rig, the l adic cohomology of x and x rig agree. All of the p adic algorithms i discuss rely upon rigid cohomology in some incarnation, and the alternative approaches pioneered by mestre and satoh are not touched upon. That will get us through the important smooth and proper base change theorems, as well as the basic formalism of l adic cohomology. By delignes proof of the weil conjectures d4, all this is known if one. For each group gand representation mof gthere are abelian groups hng,m and hng,m where n 0,1,2,3.
The speci c version of l adic cohomology used in this paper, hix. In this article we give a survey of two relatively recent developments in number theory. Recent progress on the tate conjecture ucla mathematics. Automorphy for some l adic lifts of automorphic mod l representations. Moreover, little attention is paid to the precise running times of algorithms, the. L functions and modular forms fall 2011 lecture 14. There is also a construction of rigid cohomology with supports in a closed subscheme, and of cohomology with compact supports. Algebraic geometry i shall assume familiarity with the theory of algebraic varieties, for. Lie algebra n, by extending a classical result of kostant to our integral p adic setup.
Much of the material in these notes parallels that in, for example, iversen, b. Some years ago a conference on l adic cohomology in oberwolfach was held with the aim of reaching an understanding of delignes proof of the weil conjec tures. The corresponding statement for intersection cohomology was originally proved this way by gabber. We will also explain how to define cycle class maps from chow groups to etale cohomology groups with coefficients in p adic etale tate twists and state injectivity and. This text is an extended version of a series of lectures given by the third and fourth. In this paper we consider period domains over local fields for quasisplit reductive groups. Pdf values of lfunctions and padic cohomology semantic.
As far as i know, the aim was to prove the weil conjectures, which deligne did in 1974. To give an example where this galois representation has a simple interpretation. Let x be an algebraic variety over an algebraically closed. Weil cohomology theory when chark p 0, the ladic cohomology with k ql, for. We will explain how to construct p adic etale tate twists on regular arithmetic schemes with semistable reduction, and state some fundamental properties of those objects. Cohomology of p adic symmetric spaces moduli spaces of weakly admissible. On the cohomology of integral padic unipotent radicals. The representation v is said to be rational if there exists a nite set s of primes such that v is unrami ed outside of s fgthat is, for all primes p 2 s fg, the action of. We compute the continuous l adic cohomology with compact support of them in the case of a basic isocrystal.
Higherdimensional arithmetic using padic etale tate. We describe how a systematic use of the deep methods from ladic cohomology pio neered by grothendieck and deligne and further developed. Unlike more subtle invariants such as homotopy groups, the cohomology ring tends to be. Bounding picard numbers of surfaces using padic cohomology. Michael harris eisenstein cohomology and special values of l functions. We have recapitulated the definition of constructible l adic sheaves over. Kedlaya, p adic cohomology from theory to practice arizona winter school lecture notes. We describe how a systematic use of the deep methods from l adic cohomology pio neered by grothendieck and deligne and further developed. The obstruction to defining ladic cohomology using sheaves is a shortage of open subsets in the zariski topology. A family of calabiyau varieties and potential automorphy. In previous lectures, we saw that the l function of such a modular form has a nice product expansion, namely, l f.
Singular cohomology is a powerful invariant in topology, associating a gradedcommutative ring to any topological space. Rigidity of p adic cohomology classes of congruence subgroups of gln, z proc. L functions, weights, and the decomposition theorem by shenghao sun doctor of philosophy in mathematics university of california, berkeley professor martin c. But even though we get to this at the very end, the bulk of the paper is devoted. On ladic cohomology of artin stacks university of california. Deuxmeth odesdistinctessonte laborees a l aide dun exemple. The third part sketches a particular algorithm for producing an approximate frobenius matrix on the cohomology of a smooth hypersurface, using p adic cohomology and a description of the cohomology of a smooth hypersurface due to gri. This book covers the main materials in sga 1, sga 4, sga 4 12 and sga 5 on etale cohomology theory, which includes decent theory, etale fundamental groups, galois cohomology, etale cohomology, derived categories, base change theorems, duality, and l adic cohomology. Sage for the p adic regulator of an elliptic curve over q, for p a good ordinary prime. X,athat have good functorial properties and in case when l c. Let kbe a pure lisse adic sheaf on a smooth connected variety x. Oct 26, 2016 an adic gk representation v is geometric if it is semisimple and decomposes v l w2z vw where vw is pure of weight w and almost all vw 0. Introduction in this paper we investigate the theory of p adic families of automorphic forms for the group gsp 4q whose component at in nity has singular harishchandra parameter and.
We will also investigate the relation of these classes to special values of p adic hecke l functions associated to the totally real eld. Eisenstein cohomology and special values of lfunctions. The syntomic cohomology, more precisely the cohomology of the sheaves sn on the syntomic site of a scheme, where introduced in fm87 in order to prove comparison isomorphisms between crystalline and p adic etale cohomology. V on the inertia group ip is trivial, and furthermore. Sage for the padic regulator of an elliptic curve over q, for p a good ordinary prime. Usually we prefer the intersection cohomology, since it has a close relation to the automorphic spectrum of ga. For the convenience of the speakers the present authors who were also the organisers of that meeting prepared short notes containing the central definitions and ideas of the proofs. A comparison theorem for ladic cohomology compositio. The third appendix relates hecke pollomisls mod p to l series, coherent cohomology and the fredholm determinant of u.
A comparison theorem for ladic cohomology springerlink. The arizona winter school 2007, on which the current book is based, was a unique opportunity to introduce graduate students to this subject. X y determines a homomorphism from the cohomology ring of y to that of x. Sheaf theory etale cohomology is modelled on the cohomology theory of sheaves in the usual topological sense. We fix an algebraic closure f of our ground field k. As a consequence of these computations, also in section 9, we deduce the 3 adic rigidity of the cuspidal cohomology with trivial coe.
In the winter we will delve further into the cohomology theory especially to duality theorems and kunneth formulas, and then move on to laumons technique of l adic fourier transforms in the sheaf setting. Olsson, chair we develop the notion of strati ability in the context of derived categories and the six operations for stacks in 26, 27. Etale cohomology theory nankai tracts in mathematics. Betti cohomology, etale cohomology and l adic cohomology.
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