To reiterate, a -representation of is if there exists a filtered (graded if ) -vector space with trivial Galois action such that is an isomorphism (preserving Galois action and filtration). The last statement implies that .
Fact: (Tate-Sen) For every finite extension we have if and otherwise. More generally, if is a character, and set , then if is finite and zero otherwise. This follows from the characterization of -admissibility. The proof starts by proving that the study of -semilinear representations of can be reduced to that -semilinear representations of where is a totally ramified -extension of . To from to we use almost etale descent; to go from to we use some decompletion process (note that if is a Galois extension, we know that there is a oneone correspondence between the elements of and the isomorphism classes of -semilinear representations of dimension of ). Then we can define Sen’s operator, which is morally ‘’ (indeed it is related to the Lie algebra of ). In particular it allows us to extract the Hodge-Tate weight. For more details see Fontaine’s note, chapter 3.
Faltings’s version of -adic Hodge decomposition of etale cohomology of a proper smooth variety for a finite extension :
Moreover, this isomorphism preserves Galois action. The theorem implies that etale cohomology is Hodge-Tate.
To show it is further de Rham (i.e. to capture the de Rham filtration not just the Hodge-Tate weight), we can tensor with and get an isomorphism that is compatible with filtration The only problem is that after tensoring with we don’t know how to undo it. A lot more can be said if we know the existence of some (-)integral model of over , i.e. a proper flat scheme satisfying some good properties, e.g. smooth or semi-stable over . In the former case we can define crystalline cohomology and in the latter case we can define log crystalline cohomology (since semi-stable implies log smooth over with certain conditions like ‘of Cartier type’). Th reason why such an integral model is useful is because of standard machinary like proper/smooth base change.
The Fontaine-Mazur conjecture states that a -representation of that is unramified at all but finitely many places and de Rham at places above . Then there exists projective smooth algebraic varieties and integers and such that is isomorphic to a subquotient of . All known cases involve the use of some modular curves or the higher dimensional generalizations plus some standard (often difficult) techniques to extrapolate from those cases plus some special cases with finite image. See this post for motivation behind this conjecture.
Now we define crystalline representations. Let be the divided power envelope of . More precisely, it is the -subalgebra of generated by for . Let be the -adic completion of and and . There are several facts:
is injective, where is the maximal unramified subextension of .
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The induced map between graded pieces is an isomorphism.
The Frobenius map on induces where is the -adic analogue of .
The induced filtration on is not -stable. In fact, on doesn’t preserve . and doesn’t extend to .
One key technical result we need is . This will allow us to recover from . Namely, we have .
Crystalline cohomology is by design an unramified object. It takes as input a scheme where and output a module over the ring of Witt vectors . Roughly speaking, crystalline cohomology of a variety in characteristic is the de Rham cohomology of a smooth lift of to characteristic , while de Rham cohomology of is the crystalline cohomology reduced mod . The idea of crystalline cohomology, roughly, is to replace the Zariski open sets of a scheme by infinitesimal thickenings of Zariski open sets with divided power structures. The motivation for this is that it can then be calculated by taking a local lifting of a scheme from characteristic to characteristic and employing an appropriate version of algebraic de Rham cohomology.
Crystalline comparison theorem (Faltings): Let be a proper smooth variety over with good reduction and let be the reduction of mod . Then
If does have a proper smooth model over , then the -adic etale cohomology is crystalline. Thus bad Galois representation forces bad geometry. But there are situations where we know the Galois representation is crystalline but it is not known whether a proper smooth model exists.
Semistable representations: There exists a unique homomorphism such that is the usual log, and and (Iwasawa’s convention).
For every , the element lies in for and converges to zero in the -adic topology. Thus we can make sense of . We extend this to all by setting . Then we have by composing the Teichmuller map with . The map induces a -algebra homomorphism . Let , which is noncanonically isomorphic to by matching some nonzero with . Finally define nononically isomorphic to .
The crystaline Frobenius mapping to extends to by mapping to . Noncanonically it is .
The new structure we get on is that for any nonzero such that so that , if we set (‘monodromy operator’), then is an endomorphism of that is independent of the choice of and the isomorphism . The map is -linear and and it satifies , and it is compatible with -actions.
Every coset in is represented by some such that (which implies makes sense) and we can define . For such that , we can define in the same way. Then is a Galois-equivariant map from to over and is injective.
Similar to crystalline representations, for semistable representations we have .
To summarize and and crystalline implies semistable implies de Rham implies Hodge-Tate.
Let be the category of filtered -modules . The tuple is isocrystal over , where is a finite-dimensional -vector space with a Frobenius-semi-linear (i.e. where is a lift of ). The filtration is on . Similarly let be the category of filtered -modules equipped with -linear such that .
Let be the category of finite-dimensional crystalline representation of over .
Fact: The functor from to is fully faithful and provides a quasi-inverse on the essential image. Similarly for .
The essential image is the so-called weakly admissible modules (a result of Colmez-Fontaine). The condition is for any subobject , we have . Roughly speaking, it means the Newton polygon defined by (over ) is above the ‘Hodge polygon’ defined by , and the two endpoints are the same.
Fact: Subquotient of cryst/st/dR/HT are cryst/st/dR/HT. But extensions need not preserve these properties. However, if is dR and and are st, then is st (Hyodo, Nekovar).
Semistable comparison: semistable over means is etale locally of the form . In this case is flat over , and it is regular. Also, the special fiber is a vertical divisor on with normal crosing.
We say in this case has semistable reduction. Kato, Hyodo define log-smoothness. With this setup, we can define log-crystalline cohomology (we can also do for all ), which is equipped with crystalline Frobenius and monodromy operator . Roughly speaking, log structure systematically equips varieties with log poles and residues along give the monodromy operator. For we define and and are analogous.
If smooth over , then log-crystalline cohomology agrees with usual crystalline cohomology (i.e. ).
Theorem (Tsuji, Faltings):
In crystalline and semistable comaprisons, the existence of some integral model proper smooth or sst over is a highly nontrivial condition to check. But the comparison theorem still tell us the geometry can’t be better than what the Galois representations reflect.
We will say a galois represetntation is potentially crys/sst if there exists finite extension such that is crys/sst.
Theorem (Berger, Andre-Kedlaya-Mebkhout) dR iff potentially sst.
This doesn’t imply the ‘geometric potentially semistable conjecture’, i.e. whether there exists finite extension such that there exists proper sst model over . If we replace number field by function field of a curve, then it has been proved by Mumford. A base change from to is necessary because there exists eliiptic curve with good reduction but a twist of it doesn’t.
Nevertheless, de Jong shows the following: There exists finite extension and a proper sst such that is an alteration of , i.e. there exists proper generically finite surjective map from to which implies that BAKM’s result using some generic etaleness and excision technique and induction argument plus the fact that dR extension of sst representation is sst.
Weil-Deligne representation: Let be a -representation over that is dR. Let be a finite Galois extension such that is sst. Consider . This is an -vector space with an action of and a semilinear and a nilpotent linear operator such that . Define a linear action sending to for each with image in and in . Since acts trivially, this is a continuous representation using discrete topology on the target. Then we take any , and set , which is unique up to isomorphism. Set for any .
Fact/example: where and and . If is a Galois character then with . For 1-dimensional representation, we also have semistable iff crystalline since there is no nonzero 1-dimensional nilpotent matrix.
Compatible system: A weakly compatible system is a collection of -dimensional representations of and isomorphism such that 1. is semistable and continuous with respect to the -adic topology on . 2. They have the same Hodge-Tate weights and the same Frobenius-semisimple -representations (which is unramified) across all but finitely many places.
is said to be strongly compatible if exists and for all finite places.
is irreducible if is irreducible for all and it is strictly pure of weight if for all but finitely places with defined and for all eigenvalues of where is any lift of the geometric Frobenius we have and all the Galois conjugate of is . In this case is force to be zero because of the relation between and .
More generally, we say a WD-rep of over is mixed if there exists an increasing filtration on by WD-subrep such that for and for and the graded piece is strictly pure of weight . The monodromy will map to . We say a WD-representation of is pure of weight if it is mixed and if for every , then is an isomorphism. In general, any nilpotent monodromy operator of defines (up to degree shifting a ‘mondromy filtration’ with symmetric properties. The condition is that the weight filtration coincides (up to degree shifting) with the monodromy filtration.
we say is geometric if there exists projective smooth variety and integer and and a subspace such that for every prime and the pullback of via the comparison between etale cohomology and singular cohomology is a subrepresentation of and .
The definitions are meaningful because for places having good reduction the representation are unramified and any of them are de Rham. Any are pure of weight (as global Galois representation so not touching the problem of bad reduction; use projecrive smooth models over at all but finitely many places which gives and then use Weil conjecture for RHS over finite field to show that is unramified and is strictly pure of weight .)
At places of when we don’t know has good reduction at , Deligne’s weight-monodromy conjecture still predicts that the above is pure (but not necessarily strictly pure) of weight when . If with having sst model over there are still analogues of weight-monodromy conjecture with log-crystalline cohomology (based on semistable comparison) and once we have the weight filtration (using mokrane’s work) we can compare it with the monodromy filtration.
(Tate conjecture) The cohomology of analytification with the following property:
For each prime and each embedding of of into , the pullback of of under comparison isomorphism is an irreducible subrepresentation of over .
For each and each place of , there is a WD-representation of of over and the base change of to is isomorphic to .
There is a tuple of multi-sets of integers indexed by such that for every with , we have and the multiplicity of is using Hodge decomposition and GAGA.
If we ignore 1 then 2 is known up to Frobenius-semisimplification for all but finitely many places and 3 is known (Katz-Messing).
Conjecture:
If is semisimple and algebraic, then is part of a weakly compatible system.
A weakly compatible system is strongly compatible.
An irreducible weakly compatible system is geometric and pure of weight for any .
Conjecture 2 and 3 are known when (Serre’s green book).