Holmstrom l-adic cohomology

l-adic cohomology

MathSciNet

Google Scholar

Google

arXiv: Experimental full text search

arXiv: Abstract search

---

l-adic cohomology

AG (Algebraic geometry), AAG (Arithmetic algebraic geometry)

---

l-adic cohomology

Pure

---

l-adic cohomology

Katz: Review of $\ell$-adic cohomology (in the Motives volumes)

Katz: Independence of $\ell$ and Weak Lefschetz

Tate: Conjectures on Algebraic Cycles in $\ell$-adic cohomology (in the Motives volumes)

---

l-adic cohomology

Have not checked search results for this page.

---

l-adic cohomology

Deligne: Decompositions… (Motives vol) has a section.

Ekedahl: On the adic formalism

Kahn: We prove some finiteness theorems for the 'étale cohomology, Borel-Moore homology and cohomology with proper supports of schemes of finite type over a finite or p-adic field. This yields vanishing results for their l-adic cohomology, proving part of a conjecture of Jannsen.

Jannsen: On the $l$-adic cohomology of varieties over number fields and its Galois cohomology. Galois groups over $Q$ (Berkeley, CA, 1987), 315–360.

Jannsen: On the Galois cohomology of $l$-adic representations attached to varieties over local or global fields. Séminaire de Théorie des Nombres, Paris 1986–87, 165–182. Let $K$ be a local or global field, $\overline K$ its separable closure, and $X$ an algebraic variety over $K$. The author presents three conjectures on Galois cohomology of the étale cohomology groups of $\overline X=X\otimes\overline K$. Statements of conjectures: see Mathscinet - they failed to save.

He states with or without proofs that all conjectures except (1b) are true for $\roman{spec}\,K$ or a form of the standard cellular varieties, that (3) is true for $i=1$, and that (1) holds for $i=1$ if $X$ is an elliptic curve defined over, and having CM by, $K$ and $l$ is a regular prime for $X$.

Analogues of (1) and (3) in the function field case are proved and interrelations between the three conjectures and those of \n A. Be\u\i linson\en [in Current problems in mathematics, Vol. 24 (Russian), 181238, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984; MR0760999 (86h:11103)] and \n J.-M. Fontaine\en [Ann. of Math. (2) 115 (1982), no. 3, 529577; MR0657238 (84d:14010)] are discussed.

---

l-adic cohomology

See also Sheaf cohomology, Weil cohomology, Etale cohomology

---

l-adic cohomology

Notes from Katz in Motives vol

(first part of these notes under étale cohomology)

See Etale cohomology for the def of $H^i(X \otimes_k \bar{k}, \mathbb{Z} / N \mathbb{Z})$ and the compact support version of the same group. Taking inverse limits gives $\ell$-adic cohomology groups.

Here we work with a scheme separated and of finite type over a field $k$, and a prime $\ell$ invertible in $k$. The groups above are finitely gen $\mathbb{Z}_{\ell}$-modules with a continuous action of the Galois group, which vanish for $i > 2 \ dim(X)$ and which satisfies the expected universal coefficient long exact seq.

Tensoring with $\mathbb{Q}_{\ell}$ gives finite-dim vector spaces, again with a cont Galois action and vanishing thm. We also get a Kunneth formula both for ordinary cohomology groups and compact support cohomology groups. If $X$ is smooth and geometrically connected, PD holds, replace $\mathbb{Z} / N \mathbb{Z}$ by $\mathbb{Q}_{\ell}$ in the formula above.

From here, Katz treats the case of a finite ground field with $q = p^n$ elements. Details omitted here. Summary: Def of zeta function. Explanation of why/when Frob acts as identity on cohomology. Geometric, absolute and relative Frob. For any $X$ sep of finite type, and any $\ell \neq p$, expression of the zeta function as the alternating product of characteristic polynomials of geometric Frob acting on cohomology with compact support. Statement of Weil conjectures, the results hold for any $X$ sep of finite type.

Corollary: Let $X$ be proper and smooth. Then each such characteristic polynomial lies in $\mathbb{Z}[T]$ and has coeffs indep of $\ell = p$. Consequence: Betti numbers are indep of $\ell \neq p$.

Things we don’t know:

• Semisimplicity of Frobenius: For $X$ proper and smooth, the action of Frob on $H^i_c$ is diagonalizable over the algebraic closure of $\mathbb{Q}_{\ell}$, conjecturally. This is known for abelian varieties
• Conj: Independence of $\ell$ of characteristic polynomials, for a general scheme $X/k$ separated and of finite type. Similarly indep of Betti numbers is not known. However, indep of Euler characteristic is known.
• “Indep of $\ell$ of weight $j$ Betti numbers”. Similar to the previous, we know only indep of $j$-th virtual Betti number
• Indep of $\ell$ for kernels and cokernels of maps. This would follow from the standard conjectures
• Indep of $\ell$ of Leray spectral sequences. Explanation: Let $X/k$ and $S/k$ be separeated of finite type, and $f: X \to S$ a $k$-morphism. For any $\ell \neq p$, we have the Leray spectral sequence $E^{a,b}_2 = H^a(S \otimes_k \bar{k}, R^b f_* \mathbb{Q}_{\ell}) \implies H^{a+b}(X \otimes_k \bar{k}, \mathbb{Q}_{\ell})$

and its compact support analogue

and it is hoped that the characteristic polynomial of Frob on each $E^{a,b}_r$ has $\mathbb{Z}$ coeffs, indep of $\ell$.

---

Notes on Galois representations

Following Ito

General definitions

Let $F$ be a field, with absolute Galois group $\Gamma_F$. An Artin represenation of $\Gamma_F$ is a continuous HM to $GL(V)$, where $V$ is a complex vector space of finite dimension. An $\ell$-adic rep is the same thing with $\mathbb{C}$ replaced by a finite extension of $\mathbb{Q}_{\ell}$. Any Artin rep has finite image, so can be regarded as an $\ell$-adic rep after fixing an isomorphism $\mathbb{C} \to \bar{\mathbb{Q}}_{\ell}$.

Remark: Any continuous HM $\Gamma_F \to GL(n, \bar{\mathbb{Q}}_{\ell})$ has image contained in $GL(n, K)$ for some finite extension of $\mathbb{Q}_{\ell}$.

Example: $\mathbb{Z}_{\ell}(1)$, defined as the inverse limit of the $\mu_{\ell^n}$, gives the cyclotomic character $\chi_{\ell}$, for any $\ell$ invertible in $F$. Tensoring any $\ell$-adic representation with this character defines the Tate twist of the rep.

Example: The Tate module of an elliptic curve over $F$, for any $\ell$ invertible in $F$. In the case where $F$ is a number field and $E$ has no CM, the image of this rep is if finite index in $GL(2, \mathbb{Z}_{\ell}$ for all $\ell$, and is the whole thing for almost all $\ell$.

Example: For any variety $X$ defined over $F$, and any prime number $\ell$ invertible in $F$, we have the $\ell$-adic cohomology group $H^i(X_{\bar{F}}, {\mathbb{Q}}_{\ell} )$.

Case of local fields

Let $F$ be local, with residue field $\mathbb{F}_q$. Def: inertia subgp of $\Gamma_F$. Def: Weil group $W_F$. LCFT gives the reciprocity isomorphism $F^{\times} \to W_F^{ab}$. Here a uniformiser of $F$ corresponds to a lifting of the geometric (or arithmetic) Frobenius.

Can consider $\ell$-adic reps of $W_F$. Any $\ell$-adic rep of $\Gamma_F$ defines an $\ell$-adic rep of $W_F$, but not every such rep arises like this, because the topology we use on $W_F$ (in which intertia is an open subgroup) is stronger than the induced topology. An $\ell$-adic rep of any of these two groups is said to be unramified if the image of inertia is trivial.

Assume that $\ell$ does not divide $q$. Then can define the L-function of a representation $\rho: W_F \to GL(V)$ by

When $\ell$ divides $q$, can define the L-function using $p$-adic Hodge theory.

Global fields

Consider a global field $F$ and a prime number $\ell$ invertible in $F$. An Artin or $\ell$-adic representation $\rho$ induces a representation of $\Gamma_{F_v}$ for every finite place $v$ of $F$. We define the L-function of $\rho$ as

the product taken over all finite places.

We say that $\rho$ is pure of weight $w$ if there is a finite set $S$ of finte places such that, for each finite place $v$ outside $S$, $\rho_v$ is unramified, and the eigenvalues of $\rho(Frob_{q_v})$ are algebraic integers whose complex conjugates (?) have complex absolute values $q_v^{w/2}$.

Examples: Artin reps are pure of weight $0$. The $\ell$-adic cyclotomic character is pure of weight $-2$. Etale cohomology groups $H^i$ as above are pure of weight $i$, by basic properties of étale cohomology together with the Weil conjectures.

Examples: The above definition of L-function gives the Riemann zeta function and Dirichlet L-functions in the obvious cases.

Example: For an elliptic curve over a global field, the L-function attached to $H^1_{\ell}$ is independent of $\ell$, so can write $L(s, E)$. Interpretation in terms of number of rational points on reductions. More generally, for a projective smooth variety $X$ over $F$, the Hasse-Weil zeta function of $X$, at good places, can be written as a product of L-functions and inverses of L-functions attached to the $\ell$-adic cohomology groups of $X$.

Example: The $\ell$-adic representation attached to a modular form for $\Gamma_1(N)$ (of level $k$, character $\epsilon$, which is a normalized cusp form and a simultaneous eigenvector for all the Hecke operators). (Eichler, Shimura, Deligne, Serre.) In the case $k=2$, $\epsilon$ is trivial, and $a_k \in \mathbb{Z}$ for all $k$, then the Galois representation is constructed from the Tate module of an elliptic curve.

Conversely, given a 2-dimensional Galois rep, one wants to know if it comes from a modular form. The strong Artin conjecture predicts that any irreducible, odd, 2-dimensional Artin rep comes from a modular form of weight 1. This is true when the image is solvable, and perhaps true in general by recent work of Khare and Wintenberger. Taniyama-Shimura-Wiles says that the Galois rep ass to an elliptic curve over $\mathbb{Q}$ is associated to some modular form of weight 2.

The Langlands correspondence for $GL(n)$

Consider a global field $F$, its adele ring $A_F$, and a prime $\ell$ invertible in $F$. Roughly speaking, the global Langlands correspondence for $GL(n)$ over $F$ predicts an L-function-preserving correspondence between

1. Algebraic automorphic reps of $GL(n, A_F)$, and
2. Geometric $n$-dimensional $\ell$-adic reps of $\Gamma_F$

Say that an automorphic rep $\pi$ and a Galois rep $\rho$ are associated if their L-functions agree, up to a shift, and at almost all places. Examples come from class field theory and Galois reps attached to modular forms. Remark: For each $\pi$, there exists at most one associated $\rho$ (by Chebotarev density thm), and for each $\rho$, there exists at most one $\pi$ (by strong multiplicity one).

One expects that cuspidal automorphic reps correspond to irreducible Galois reps.

Ramanujan conjecture predicts that the $\ell$-adic rep associated to a cuspidal automorphic rep is pure.

The word $geometric$ in the above correspondence should mean: Unramified at all but finitely many places, and de Rham at $p = \ell$. (At least for irreducible Galois reps of $\Gamma_{\mathbb{Q}}$)

For certain number fields and certin automorphic reps, the associated Galois rep was constructed by Clozel, Kottwitz, Harris-Taylor.

One can also express the Langlands correspondence in terms of the conjectural Langlands dual group $L_F$. This group should be an extension of $\Gamma_F$ by a compact group. For a local field, there are definitions of $L_F$, namely as $W_F$ when $F$ is archimedean, and as $W_F \times SU(2)$ when $F$ is nonarch. Very roughly, complex representations of the Langlands dual group correspond to $\ell$-adic reps of $\Gamma_F$ (taking into account some p-adic Hodge theory).

The local Langlands correspondence says, roughly, that for a local field $F$ there exists a “natural” bijection between

• Equivalence classes (over $\bar{\mathbb{Q}}_{\ell}$) of Frobenius semisimple n-dimensional $\ell$-adic reps of the Weil group $W_F$, and
• Equivalence classes (over $\mathbb{C}$) of irreducible admissible reps of $GL(n, F)$

This can also be formulated in terms of complex reps of the Langlands dual group, or in terms of the Weil-Deligne group.

Remark: In contrast to the global case, the local L-function does not characterise representations.

Remark: We do not treat the archimedean case here. There is a formulation of local Langlands correspondence, which was proved by Langlands.

The case of a general reductive group

Let $F$ be a global field and $G$ a reductive connected group over $F$.

The L-group ${}^L G$ of $G$ is the semidirect product of a $\hat{G}$ with $\Gamma_F$, where $\hat{G}$ is (the complex points of) the so called dual of $G$. Roughly speaking, this dual is obtained from $G$ by interchanging roots and coroots. If $G$ is simply connected (adjoint), then $\hat{G}$ is adjoint (simply connected). In the case where $G$ is an inner form of a split group, the above semidirect product is just a direct product.

There should be a Langlands correspondence between:

• Automorphic side: Certain automorphic reps of $G(A_F)$, and
• Galois side: Certain HMs $\rho: L_F \to {}^G L$, called L-parameters, such that the composite $pr_2 \circ \rho: L_F \to \Gamma_F$ is the canonical one.

This correspondence is not bijective. In particular, objects on the automorphic side should be partitioned into L-packets.

To obtain precise statements, work with $\ell$-adic reps of $\Gamma_F$ instead of complex reps of $L_F$. The maps $\rho$ above can be replaced by maps $\rho: \Gamma_F \to \hat{G}(\bar{\mathbb{Q}}_{\ell})$

Def: L-functions on the automorphic side (through Satake parameters)

Def: L-function on Galois side

Both L-functions are only defined after fixing a rep $\hat{G} \to GL(n)$.

Formulation of Langlands correspondence, roughly as above.

Local versions: For archimedean, this was done by Langlands (Langlands classification). For nonarchimedean, it is known for unramified reps (Satake parameters).

Principle of functoriality

One expects the following: A homomorphism $f: \hat{G} \to \hat{G}'$ should induce an operation $\pi \mapsto f_* \pi$ from the set of (L-packets of) automorphic reps of $G(A_F)$ to that of $G'(A_F)$, satisfying

for all $r: \hat{G}' \to GL(n)$. In this situation, $f_* \pi$ is called a lifting, or transfer, of $\pi$. This operation can be described conjecturally, at least for almost all places. If we believe the Langlands correspondence, we can construct the operation by passing to the Galois side, composing with $f$, and then going back to the automorphic side.

There is a “fuller” version of the Principle of Functoriality (POF) expressed in terms of L-groups rather than dual groups.

Exampe: The Langlands correspondence itself follows from the POF between the trivial group and $G$.

Example: Base change: An operation from automorphic reps for $F$ to automorphic reps for a finite extension of $F$.

Example: Automorphic induction; take $GL(n)$ for simplicity: An operation from automorphic reps of $GL(n, A_{F'})$ to automorphic reps of $GL(nd, A_F)$, where $F'/F$ is an extension of degree $d$.

Example: Transfer to quasi-split inner forms. As a special case, obtain the Jacquet-Langlands correspondence.

Example: Symmetric power lifting. Let $G= GL(2)$ and $G' = GL(n+1)$. Have the symmetric power homomorphism $Sym^n: GL(2. \mathbb{C}) \to GL(n+1, \mathbb{C})$. Should get operation on automorphic reps.

Example: The (weak) Artin conjecture says that the L-function of an irreducible nontrivial Artin rep is entire. This is a consequence of the Langlands correspondence for $GL(n)$ for Artin reps.

Example: FLT follows from Taniyama-Shimura. Sato-Tate follows from Taniyama-Shimura together with existence of symmetric power liftings.

Techniques for constructing automorphic reps:

• Explicit constructions (Eisenstein series, Ramanujan’s Delta function)
• Theta series (applicable only for special pairs of groups)
• For $GL(n)$, can use Converse theorem, once we know nice analytic properties of L-functions (used for example by Lafforgue)
• Trace formulae (stabilized or twisted). To apply this, must prove “Fundamental lemma”. Works for $GL(n)$ and related groups; recent work on $GSp(4)$ and unitary groups.
• Sometimes we can use Galois reps, e.g. the $R=T$ method, for FLT and Sato-Tate; also Khare-Wintenberger for Artin conjecture for $GL(2)$ over $\mathbb{Q}$.

The case of GSp(4)

Much recent progress. Details…

---

l-adic cohomology

Gabber, Ofer (F-IHES): Sur la torsion dans la cohomologie l-adique d’une vari´et´e. C. R. Acad. Sci. Paris S´er. I Math. 297 (1983), no. 3, 179–182. The main result is that the etale cohomology of a smooth projective variety over a separably closed field is torsion free except at a finite number of primes. The proof depends on the cohomology of Lefschetz pencils and there is a complement on the Lefschetz morphisms.

---

l-adic cohomology

http://mathoverflow.net/questions/805/cohomology-of-moduli-spaces

nLab page on l-adic cohomology