Selected publications

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(1) The discriminant of a symplectic involution (with M. Monsurr`o and J.-P. Tignol). Pacific J. of Math. 209, 201-218 (2003) (2) Essential dimension: a functorial point of view (with G. Favi).

Documenta Math. 8 , 279–330 (2003)

(3) Cohomological invariants and R-triviality of adjoint classical groups (with M. Monsurr`o and J.-P. Tignol). Math. Z. 248, No 2, 313–323 (2004)

(4) Essential dimension of cubics (with G. Favi). J. of Algebra 278, No 1, 199-216 (2004)

(5) On the notion of canonical dimension for algebraic groups (with Z. Reichstein). Adv. in Maths198, No. 1, 128–171 (2005) (6) Finiteness of R-equivalence groups of some adjoint classical groups

of type 2D3. To appear in J. of Algebra (2006)

(7) Cohomological invariants of quaternionic skew-hermitian forms.

To appear in Archiv der Math. (2006)

(8) Serre’s Conjecture II for classical groups over imperfect fields.

(with C. Frings and J.-P. Tignol). To appear in J. of Pure and Applied algebra (2006)

Books

(9) Introduction to central simple algebras and their applications to wireless communication. (with F.Oggier). In preparation.

(10) An introduction to Galois cohomology and applications. In prepa- ration.

Other publications

(11) Calcul des classes de Stiefel-Whitney des formes de Pfister, Pub.Math. Besan¸con, Th´eorie des nombres (96/97-97/98) (12) Characterization of hermitian trace forms, J. of Algebra 210,

690-696 (1998)

(13) R´ealisation de formes Z-bilin´eaires sym´etriques comme formes trace hermitiennes amplifi´ees . J. Th. des Nombres de Bor- deaux 12, 25-36 (2000)

(14) On hermitian trace forms over hilbertian fields. Math.Z. 237, 561-570 (2001)

(15) On the computation of trace forms of algebras with involution.

Comm. in Algebra29 (1), 457-463 (2001)

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(16) Autour des formes trace des alg`ebres cycliques. Pub. Math.

Besan¸con, Th´eorie des nombres, 1-9 (1998/2001)

(17) Divisible subgroups of Brauer groups and trace forms of central simple algebras (with D.B Leep). Documenta Math. 6, 486- 500 (2001)

(18) On the second trace form of central simple algebras in charac- teristic two (with C. Frings). Manuscripta. Math. 106, 1-12 (2001)

(19) On the set of discriminants of quadratic pairs. J. Pure Appl.

Algebra 188/1-3, 33-44 (2004) (Erratum: J. Pure and Appl.

Algebra195, No 1, 125–126 (2005))

(20) CM-fields and skew-symmetric matrices. (with E. Bayer and P. Chuard). Manuscripta Math. 114, No 3, 351–359 (2004) (21) (with F.Oggier)On Improving4×4 Space-Time Codes. To ap-

pear in Proceeding of the 40th Asilomar Conference on Signals, Systems and Computers(2006)

Preprints

(22) (with F.Oggier)On the existence of Perfect Space-Time Codes.

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1. Research work

1.1. Motivations. My research mostly concerns invariants of alge- braic structures, with a particular interest for invariants of algebraic groups. Recently, I also became interested in the applications of central division algebras to wireless communication (e.g. cell phones, wireless internet connection).

1.1.1. Invariants of algebraic structures. Given a class of algebraic struc- tures (e.g. quadratic forms, cubic forms, torsors under a given algebraic group, central simple algebras...), it is natural to try to classify them up to isomorphism. Since this kind of question is very difficult to solve di- rectly, one natural method to attack this problem is to define invariants attached to these structures, and then prove that these invariants are sufficient for the classication. This approach has been fruitful through the years, from Minkowski’s classification of rational quadratic forms to the recent breakthrough of Voevodsky’s proof of Milnor’s conjec- ture which can be considered as solving the classification problem of quadratic forms over arbitrary fields of characteristic different from 2.

In order to study a given algebraic structure, it is often more conve- nient to study its group of automorphisms, and try to deduce some results on the structure itself. The advantage of this approach is that this group has (in most of cases) a structure of algebraic group, and the machinery of algebraic geometry can be used to attack the classifi- cation problem. Moreover, this approach allows us to reinterpret some questions in cohomological terms. For example, ifq is a quadratic form of rank n over a field F, the associated automorphism group is the or- thogonal groupO(q), and the pointed set H1(F,O(q)) is in one-to-one correspondence with the isomorphism classes of quadratic forms of rank n. If G is the automorphism group of a central simple F-algebra with involution (A, σ), the pointed set H1(F, G) is in one-to-one correspon- dence with the isomorphism classes of algebras with involutions which are isomorphic to (A, σ) over Fsep. Moreover, by the fundamental work of Weil, any absolutely simple adjoint classical groupGdefined overF is isomorphic to the connected component of the automorphism group of some central simple F-algebra with involution (A, σ) when the char- acteristic of F is different from 2.

In this setting, many classical invariants can be viewed as cohomologi- cal invariants of a suitable linear algebraic group. For example, the dis- criminant of quadratic forms can be considered as a mapH1(,O(q))→ H1(, µ2), that is a cohomological invariant ofO(q) of dimension 1 with

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values in µ2. Other types of invariants can be naturally attached to algebraic structures, as GW-invariants, which associate to each struc- ture a quadratic form (e.g. the trace form of ´etale algebras or central simple algebras) or as discrete invariants (e.g. the essential dimension).

A crucial problem is the construction of non trivial invariants for a given class of algebraic structures. This can be extremely difficult, as the construction of the Rost invariant of an absolutely simple simply connected algebraic group shows. For a given invariant, many im- portant problems naturally come up: description of the set of values of this invariant, explicit computation of a given algebraic structure, solvability of the classification question. Some of these questions are not independent. For example, the construction of a non trivial (in a strong sense) cohomological invariant permits to give a lower bound to the essential dimension of an algebraic group. A supplementary interest of the study of these invariants is their possible interaction with other problems, as we will see later. For example, the theory of scaled hermitian trace forms of ´etale algebras can be used to give some structure theorems for CM fields, and cohomological invariants of algebraic groups give obstructions to the stable rationality of these groups. Finally, the canonical dimension of an algebraic group G is strongly related to the transcendence degree of generic splitting fields of G-torsors.

Let us also point out that Serre proved that the Witt invariant of the trace form of a Galois extension E/k of group G characterizes the obstruction to a certain Galois embedding problem ; this result has been used by several mathematicians to solve the inverse Galois problem in many new cases. Moreover, the existence of a non zero unramified invariant for a finite groupGimplies that Noether’s problem has a negative solution for G. Using this fact, Serre provided new families of groups for which Noether’s problem has a negative solution overQ. Finally the Rost invariant and its properties allowed Bayer and Parimala to prove the Hasse principle for classical simply connected groups over fields of virtual cohomological dimension at most 2.

To end this section, let us point out that the notion of invariant can be naturally defined more generally for functors. Since the functorial point of view is used in the study of essential dimension, we recall now the definition of the various types of invariants mentioned above in this more general setting.

Let k be any field, let Ck be the category of field extensions of k, and letF:Ck →Sets be a covariant functor.

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A cohomological invariant of F of dimension d with values in a dis- crete Galois module M is a natural transformation of functors F → Hd(, M). When F= H1(, G), we will assume that this invariant is normalized, that is it maps the trivial cocycle to the trivial cocycle.

We denote by Invd(G, M) the group of normalized invariants.

A GW-invariant of F is a natural transformation of functors F → GW(), where GW(E) is the Grothendieck-Witt group of the fieldE.

A discrete invariant of F is a map F→Z∪{±∞}.

1.1.2. Central simple algebras and wireless communication. Suppose that we want to transmit information symbols without using any wire.

Typically, it is the case when you are using wireless internet connections or cellular phones. During transmission via the channel (the air for example, in the case of cellular phones), two phenomenons may occur:

fading, that is a loss of intensity of the transmitted information, due to the fact that it may go through obstacles, such as trees or buildings (this is way your voice may not appear as loud as it is actually is to your interlocutor), and noise, due for example to interferences with other waves (this is why your interlocutor may not hear you properly sometimes). This is why the information transmitted to the receiver is not the original one.

The problem is to encode your information and transmit it in such a way that the probablity error is minimal, that is only very few errors occur during transmission. Of course, one way to proceed to send the same information several times, but it costs computer memory, it increases the amount of energy necessary for transmission, and no so much information is transmitted, so it is not worth it.

Suppose that we have two transmitting antennas and two receiving antennas. The information symbols we want to transmit are com- plex numbers. Each transmitting antenna sends an information sym- bol which will be received by each of the two receiving antennas, the information symbol going through two different paths. We will assume that the channel does not have time to change during two successive uses.

During the first use, the first antenna transmitsx0 and the second one transmits x2. Each of these two symbols go through the two possible paths and are received by the receiving antennas.

The symbol x0 arrives to the first receiving antenna as h1x0 and to the second one as h3, where h1, h3 are coefficients representing fading.

The symbolx2 arrives to the first receiving antenna as h2x0 and to the

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transmitter channel path receiver x0

x1

x3 x2

h4

h1

h3

h2

y0

y1

y3 y2

second one as h4, where h2, h4 are once again coefficients representing fading.

Therefore, the first receiving antenna receives a signal y0 which is the sum of 3 different signals: h1x0, h3x1 and some noise ν1, so

y0 =h1x0+h3x21

Similarly, the second receiving antenna receives a signal y2 of the form y2 =h2x0+h4x22

During the second use, the first transmitting antenna sendsx1, and the second one sends x3. Since the channel does not have time to change between the two uses, the fading coefficients will remain the same, and the first and second receiving antennas will receveive signals y1 and y3

of the form

y1 =h1x1+h3x33

and

y3 =h2x1+h4x34

Therefore, setting H =

h1 h3

h2 h4

, N =

ν1 ν3

ν2 ν4

and

X =

x0 x1

x2 x3

, Y =

y0 y1

y2 y3

,

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we get the following matrix equation Y =HX+N

The matrices H and N are random matrices following a Gaussian law.

We send in fact a matrix X ∈ M2(C), and we receive a matrix Y ∈ M2(C). The receiver is supposed to know the setC of all matricesX we send, called thecodebook . An element X ∈ C is called acodeword. He is supposed to know also the channel, that is the matrix H. The main problem is that Y /∈ C in general. How to decode? That is, how to recover a codeword ˆX ∈ C from Y, in such a way that the probability P(X →X) of sendingˆ X and decoding ˆX 6=X is as small as possible?

The recipe is as follows: for any M =

a b c d

, set

||M||2 =p

|a|2+|b|2+|c|2+|d|2

The codeword ˆX will be a codeword such that||Y −HX||2 is minimal among all the codewords X ∈ C (if there is more than one codeword with this property, one is chosen at random). The receiver can always compute ˆX since he knows C and H.

With this way of decoding, we have

P(X →X)ˆ ≤ C

Xmin6=X∈C|det(X−X)|4,

where C is a constant depending on the channel and X, X describe C. So the main question is now: how to design the codebook C? The criterion is: reliability ! To have an interesting upper bound, and ensure that P(X →X) is small, we need to maximize minˆ

X6=X∈C|det(X−X)|4, the first step being that we need to ensure that C is chosen in such a way that det(X−X)6= 0 for allX 6=X.

The main difficulty to achieve this is the non-linearity of the deter- minant. The idea is then to take for C a finite subset of a subring D of M2(C) which is also a division ring. In this way, we will have X −X ∈ D\{0} since R is a ring and the fact that D is a division algebra will ensure that every X −X is invertible in D, and there- fore in M2(C), which means that we will have det(X−X)6= 0 for all X 6=X ∈C.

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Everything may be generalized to a higher number of antennas, and the question becomes: Construct a codebook on a division subalgebra of Mn(C), such that min

X∈C|det(X)| is as large as possible?

A possible way to achieve this is to construct codes on central division algebras over a number fieldK. Indeed, IfDis such a central simpleK- algebra, then there exists a number fieldLcontainingKsuch thatD⊗K

L ≃ Mn(L). Then we immediately get an injection D ֒→ Mn(L) ⊆ Mn(C).

However, let us first emphasize that the above characterization of the coding problem is just a starting point. Coding involves many more issues, that we will discuss in the later chapters. Those include, to name a few

(1) encoding, that is, how do we map the information symbols into the matrices,

(2) rate, that is, how many matrices can we built, as a function of the number of information symbols and number of transmit antennas,

(3) decoding, which is the question of how to retrieve the informa- tion symbols from the received matrix.

Being able to design good codes consists not only of building the linear subspaces whose matrices are invertible, those matrices need to be easy to encode, to decode, and they should be available at high rate.

The construction of the codebook C on division algebras is more del- icate, but interesting codes have been constructed already on cyclic algebras.

1.2. Thesis work. In my thesis, I was interested in the possible values of some GW-invariants attached to ´etale algebras with involutions.

The framework of my thesis is the following:

let F be a field of characteristic different from 2. If E/F is a finite separable extension, with aF-linear nontrivial involution σ (so [E :F] is even), one can define a quadratic form for any symmetric elementλ, called hermitian scaled trace form, as follows:

x∈E →F, x7→TrE/F(λxxσ).

If λ= 1, we will call it hermitian trace form.

A natural question is to know which quadratic forms areF-isomorphic, or more reasonably Witt-equivalent, to such forms. The articles [12],

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[13], [14], [15] and [16] contain my thesis work and describe the original results I’ve obtained.

In [14], we prove the following realization theorem for hermitian scaled trace forms over hilbertian fields:

Theorem 1. If F is a hilbertian field, every even-dimensional qua- dratic form over F, which is not isomorphic to the hyperbolic plane H, is isomorphic to a hermitian scaled trace form.

We also give a theorem concerning Witt classes of hermitian trace forms:

Theorem 2. If F is a hilbertian field with stability index at most 2, then an even-dimensional quadratic form is Witt-equivalent to a her- mitian trace form if and only if it is positive (that is, all its signatures are non-negative integers).

In the case of algebraic number fields, we have a more precise result, proved in [12]:

Theorem 3. If F is an algebraic number field, an even-dimensional quadratic form of dimension at least 4 is isomorphic to a hermitian trace form if and only if it is positive.

We also give in [13] the following realization result for integral trace lattices:

Theorem 4. If b : M →Z is a Z-bilinear non degenerate symmetric form over a finitely generated free un Z-module of even rank, which is not Q-isomorphic to H, then b can be realized explicitely as a form (x, y)∈A×A7→TrQ(α)/Q(λxyσ), where α∈C is an algebraic integer, A is an ideal of Z[α], σ is a Q-linear non trivial involution on Q(α) and λ isσ-symmetric.

The proof of this theorem is explicit and algorithmic.

In my thesis, I was also interested in trace forms and hermitian trace forms of central simple algebras with and without involution (defined in an analogous way with the reduced trace). I’ve computed the trace forms of cyclic algebras in some cases, and the hermitian trace forms attached to algebras with involution over some fields, especially fields satisfying I3(F) = 0 (see [15] and [16]).

In [11], I’ve computed the total Stiefel-Whitney class of Pfister forms.

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1.3. Postdoctoral research. After my thesis, I’ve diversified my re- search interests. I’ve been naturally interested in trace forms of central simple algebras, but mostly in various questions concerning linear alge- braic groups: finding cohomological invariants, computation of essen- tial dimension over any field (in particular for finite groups), rationality of algebraic groups, finiteness of R-equivalence groups.

Thus, I’ve got the opportunity to work jointly with young researchers as M. Monsurr`o (EPFL), C.Frings (Besan¸con), G.Favi (Lausanne) or P.Chuard (Neuchatel), or some established researchers as J.-P.Tignol (Louvain-La-Neuve), E.Bayer (EPFL), D.Leep (Kentucky) or Z.Reichstein (Vancouver).

The results mentionned above can be found in the articles [17], [18], [1], [2], [19], [20], [3], [4] and [5].

7.3.1GW-invariants. LetAbe a central simple algebra over an arbi- trary fieldF. If a∈A, and if PrdA(a) := Xn−s1Xn−1+s2Xn−2+· · · is its reduced characteristic polynomial, we define respectively the re- duced trace of a and the second reduced trace of a by TrdA(a) = s1

and SrdA(a) :=s2. We then denote by TA the quadratic form a∈A7→TrdA(x2)

and by T2,A the quadratic form

a∈A7→SrdA(a).

This providesGW-invariants of central simple algebras as soon as these forms are non degenerate.

• In [17], we prove some realization results for trace forms of central simple algebras. In the following, F will always denote a field of char- acteristic different from 2, and K = F(√

−1). If F is a field, In(F) denotes the nth power of the fundamental ideal of the Witt ring of F, and ΩF will denote the space of orderings of F.

In a first part, we study the structure of the torsion group of the 2- primary component of the Brauer group. We prove the following result, which generalizes the result already known for number fields:

Theorem 5. LetF be a field satisfying the SAP property and such that I3(F)t = 0. Let T (resp. Λ) a set indexing a Z/2Z-base of Br2(F)t

(resp. of F×/P

F×2). Then we have the following isomorphism of groups:

2Br(F)≃Z(2)(T)×(Z/2Z)(Λ).

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Recall that a fieldF satisfies the SAP property if for any clopen subset X ⊂ ΩF there exists a ∈ F× which is positive with respect to any ordering lying in X and negative elsewhere.

As an application, we characterize completely the quadratic forms which can be realized as the trace form of some central simple F- algebra over some fields. We then obtain the following results:

Theorem 6. Letn= 2m ≥2be an even integer. Assume thatI2(K) = 0. Then a quadratic formq is isomorphic to the trace form of a central simple algebra of degree n if and only if the following conditions are satisfied :

(1) dimq=n2

(2) detq= (−1)n(n2−1)

(3) signvq =±n, for all v ∈ΩF.

Theorem 7. Assume F satisfies the following conditions:

(a) I3(F) is torsion-free

(b) For every r≥0 and for every [A]∈Br(F) such that 2r+1[A] = 0, there exists A, degA = 2r+1 such that [A] = [A].

Then a quadratic form q is isomorphic to the trace form of a central simple algebra of degree n if and only if the following conditions are satisfied :

(1) dimq=n2

(2) detq= (−1)n(n2−1)

(3) signvq =±n, for all v ∈ΩF.

In particular, this theorem holds for local fields, global fields or quotient fields of excellent two-dimensional local domains with algebraically closed residue fields of characteristic zero, e.g. finite extensions of C((X, Y)).

The proofs of these realization theorems are based on classification results of quadratic forms over fields satisfying I2(F)t= 0 orI3(F)t= 0, and the previous structure theorem.

• In [18], we are interested in the second trace form of central simple F-algebras when char(F) = 2, as well as in the computation of its discriminant and its Clifford invariant. When char(F) 6= 2, this in- variant does not give much information as the trace form itself, since TA ≃ TB ⇐⇒ T2,A ≃ T2,B. When char(F) = 2, the trace form has rank zero, and the second trace form is non singular. The same phe- nomenons occur for the trace form and the second trace form of ´etale

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algebras. Moreover Berg´e and Martinet showed that the trace form of an ´etale algebra over a field of characteristic 2 was completely deter- mined by its Arf invariant, showing in particular that the trace form is a very poor substitute to the trace form in this case. One can ask if this is also true in the case of central simple algebras. When char(F)6= 2, the discriminant and the Clifford invariant of TA have been computed by several authors. In particular, the discriminant of the trace form only depends on the degree ofA, and its Clifford invariant only depends on n2[A] (whenn is even).

We prove in [18] a similar result for the second trace form in charac- teristic 2 (showing in particular that the second trace form is a good substitute to the trace form in this case).

Before stating the result, let us introduce some notation.

Assume that char(F) = 2. We denote by ℘(F) the set {x2 +x, x ∈ F}. If α ∈ F× and β ∈ K, we denote by (α, β] the class of the corresponding quaternion algebra in the Brauer group. This algebra has a F-basis 1, e, f, ef satisfying the relations e2 = α, f2 +f = β and ef +f e = e. If a, b ∈ F, we denote by [a, b] the quadratic form (x, y)∈ K2 7→ax2+xy+by2. A non-degenerate quadratic form over K has even rank and is isomorphic to an orthogonal sum of some [a, b].

Ifq≃[a1, b1]⊥ · · · ⊥[ar, br], then the Arf invariant of q is the element of F/℘(F) defined by Arf(q) := a1b1 +· · ·+arbr +℘(F). We also define the Clifford invariant of q, denoted by c(q), to be the class of the Clifford algebra of q in the Brauer group.

We then have the following result:

Theorem 8. Let F be a field of characteristic 2, and letA be a central simple F-algebra of degree n, n even. Then T2,A is a non degenerate quadratic form and we have:

(1) Arf(T2,A) = Arf(T2,Mn(F)) = [n4] (2) c(T2,A) = n2[A]

• In [20], we use the theory of hermitian scaled trace forms to show that a CM-field of degree 2n overQis generated by an eigenvalue of a skew-symmetric matrix with rational coefficients of dimension at most 2n+ 4, which improves a result of Cohen and Odoni. For this, we use a result proved in [14].

Recall that a number field is a CM field if K =F(√

−θ), where F/Q is totally real andθ ∈F× is totally positive. Cohen and Odoni showed

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that √

−θ is an eigenvalue of a skew-symmetric matrix with rational coefficients of dimension at most 4n+ 2, where [K :Q] = 2n.

In [20], we prove the following result:

Theorem 9. (1) If n ≡3[4], √

−θ is the eigenvalue of some skew- symmetric matrix with rational entriesM of dimension2n+ 1, and this bound is optimal.

(2) If n ≡ 1[4], √

−θ is the eigenvalue of some skew-symmetric matrix with rational entries M of dimension 2n+ 3, and this bound is optimal provided that the characteristic polynomial of M is separable.

(3) If n is even, √

−θ is the eigenvalue of some skew-symmetric matrix with rational entries M of dimension 2n+ 4.

The strategy of the proof is the following: if f denotes the minimal polynomial of√

−θoverQ, thenf divides the characteristic polynomial ofM, which is necessarily even or odd. The problem then is equivalent to find a polynomialP divisible byf of degree 2n+ 1,2n+ 3 or 2n+ 4, and which can be realized as the characteristic polynomial of such a matrix.

We then use the following proposition:

Proposition 10. Let k be a field of characteristic different from 2, and let P ∈ k[X] be a separable polynomial of degree m, such that P(−X) = ±P(X). Let E = k[X]/(P) and let σ be the involution on E defined by σ(X) =−X.

Then P is the characteristic polynomial of a skew-symmetric matrix with coefficients in k of dimension m if and only if there exists a σ- symmetric elementλ ∈E× such that TrE/k(λxxσ) is isomorphic to the unit form.

7.3.2 Cohomological invariants. In this section we describe the re- sults on cohomological invariants which we have obtained.

First of all, let us recall some definitions. Let A be a central simple algebra with center K. An involution σ on A is a non trivial anti- automorphism of order 2.

We denote byF the subfield ofK consisting of σ-symmetric elements.

An involution is of first kind if σ|K = IdK, and unitary (or of unitary type) otherwise. In the first case,K =F, and in the second case,K/F is a quadratic separable extension and σ|K is the unique non trivial F-automorphism of K.

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If A = End(V), any involution of first kind is adjoint to a non alter- nating symmetric bilinear form or to an alternating bilinear form. IfA is arbitrary and if σ is of first kind, we say that σ is orthogonal(or of orthogonal type), resp. symplectic (or of symplectic type) if σ ⊗IdFs is adjoint to a non alternating symmetric bilinear form (resp. to an alternating bilinear form).

Assume that char(F) 6= 2. If σ is orthogonal and A has even degree, we set

disc(σ) = NrdA(u)∈F×/F×2,

where u ∈ A× satisfy σ(u) = −u. The class disc(σ) is called the discriminant of σ.

If σ is unitary, one can define a F-central simple algebra D(A, σ) of exponent at most 2, called the discriminant algebra. The map σ 7→

[D(A, σ)] then defines a cohomological invariant of dimension 2 with values in Z/2Z of unitary involutions on A. One can show that there is no cohomological invariants of dimension 1.

If (A, σ) is an algebra of involution of any type, we say that σ is hy- perbolicif there exists e∈A satisfying e2 =e and σ(e) = 1−e.

For any algebra of involution, we define the group scheme of similitudes of (A, σ), denoted by Sim(A, σ), by:

Sim(A, σ)(R) = {u∈A⊗F R |(σ⊗Id)(u)u∈R×}

for any commutative F-algebra R, and we define the group scheme of projective similitudes of (A, σ), denoted by PSim(A, σ), by:

PSim(A, σ) = Sim(A, σ)/RK/F(Gm,K)

We denote bySim+(A, σ) andPSim+(A, σ) the connected component of the unit element of these groups.

If g ∈ Sim(A, σ)(F), we set µ(g) = σ(g)g ∈ F×. We say that g is proper if g ∈ Sim+(A, σ)(F) and improper otherwise. Improper similitudes only exist eventually for orthogonal involutions, since the group of similitudes is connected in the other cases.

Two involutions σ and σ on A are called conjugate if there exists a∈A× such thatσ = Int(a)◦σ◦Int(a)−1.

The cohomology setH1(F,Sim(A, σ)) is then in one-to-one correspon- dence with the conjugacy classes of involutions on A of same type as σ.

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• Contrary to the case of orthogonal involutions, where analogues of discriminant and Clifford algebra are defined, no cohomological invari- ant attached to symplectic involutions was defined until now.

In [1], we define a cohomological invariant of dimension 3 with values inµ2 attached to a symplectic involution on a central simpleF-algebra of degree 2m, calleddiscriminant, and denoted by ∆(σ), as follows:

Recall first that, if θ be a symplectic involution on A, the pfaffian reduced norm is the homogeneous polynomial function of degree m

Nrpθ: Sym(A, θ)→F uniquely determined by the following conditions:

Nrpθ(1) = 1 and Nrpθ(x)2 = NrdA(x) for x∈Sym(A, θ).

We assume that char(F)6= 2, and thatA carries a hyperbolic symplec- tic involutionθ, which is equivalent to say thatA ≃M2(A0), whereA0

is a central simple F-algebra of exponent at most 2.

Definition. Let A be a central simple algebra over F of degree n = 2m≡0 mod 4, carrying a hyperbolic symplectic involutionθ. Letσ be symplectic involutions on A. There exists s∈Sym(A, θ)× such that

σ = Int(s)◦θ.

We define the discriminant of σ and denote it by ∆(σ), setting

∆(σ) := Nrpθ(s)

2∪[A]∈H3(F, µ2).

It only depends on the conjugacy class of θ. This invariant is the first non trivial invariant associated to symplectic involutions. Moreover, one can show that there is no other invariant of dimension 3 with values in Z/2Z associated to symplectic involutions. More precisely, we have:

Proposition 11. Let (A, θ) be a F-algebra with a symplectic involu- tion.

If A is split, we have H1 L,Sim(A, θ)

= 1 for all L∈CF, and then Invd(Sim(A, θ), M) = 0 for all d.

If A is not split, we have

Invd(Sim(A, θ), M) = 0 for d= 1, 2 and

Inv3((Sim(A, θ), M) =

(0 if degA≡2 mod 4, Z/2Z if degA≡0 mod 4.

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In the case of orthogonal and unitary involutions of degree 4, the trivi- ality of the discriminant and of the discriminant algebra respectively is equivalent to the decomposability of these involutions in tensor product of two algebras with involutions. In the symplectic case, this decom- position always exists in the degree 4 case. One can then asks if the triviality of the discriminant is related to the decomposability of sym- plectic involutions on algebras of degree 8. The answer is affirmative, and we show in [1] the following result:

Theorem 12. Let A be a central simple F-algebra of degree 8 with index dividing 4. For any symplectic involution σ on A, ∆(σ) = 0 if and only there is a decomposition

(A, σ) = (A1, σ1)⊗F (A2, σ2)⊗F (A3, γ3)

whereA1, A2,A3are quaternion subalgebras ofA, σ12 are orthogonal involutions on A1 and A2 respectively, γ3 is the conjugation involution on A3, and A1 is split,

A1 ≃M2(F).

•IfXis an irreducible variety defined overF, we say thatX isrational if X is birationally isomorphic to an affine space, and stably rational if there exists m≥0 such thatX×AmF is rational. A natural problem is to know whether or not a given variety X is (stably) rational.

When X is a linear algebraic group, Manin defined an obstruction to rationality as follows:

Let G be a linear algebraic group defined over F. We say that g ∈ G(F) is R-trivial if there exists a rational map f : A1R 99K G(F), defined at the points 0 and 1, such that f(0) = 1 and f(1) = g. We denote by RG(F) the normal subgroup of R-trivial elements of G(F).

The quotient group is simply denoted by G(F)/R, and called the R- equivalence group of G. It is known that, if a connected group G defined over F is stably rational as a variety, then Gis R-trivial, that is G(E)/R= 1 for any field extension E/F.

Only few results on rationality of absolutely simple adjoint groups were known until now. Adjoint groups of type 1An−1 and Bn are rational and adjoint groups of type 2An−1 and Cn are stably rational for n odd. Moreover, Merkurjev gave examples of adjoint groups of type

2Dn which are not R-trivial (hence not stably rational) for any n≥3.

In [3], we give the first families of absolutely simple adjoint groups of type Cn, 2An−1 et 1Dn for n even, which are not R-trivial, hence not stably rational.

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Recall that an absolutely simple adjoint group of classical type is iso- morphic to PSim+(A, σ). Type 2An−1 corresponds to degA = n and σ unitary, type Cn to degA = 2n and σ symplectic, and 1Dn to degA= 2n and σ orthogonal with trivial discriminant.

We then have precisely the following results:

Theorem 13. Let Q, H be two quaternion F-algebras (char(F) 6= 2) satisfying

(−1)∪[H] = 0 in H3(F, µ2) and [H]∪[Q]6= 0 in H4(F, µ2).

Let A =M2r(H)⊗Ms(Q), where r is arbitrary and s is odd. Let ρ be an orthogonal involution on M2r(H)carrying improper similitudes and let τ any involution of first kind on Ms(Q). Then PSim+(A, ρ⊗τ) is not R-trivial.

Theorem 14. Let r be an arbitrary integer. Let H be a quaternion F-algebra (char(F) 6= 2), α ∈ F×, K = F[X]/(X2−α), and let ι be the non trivial automorphism of K/F. Assume that

(−1)∪[H] = 0 in H3(F, µ2) and (α)∪[H]6= 0 in H3(F, µ2).

Let ρ be an orthogonal involution on M2r(H) carrying improper simil- itudes. Then PSim+(M2r(H)⊗F K, ρ⊗ι) is not R-trivial.

The existence of an orthogonal involution ρ on M2r(H) carrying im- proper similitudes is insured by the condition (−1)∪[H] = 0. In fact, one can show that a central simple algebra A with an involution of first kind has an orthogonal involutionρ carrying improper similitudes if and only if A has index at most 2 and if (−1)∪[A] = 0 if n ≡0[4].

The strategy of construction of these examples is the following: the idea is to define a non trivial morphism θ : PSim+(A, σ)→ H(,Z/2Z) which is zero on the group of R-trivial elements. This morphism then factors into a morphismθ :PSim+(A, σ)()/R→H(,Z/2Z) which has a non trivial image, which implies that PSim+(A, σ) is not R- trivial.

In order to define this morphismθ, we use some cohomological invari- ants of the group Sim(A, σ).

If σ is unitary, the morphism θ is given by

θE :g ∈PSim+(A, σ)(E)→(µ(g))∪[D(A, σ)]∈H3(E,Z/2Z), for any field extension E/F.

Ifσ is of first kind, we define a new cohomological invariant of dimen- sion 3.

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Let Tσ+ the restriction of the trace form to the space of σ-symmetric elements. We then have the following lemma:

Lemma 15. Let σ, σ0 be two involutions of first kind on A.

• If σ and σ0 are both symplectic, then Tσ+−Tσ+0 ∈I3F.

• Ifσ andσ0 are both orthogonal, and discσ = discσ0, thenTσ+− Tσ+0 ∈I3F.

We then obtain a non trivial cohomological invariant taking the Arason invariant of Tσ+−Tσ+0.

We then define θ by

θE :g ∈PSim+(A, σ)(E)→(µ(g))∪e3(Tσ+−Tσ+0)∈H4(E,Z/2Z), for any field E/F, whereσ0 is hyperbolic and disc(σ) = 1.

We then show that the morphism θ is non trivial proving that θ(g0⊗ 1)6= 0 ifg0 is an improper similitude of ρ, and that it factors through R-equivalence using the explicit description of PSim+(A, σ) obtained by Merkurjev.

In [6], we construct families of adjoint groups G of type 2D3 defined overF (but not overk) such thatG(F)/Ris finite for various fields F which are finitely generated over their prime subfield. We also construct families of examples of such groupsGfor whichG(F)/R ≃Z/2Zwhen F = k(t), and k is (almost) arbitrary. This gives the first examples of adjoint groupsG which are not quasi-split nor defined over a global field, such that G(F)/R is a non-trivial finite group.

• Let F be a field of characteristic 2, and let A be a central simple F-algebra with an involution of first kind.

A quadratic pair onAis a pair (σ, f), whereσis a symplectic involution onA and f : Sym(A, σ)→F is a linear map satisfying

f(a+σ(a)) = TrdA(a) for all a∈A.

For example, ifσ is a symplectic involution on A, then for every ℓ∈A satisfying ℓ+σ(ℓ) = 1, the pair (σ, f), where f : s ∈ Sym(A, σ) 7→

TrdA(ℓs), is a quadratic pair onA. Conversely, for any quadratic pair (σ, f) on A, there exists an element ℓ ∈ A, uniquely determined up to addition of an element of Alt(A, σ), such that f(s) = TrdA(ℓs) for all s ∈ Sym(A, σ). Moreover, this element satisfies ℓ+σ(ℓ) = 1. The discriminant of (σ, f), denoted by disc(σ, f), is the element of F/℘(F) defined by disc(σ, f) = SrdA(ℓ) + m(m−1)2 +℘(F).

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The quadratic pairs play the same role as orthogonal involutions in characteristic not 2, since any quadratic pair is adjoint to a quadratic formq whenAis split, and the discriminant in this last case is the Arf invariant of q.

If char F 6= 2, Knus, Parimala and Sridharan proved that the set of discriminants of orthogonal involutions on a division algebra D of degree at least 4 equals to NrdD(D×). The previous considerations show that it is natural to ask for an analoguous result for quadratic pairs.

For any symplectic involutionσ, we setd(A, σ) :={SrdA(ℓ) +m(m−1)2 +

℘(F), ℓ+σ(ℓ) = 1}.

In other words,d(A, σ) is the set of discriminants of quadratic pairs on A of the form (σ, f).

We prove the following result (see [19]):

Theorem 16. Let F be any field of characteristic 2, and let A be a central simple F-algebra carrying symplectic involutions, which is not a division quaternion algebra. Then there exists a symplectic involution σ on A such that d(A, σ) = F/℘(F). In particular, any element α ∈ F/℘(F) is the discriminant of a quadratic pair on A.

• In [7], we are interested in the classification of skew-hermitian forms over quaternion algebras. Contrary to the case of quadratic forms., no complete system of invariants for (skew)-hermitian forms over a division algebra with involution is known. In fact, very few invariants have been constructed. The major advance in this direction is the construction by Rost of a cohomological invariant H1(, G) → H3(,Q/Z(2)), where Gis a semi-simple simply connected linear algebraic group, which then has been used by Bayer-Fluckiger and Parimala in to construct a Rost invariant for skew-hermitian forms over a division algebra with a sym- plectic involution.

In this paper, we define some invariants en,Q for skew-hermitian forms over quaternion algebras, which are twisted versions of the invariants en. We then prove that these invariantsen,Q form a complete system of invariants of quaternionic skew-hermitian forms. As an application, we show that quaternionic skew-hermitian forms defined over a field of 2- cohomological dimension at most 3 are classified by rank, discriminant, Clifford invariant and Rost invariant, defined by Bayer-Fluckiger and Parimala.

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More precisely, we show the following result. If h is a skew-hermitian form of rank n over a quaternion k-algebra (char(k) 6= 2), then hk(Q) corresponds to a quadratic form qhk(Q) over rank 2n over k(Q) (the function field of the conic corresponding to Q). We then have the following proposition:

Proposition 17. Let h be a skew-hermitian form over Q, and assume that qhk(Q) ∈In(k(Q)), n ≥1. Then the following holds:

1) If d = 1, there exists a unique element en,Q(h) ∈ H1(k,Z/2Z), satisfying

Resk(Q)/k(en,Q(h)) =en(qhk(Q)).

2) If d ≥ 2, there exists a unique element en,Q(h) ∈ Hd(k, µ⊗d4 )/[Q]∪ Hd−2(k, µ2) satisfying

Resk(Q)/k(en,Q(h)) =en(qhk(Q)).

Here [Q]∪Hd−2(k, µ2) is identified with a subgroup of Hd(k, µ⊗d−14 ).

The proof is based heavily on the fact that the unramified cohomology ofk(Q) comes fromH(k, µ2). This proposition is not specific toenand may be generalized to cohomological invariants of functors satisfying reasonable properties. See [7] for more details.

We also define an invariant e0,Q by the formula e0,Q(h) = 2rk(h)∈Z/4Z

We then get an analogue of the classification theorem for quadratic forms:

Theorem 18. Let h be a skew-hermitian form over Q. Then h is hyperbolic if and only en,Q(h) = 0 for all n ≥0.

Finally, in [8], we proved the strong version of Serre’s conjecture II for absolutely simple simply connected groups of classical type. More precisely, we prove:

Theorem 19. Let G be a simply connected absolutely simple group G of type A, B, C or D (trialitarian case excluded) defined over a field F of arbitrary characteristic. Assume that for every torsion prime p of the root system of G, and for every finite separable extension E/F, the reduced norm map of every p-primary central simple E-algebra is surjective. Then H1(F, G) = 1.

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This result has been proved by bayer and Parimala when char(F)6= 2 or for perfect fieldsF of characteristic 2. This result is then new when F is imperfect of characteristic 2. Our proof is characteristic free. In particular, we recover the result of Bayer and Parimala.

7.3.3. Essential dimension. The notion of essential dimension of an algebraic group has been introduced by Reichstein. Rost also de- fined the essential dimension of some subfunctors of Milnor’sK-theory.

Finally, in some unpublished notes, Merkurjev defined more generally the essential dimension of a functor F:Ck →Sets.

Letk be a field, and letCk be the category of field extensions ofk. We also denote by Sets, respectively Sets*, the category of sets, respec- tively the category of pointed sets.

LetF:Ck →Sets be any covariant functor.

Let K/k be a field extension, and a∈F(K). Let n ≥0 be an integer.

We say that the essential dimension of a is less or equal to n, and we denote it by edk(a)≤n if there exist a field E, k ⊆E ⊆K such that:

(1) trdegkE =n

(2) a∈Im(F)(E)→F(K)).

We say that edk(a) =n if edF(a)≤n and edk(a)6≤n−1.

The essential dimension of F, denoted by edk(F) is the supremum of the essential dimensions edk(a), for all a ∈ F(K) and all extensions K/k.

IfGis a linear algebraic group defined overk, we denote by edk(G) the essential dimension of the functor H1(, G), and call it the essential dimension of G.

The notion of essential dimension is a new one, and very few results are known. This invariant is particularly interesting but also difficult to compute for a given algebraic because it contains a lot of arith- metic and algebraic information. For example if Gis a finite constant group, ed(G) is the minimal number of parameters needed to construct a generic Galois extension of groupG. This explains why it is even dif- ficult to give good upper bounds for this number, since it would imply to know good parametrizations of Galois extensions over a general field.

The computation of this number is not even known for G=Z/8Z for k =Q(which turns out to be the minimal counterexample to Noether’s problem) or G= Q8. If G=Sn, then H1(, G) classify ´etale algebras

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rank n, so ed(Sn) is the minimal number of algebraic independent co- efficients in a minimal polynomial defining such an extension; therefore this question is related to the problem of reduction of polynomial equa- tions, which is an old problem interesting in itself, far from being solved.

Essential dimension is also related to one of the most famous questions in non commutative algebra: an old conjecture of Albert states that any central simplek-algebra of prime degreepcontains a commutative Galois subextension L/k of degree p. This conjecture is still open for p≥5. However, it is easy to show that if this conjecture is true, then edC(PGLp) = 2. Therefore, a proof of the inequality edC(PGLp) ≥3 would immediately disprove Albert’s conjecture. Finally, essential di- mension of finite groups is related to splitting properties of torsors, as well as to important arithmetic conjectures as we explain now. Let k be an algebrically closed field of characteristic zero. For any field extension K/k, denote by Kab (resp. Ksol) the maximal abelian (resp.

the maximal solvable) extension of K. Suppose that one could prove the following property: for any finite group S, any field extension K/k and any α ∈ H1(K, S), there exists an abelian (resp. solvable) field extensionL/K such that edkL)≤1. Then this would imply thatKab

(resp. Ksol) has cohomological dimension less or equal to one, which is a very difficult open conjecture. A direct consequence of this re- sult would be that for any connected group Gdefined over k, and any α ∈ H1(K, G), the class α can be split by an abelian (resp. solvable) extension of K; in particular, it would imply that Serre’s conjecture II is true for groups of type E8, which is a very difficult (and still open) case ).

In [2], jointly with G.Favi, we are interested in studying the basic properties of essential dimension and reconciliate the three different notions. This paper is divided in eight sections.

In Section 1, we study the behaviour of this notion under products, coproducts and field extension. Along the way, we define the notion of fibrations of functors.

In Section 2, we define the essential dimension of an algebraic group G defined over an arbitrary field k. They we give some examples of computation of this essential dimension, including the case of the circle group.

In Section 3, we are interested in group actions on algebraick-schemes.

Recall that an algebraic group Gacts generically freely on a k-scheme X if the scheme-theoretic stabilizer of any point of X is trivial.

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We then prove the following result:

Theorem 20. Let G be an algebraic group defined over k, acting lin- early and generically freely on the affine spaceA(V)(whereV is a finite dimensional k-vector space). Then we have

edk(G)≤dimV −dimG.

As an application, we show that the essential dimension of any algebraic group is finite. We also show that an ´etale group scheme acts linearly and generically freely on A(V) if and only if it acts faithfully onA(V).

We then apply the previous results to estimate the essential dimension of finite abelian groups and dihedral groups when the base field is large enough.

In Section 4, we introduce Merkurjev’s notion ofn-simple functors and apply it to give lower bounds of essential dimension of some algebraic groups (e.g. symmetric groups).

We then obtain the following result, which gives a supplementary in- terest to the study of cohomological invariants:

Proposition 21. If F has a non trivial cohomological invariant of dimension n, then edk(F)≥n.

Recall that a cohomological invariant is said to be non-trivialif si for any field extension K/k there exists L ⊇ K and a ∈ F(L) such that ϕL(a)6= 0.

In Section 5, inspired by Rost’s definition of essential dimension for some subfunctors of Milnor’s K-theory, we define the notion of ver- sal pair for functors from the category of commutative and unital k- algebras to the category of sets. We then define a new kind of essential dimension of functors having a versal pair, that we call Rost’s essential dimension, and compare it to Merkurjev’s essential dimension.

More precisely, let k be any field and let Ak be the category of unital commutative k-algebras.

Let K/k be a field extension. For a local k-subalgebra O of K, with maximal ideal m, we will write κ(O) = O/m for its residue field and π :O →κ(O) for the quotient map.

LetK andLbe two extensions ofk. Apseudok-placef : K Lis a pair (Of, αf) whereOf is a localk-subalgebra ofKandαf :κ(Of)→L is a morphism in Ck.

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LetF:Ak →Sets be a functor and take f :K La pseudo k-place.

We say that an element a ∈ F(K) is unramified in f if a belongs to the image of the map F(Of)→F(K). In this case we define the set of specializations of a to be

f(a) =

F(αf ◦π)(c)| c∈F(Of) with cK =a .

We say that a pair (a, K) with a ∈ F(K) is a versal pair for F (over k) if for every extension L/k and every element b ∈ F(L) there exists a pseudo k-place f : K L such that a is unramified in f and such that b ∈f(a).

LetF:Ak →Sets be a functor which has a versal pair. We define its (Rost’s) essential dimension (denoted by ed(F)) to be the minimum of the transcendence degree of the field of definition for versal pairs.

More precisely ed(F) = min trdeg(K : k) for all K/k such that there exists an element a∈F(K) making (a, K) into a versal pair for F.

We say that a versal pair (a, K) isniceif for anyL⊂K anda ∈F(L) such that a=aK, the pair (a, L) is versal. We say that F is niceif it has a nice versal pair.

We then have:

Proposition 22. Let F :Ak → Sets be a functor which has a versal pair. Then we have

edk(F)≤edk(F)

where on the left F is viewed as a functor on Ck. Moreover, if F is nice, then

edk(F) = edk(F) = ed(a), where (a, K) is any nice versal pair.

In Section 6, we establish the relation between the essential dimension of an algebraic group G and those of a generic G-torsor.

Let G be an algebraic group over k, K a field extension of k and P →Spec(K) a G-torsor. We say that P is k-generic if

i) there exists an irreducible scheme Y (whose generic point is denoted byη) with function fieldk(Y)≃K (such a scheme is called a model of K) and a G-torsor f : X →Y whose generic fiber f−1(η)→Spec(K) is isomorphic to P →Spec(K). In other words

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P //

X

Spec(K) //Y is a pull-back.

ii) For every extension k/k with k infinite, for every non-empty open set U of Y and for every G-torsor P → Spec(k), there exists a k- rational point x∈U such thatf−1(x)≃P.

If G is an algebraic group, we denote by G−Tors : Ak → Sets the functor ofG-torsors. The restriction of this functor toCk is isomorphic to the functor H1(, G). We denote by edk(G) the Rost’s essential dimension of this functor.

We then show:

Theorem 23. Let G be an algebraic group defined over k, and let T ∈H1(K, G) be a generic G-torsor. Then(T, K)is a nice versal pair for the functor G-torsor In particular, we have

edk(G) = edk(G) = edk(T).

We also relate Reichstein’s essential dimension and Merkurjev’s essen- tial dimension.

Letf :X →Y and f :X →Y be two G-torsors.

We say that f is a compressionof f if there is a diagram X

f

g _// _

_ X

f

Y _ _h _//Y

where g is a G-equivariant rational dominant morphism and h is a rational morphism too. Theessential dimension of a G-torsor f is the smallest dimension of Y in a compression f off. We still denote this by ed(f).

The Reichstein’s essential dimension of G is then the maximum of ed(f) when f ranges over all G-torsors. We then have:

Proposition 24. Let f :X →Y be a G-torsor with Y irreducible and reduced. LetT →Spec(k(Y))be its generic fiber. Thened(f) = ed(T).

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This proposition, combined with the previous theorem, shows that the three notions of essential dimension coincide.

In Section 7, we focus on essential dimension of finite constant group schemes. First of all, we prove that the essential dimension of such a groupGis the minimum of the trdeg(E :F) for all the fieldsE ⊆k(V) on which G acts faithfully, which is Reichstein’s original definition of essential dimension of finite groups. We then apply these results to compute essential dimension of cyclic and dihedral groups over real numbers, and essential dimension of cyclic groups of order at most 6 over any base field.

Finally in Section 8, we prove the following homotopy invariance the- orem:

Theorem 25. LetF be an infinite field, and let Gbe a linear algebraic group defined over F. Then edF(G) = edF(t)(GF(t)).

• In [4], we compute the essential dimension of cubic forms in 3 vari- ables. We use for this some techniques and results developped in [2].

Let P, P be two homogeneous polynomials of degree d in n variables with coefficients in L. We say that P and P are equivalent if there exists λ ∈ L× and f ∈ GLn(L) such that P = λP ◦f. The functor of equivalence classes is denoted by Fd,n. If d = 3, we denote this last functor byCubn (the functor of cubic forms).

Problem: Compute edk([P]).

This question is motivated by the following: it is easy to see that if ϕ : F → G is a natural map, then edk(a)) ≥ edkK(a)) for any a ∈ F(K). Since it often happens that one can associate functorially a homogeneous form to classical algebraic structures (the trace form, the norm form...), it is fundamental to want to learn more about the essential dimension of such forms.

In [4], we have proved the following:

Theorem 26. Let k be any field.

1. If char(k)6= 3, then edk(Cub2) = 1.

2. If char(k)6= 2,3, then edk(Cub3) = 3.

The proof of this result is based on the following geometric idea: in- formally speaking, defining a non-singular cubic in three variables over a field L up to projective equivalence is equivalent to specifying (i) a configuration of nine (unordered) flex points in P2(Ls) (where Ls is a

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separable closure ofL) and (ii) a value of thej-invariant. We show that these two choices are independent and that (i) requires two parameters.

For this, we compute the essential dimension of the functor of cubics with prescribedj-invariant and of the functor of cubics with prescribed flex points. The advantage of this approach is that these two functors can be described as Galois cohomology functors of some suitable alge- braic groups; we then use crucially generically free representations of these groups to compute their essential dimensions.

7.3.4 Canonical dimension. In [5], in collaboration with Zinovy Reinchstein, we defined the notion of canonical dimension of a G- variety X defined over an algebraically closed field k of characteris- tic zero (where G is an algebraic group defined over k) as follows: we say that F : X 99K X is a canonical form map of X if there exists f :X 99KG such that F(x) = f(x)·x. Then we set

cd(X, G) = sup

F

F(X)−dimX/G,

whereF ranges over all canonical form maps, and whereX/Gdenotes a rational quotient of X byG.

It turns out that the notion of canonical dimension gives a geometric link between the theory of generic splitting of central simple algebras and essential dimension of generic homogeneous forms; the essential dimension of the generic form can then be computed for example when gcd(n, d) is a prime power, using a degree formula proved by Merkurjev and its consequences on incompressibility of varieties.

We now describe more precisely the results obtained in [5]:

In [5], we relate this invariant to the essential dimension of the functor OrbX,G defined by OrbX,G(L) := X(L)/G(L). In fact, we show the following result:

Proposition 27. Let η∈X(k(X))the generic point of X, and let [η]

be its class in OrbX,G(k(X)). Then we have:

edk([η]) = cd(X, G) + dimX/G.

We also study the basic properties of the canonical dimension of a G-variety X.

LetS be an algebraic group andY be a generically free S-variety. We define e(Y, S) as the smallest integer e with the following property:

given a point y ∈ Y in general position, there is an S-equivariant rational map f: Y 99K Y such that f(Y) contains y and dim f(Y) ≤ e+ dim(S).

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We then have the following result:

Proposition 28. Let X be a G-variety having stabilizer S in general position, and let Ns be the normalizer of S in G. Then we have

e(G, S)≤cd(X, G)≤e(G, NS)−dim(S) + dim(NS).

We also introduce the notion of split G-variety. Let X be a G-variety having stabilizer S in general position. We say that X is split if it is birationally isomorphic as a G-variety to G/S ×X/G (where G acts by left translations on the first factor and trvially on the second one).

If X is generically free (that is, G acts generically freely on X), we recover the classical notion of splitG-torsor.

We then prove the following result:

Proposition 29. Let X be a G-variety having stabilizer S in general position. If X is a split G-variety then we have

cd(X, G) = e(G, S).

Notice that we always have the inequality e(G, S) ≥ ed(S), and that equality holds if G is special.

When X is generically free, that is when we have a G-torsor defined on Spec(k(X)G), we relate the canonical dimension of X to the tran- scendence degrees of generic splitting fields of this torsor as follows:

Theorem 30. Let G be a connected algebraic group, X be an irre- ducible generically free G-variety, E =k(X)G = k(X/G), α = [X] be the class represented byX inH1(E, G)andF: X 99KX be a canonical form map of X. Then

(1) The extension k(F(X))/E is a generic splitting field extension for α.

(2) cd(X, G) = min{trdegE(K) | K/E is a gneric splitting field extension for α}

Here the term ”generic” relative to the property of single specialization as well as to the property of rational specialization.

If X is a generically free linear representation V of G, we show that cd(V, G) does not depend on the choice of V. We call it thecanonical dimension of G, and denote it by cd(G).

We then have the following properties:

Lemma 31. (1) cd(G) = max cd(X, G), where X ranges over all generically free G-varieties.

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(2) cd(G) = cd(G0).

(3) If G is connected, cd(G) = 0 ⇐⇒ G is special.

We then relate the essential dimension of generic homogeneous forms of degree d inn variables and the canonical dimension of the algebraic group GLnd as follows:

Proposition 32. Let k be an algebraically closed field of characteristic 0. Let P0 = P

ItIXI be the generic homogeneous polynomial of degree d in n variables , and let [P0] be its equivalence class. If d ≥ 3 and (n, d)6= (2,3), (2,4) or (3,3) then

ed([P0]) =

n+d−1 d

−n2+ cd(GLnd).

This leads in particular to the following result:

Theorem 33. Assume thatgcd(n, d) =pm, m≥0, wherepis a prime.

Then

ed([P0]) =

n+d−1

d

−n2 if m= 0

n+d−1 d

−n2 +pυp(n)−1 otherwise

Finally, the interpretation of canonical dimension in terms of generic splitting fields allows us to give some examples of groups of canonical dimension at least 2 or 3, and to classify simple groups of canonical dimension 1.

We then obtain the following result:

Theorem 34. Let k be an algebraically closed field of characteristic 0, and let G be a simple group defined over k. Then we have

cd(G) = 1 ⇐⇒ G≃SO3,SO4,SL2m2 or PGSp2m, modd.

1.4. Central division algebras and wireless communication. As pointed out before, codes with good performances may be constructed on cyclic division algebras D= (a, L/K, σ). In our context K =Q(i) or Q(j), and L is stable under the complex conjugation. In this case, the codebook will be constructed as

C ={ϕ(x0+x1e+. . .+xn−1en−1)|xi ∈A},

where A is a fractional ideal of L and ϕ : D ֒→ Mn(L) is the natural injection.

In this setting, it can be shown that maximizing min

X∈C|det(X)|is equiva- lent to minimize|dL/K|. The code also has to satisfied extra conditions

(30)

to be efficient and applicable from an industrial point of view. In par- ticular, encoding must not increase the energy cost, and the code must be information lossless. If our code C satisfies these conditions, we say that C is perfect.

The condition of energy cost may be translated in mathematical terms as follows. We may findλ ∈L∩R+ such that the lattice

A×A→Z,(x, y)7→T rL/K(λxy)¯ is isomorphic to OKn.

In [22], we show that perfect codes only exists for n = 2,3,4 or 6.

In [21], we give the construction of a code for n = 4, which is not perfect but has better performances that the best perfect code known, by relaxing the ”division algebra condition”.

Figure

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