09w5026 Linear Algebraic Groups and Related Structures

Arriving Sunday, September 13 and departing Friday, September 18, 2009

Organizers: Vladimir Chernousov (University of Alberta), Alexander Merkurjev (University of California Los Angeles), Ján Minác (University of Western Ontario), Zinovy Reichstein (University of British Columbia).

Confirmed Participants

Press Release: Linear Algebraic Groups and Related Structures

Information for Participants

Schedule and Abstracts (PDF file)

Mailing List

Final Report (PDF file)

Objectives

The purpose of this workshop is to provide a forum for the experts in the field of algebraic groups and related areas, to exchange ideas, disseminate new techniques and discuss recent developments. The workshop will also be an opportunity for younger researchers to learn about open problems and state of the art techniques in this field.

We will now list a number of specific important recent developments in this area. We expect these topics to be at the center of the scientific activities at the workshop. Note that the field of algebraic groups has been experiencing rapid progress: many of the results mentioned below have been obtained in the past year!

1. The Bloch-Kato conjecture. In 1970 J. Milnor defined his $K$-theory $K_*(F)$ of fields and asked if the ring $K_*(F)/2K_*(F)$ is isomorphic to both the Galois cohomology ring $H^*(F, mathbb{F}_2)$ and the graded Witt ring of quadratic forms. This question, which became known as Milnor's conjecture, remained the preeminent open problem in the algebraic theory of quadratic forms for over 30 years. After the initial breakthrough by Merkurjev, who established this conjecture for degree $2$, several other special cases were settled in the 1980s, but the general case remained open until a complete solution was given by Voevodsky around 2000. For this achievement Voevodsky received a Fields medal in 2002. In 2003 Voevodsky announced a proof of the more general Bloch-Kato conjecture. The Bloch-Kato conjecture asks whether the natural map $K_*(F)/ pK_*(F) to H^*_{text{et}}(F, {mu_p^{otimes}}^{*})$ is an isomorphism for every prime $p$, where $F$ is a field of characteristic $ne p$. The case where $p = 2$ is covered by Milnor's conjecture; the others cases required more delicate arguments based on motivic cohomological operations and a construction of norm varieties due to M. Rost. The details are quite complicated, and some of them have not been worked out until very recently. This month (September 2007) all the details of the proof have finally become available on the $K$-theory archive and on the homepages of M. Rost and C. Weibel. The proof of the Bloch-Kato conjecture is one of the major achievements of contemporary mathematics. This result and the techniques used in its proof have already had numerous applications in the fields of number theory, quadratic form theory, and Galois theory. They are expected to have an even broader impact in the coming years.

2. Serre Conjecture II. J.-P. Serre conjectured that $H^1(K, G) =0$ for any semisimple, simply connected algebraic groups $G$ over a perfect field $k$ of cohomological dimension two. E. Bayer-Fl"uckiger and R. Parimala proved this conjecture for the classical groups; further results for groups of exceptional type were obtained by V. Chernousov and Ph. Gille. This year de Jong, He and Starr announced a proof of Serre's Conjecture II in the case, where $K$ is the field of an algebraic surface. For such $K$ the conjecture was previously known to be true for every almost simple group, except for those of type $E_8$; the case of $E_8$ remained open for many years. The result of de Jong, He and Starr represents major progress on this long-standing open problem. The techniques used in their proof have not yet been fully absorbed by the experts; it is hoped that they will have other applications.

3. The $u$-invariant. In 1953 Kaplansky conjectured that the maximal dimension of an anisotropic quadratic form $q$ over given field $F$ (called the $u$-invariant $u(F)$) has to be a power of $2$ (of $infty$). In 1991 Merkurjev showed that for every positive integer $n$ there exists a field $F$ such that $u(F) = 2n$. This construction led to the study of index-reduction formulas for division algebras and algebraic groups. The question of whether $u(F)$ can be odd (and $> 1$) remained open, until Izhboldin constructed a field $F$ of $u$-invariant $9$. (This result appeared in print in 2001.) During the 2006 BIRS workshop on Algebraic groups, quadratic forms and related topics, A. Vishik announced a uniform construction of fields with all known $u$-invariants and a construction of fields with $u$-invariants $2^r + 1$ for every $r ge 3$. Prior to his work it was not known whether the $u$-invariant could assume any odd value $> 9$. Vishik's main tools are discrete invariants of quadrics and a subtle analysis of the behavior of Chow rings under field extensions. Also very recently R. Parimala and V. Suresh succeeded in showing that the $u$-invariant of function fields of $p$-adic curves for $p > 2$ is $8$. This answers a long-standing question in quadratic from theory. The work of Parimala and Suresh is based on recent results of D. Saltman on divisors on arithmetical surfaces and cyclic algebras over $p$-adic curves.

4. Essential dimension. The notion of essential dimension was introduced by J. Buhler and Z. Reichstein in the late 1990s and subsequently refined by A. Merkurjev, G. Berhuy and G. Favi. Informally speaking, the essential dimension of an algebraic object is the minimal number of independent parameters required to define its structure. Here an "object" can be almost any algebraic structure one encounters: a polynomial, a finite-dimensional algebra, an algebraic variety, a morphism between algebraic varieties, etc. Now suppose the objects under considertaion are $G$-torsors over a field. The maximum possible value of the essential dimension over all such torsors (for a fixed algebraic group $G$) is called the essential dimension of $G$. This numbrical invariant $G$ arises in many natural settings, in particular, in connection with several classical problems in algebra. In the past 10 years it has been studied by many mathematicians, using a wide range of techniques.

There has been dramatic progress in this area over the past year. Preprints by Florence, Brosnan-Reichstein-Vistoli and Karpenko-Merkurjev have significantly improved many of the best previously known bounds on the essential dimension of algebraic groups. Florence computed exact values of the essential dimensions of all finite cyclic $p$-groups over fields containing a primitive $p$th root of unity, Brosnan-Reichstein-Vistoli found exponential (in $n$) lower bounds on the essential dimension of $operatorname{Spin}_n$, and Karpenko-Merkurjev computed the exact value of the essential dimension of any finite $p$-group over a field containing a primitive $p$th root of unity. Most of these developments resulted from the introduction of stack-theoretic techniques, originally developed by P. Deligne, D. Mumford and M. Artin, in connection with constructing moduli spaces in algebraic geometry.

5. Canonical dimension. The canonical dimension is another numerical invariant of an algebraic object. Here, instead of the minimal number of independent parameters required to define the object, we are interested in the minimal number of independent parameters required to split it, i.e., in the minimal possible transcendence degree of a generic splitting field. The notion of canonical dimension was introduced by Berhuy and Reichstein; their paper also included an application to the classical problem of finding the smallest field of definition for a generic hypersurface. In a subsequent paper Karpenko and Merkurjev refined this notion and introduced a useful variant, called canonical dimension at a prime $p$. This quantity is a lower bound on the canonical dimension and has the advantage of being more readily accessible via Chow group techniques. Karpenko and Merkurjev outlined a general approach to computing the canonical $p$-dimension and implemented it for every simple classical group and every prime $p$. For the remaining (exceptional) groups their program was recently completed by Zainoulline (his paper came out in 2007). The latest developments in this area include computations of the canonical dimensions of some spinor groups by Karpenko and of the canonical dimension of the group $operatorname{PGL}_6$ by Colliot-Th'el`ene, Karpenko and Merkurjev ($operatorname{PGL}_6$ is the first known case where the canonical dimension and the canonical $p$-dimension diverge.)

6. $K$-theory and Galois theory. As an application of the Bloch-Kato conjecture, recently proved by M. Rost and V. Voevodsky (see above), there is a now a complete description of the $K$-theory of the integers, modulo Vandiver's conjecture; see C. Weibel's homepage and the $K$-theory archive. In another recent development D. Benson, N. Lemire, J. Minac and J. Swallow found new restrictions on pro-$p$ groups which are absolute Galois groups. In particular, they have completely characterized certain small quotients of absolute Galois pro-$p$ groups which they call $T$-groups. In yet another development, F. Pop completed a program initiated by F. Bogomolov in the early 90s about recovering the field structure from the structure of pro-$p$ quotients of the absolute Galois groups.

7. Infinite dimensional Lie algebras. Recently, P. Gille and A. Pianzola discovered an interesting connection between the non-abelian Galois cohomology of Laurent polynomial rings and extended affine Lie algebras. Informally speaking, affine Lie algebras can be thought off as higher nullity analogues of the affine Kac-Moody Lie algebras. Though the algebras in question are in general infinite dimensional over the base field (say the complex numbers), they can be thought as being finite over the centroid of the algebra, which turns out to be a Laurent polynomial ring. In this setting the tools of the theory of reductive group schemes developed by M. Demazure and A. Grothendieck can be applied. % Once this point of view is taken, the language of torsors % arises naturally. This novel geometric approach has lead to unexpected interplays between infinite dimensional Lie algebras and several important topics in the theory of algebraic groups, such as the work of Raghunathan and Ramanathan on torsors over the affine space, isotriviality questions for Laurent polynomial rings, Azumaya algebras, and Serre's Conjectures I and II.

8. Grothendieck's purity conjecture. J.-P. Serre and A. Grothendieck conjectured that for a reductive group $G$ defined over a local regular ring $R$ the natural mapping $H^1(R,G)to H^1(K,G)$ where $K$ is the field of fractions of $R$ has trivial kernel. The image of this map is conjectured to consist of unramified classes. There has been significant progress towards proving this conjecture (known as the purity conjecture) in the last few years. It has been proven for the groups of the form $SL(1,A)$ where $A$ is an Azumaya algebra over $R$ (Colliot-Th'el,ene and Ojanguren), for many groups of classical types $B_n,C_n,D_n$ (Panin, Ojanguren) and for groups of type $G_2$ (Chernousov--Panin).