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WHATIS... ? Essential ? Zinovy Reichstein

Informally speaking, the essential dimension of Here, as usual, K-linear transformations are con- an algebraic object is the minimal number of sidered equivalent if their matrices are conjugate independent parameters one needs to describe over K. If T is represented by an n×n (aij ), it. This notion was introduced in [1], where the then T descends to K0 = k(aij | i, j = 1, . . . , n), so objects in question were field extensions of finite that a priori ed(T ) n2. However, this is not op- degree. The general definition below is due to timal; we can specify T more economically by its A. Merkurjev. rational canonical form R. Recall that R is a block- diagonal matrix diag(R1,...,Rm), where each Ri Essential Dimension of a Functor is a companion matrix. If m = 1 and R = R1 = 0 ... 0 c1 Fix a base field k and let F be a covariant functor 1 ... 0 c2  . . from the category of field extensions K/k to the . . , then T descends to k(c1, . . . , cn)  . .  category of sets. We think of F as specifying 0 ... 1 cn the type of algebraic object under consideration and thus ed(T ) n. A similar argument shows and F(K) as the set of algebraic objects of this that ed(T ) n for any m. type defined over K. For a field extension K/K0, Example 3. Let F(K) be the set of the natural (“base change”) map F(K0) → F(K) classes of elliptic curves defined over K. Every el- allows us to view an object defined over K0 as liptic curve X over K is isomorphic to the plane also being defined over the larger field K. Any curve cut out by a Weierstrass equation y 2 = x3 + object α ∈ F(K) in the image of this map is said ax + b, for some a, b ∈ K. Hence, X descends to to descend to K0. The essential dimension ed(α) is K0 = k(a, b) and ed(X) 2. defined as the minimal of In many instances one is interested in the “worst K /k, where α descends to K . 0 0 case scenario”, i.e., in the number of independent For simplicity we will assume that char(k) = 0 parameters that may be required to describe the from now on. Much of what follows remains true “most complicated” objects of a particular kind. in prime characteristic (with some modifications). With this in mind, we define the essential dimen- Example 1. Let F(K) be the set of isomorphism sion ed(F) of the functor F as the supremum of classes of nondegenerate n-dimensional quadratic ed(α) taken over all α ∈ F(K) and all K. We have forms defined over a field K. Every quadratic shown that ed(F) n in Examples 1 and 2 and form over K can be diagonalized. That is, q is K- ed(F) 2 in Example 3. We will later see that, in isomorphic to the quadratic form (x1, . . . , xn) ֏ fact, ed(F) = n in Example 1. One can also show 2 2 ∗ that ed(F) = n in Example 2 and ed(F) = 2 in a1x1 + ... + anxn for some a1, . . . , an ∈ K . Hence, q descends to K0 = k(a1, . . . , an) and ed(q) n. Example 3.

Example 2. Let F(K) be the set of equivalence Essential Dimension of an Algebraic classes of K-linear transformations T : Kn → Kn. Of particular interest are the Galois cohomology 1 Zinovy Reichstein is professor of mathematics at the Uni- functors FG given by K ֏ H (K, G), where G is versity of British Columbia. His email address is reichst@ an algebraic group over k. Here, H1(K, G) denotes math.ubc.ca. the set of isomorphism classes of G-torsors (oth- DOI: http://dx.doi.org/10.1090/noti909 erwise known as principal homogeneous spaces)

1432 Notices of the AMS Volume 59, Number 10 over Spec(K). For many groups G this functor is cyclic. Suppose that the base field k contains parametrizes interesting algebraic objects. For a primitive pth root of unity. Then the essential 1 example, H (K, On) is the set of isomorphism dimension of a cyclic is easily seen classes of n-dimensional quadratic forms over K to be 2. Thus ed(PGLp) 2, and if this inequality

(in other words, FOn is the functor of Example 1), happens to be strict for some p, then Albert’s 1 H (K, PGLn) is the set of isomorphism classes conjecture fails. The value of ed(PGLp) is 2 for of central simple algebras of degree n over K, p = 2 or 3 and is unknown for any other prime. 1 H (K, G2) is the set of isomorphism classes of The problem of computing ed(PGLn) first arose algebras over K, etc. (On the other hand, in C. Procesi’s pioneering work on universal the functors F in Examples 2 and 3 are not of the division algebras in the 1960s. The inequality 2 form FG for any algebraic group G.) The essential ed(PGLn) n proved by Procesi has since been dimension of FG is called the essential dimension strengthened (see [3]), but the new upper bounds of G and is denoted by ed(G). are still quadratic in n. In the 1990s B. Kahn Algebraic groups of essential dimension zero asked if ed(PGLn) grows sublinearly in n, i.e., if are precisely the special groups, introduced by there exists a C > 0 such that ed(PGLn) Cn J-P. Serre in the 1950s. An algebraic group G over for every n. By the primary decomposition the- k is called special if H1(K, G) = {1} for every orem we lose little if we assume that n is a field K/k. For example, SLn and Sp2n are special prime power. Until recently, the best known lower for every n. Over an algebraically closed field of bound was ed(PGLpr ) 2r. This has been dramat- characteristic zero, special groups were classified ically improved by A. Merkurjev, who showed that r by A. Grothendieck. The essential dimension ed(G) ed(PGLpr ) (r − 1)p + 1 for any prime p and may be viewed as a numerical measure of how any r 2. In particular, this inequality answers much G differs from being special. Kahn’s question in the negative. Surprisingly, es- sential dimension is not a Brauer invariant; there Symmetric Groups are central simple algebras A of degree 4 such

Computing ed(Sn) is closely linked to the classi- that ed(M2(A)) < ed(A). Little is known about cal problem of simplifying polynomials of degree essential dimension of Brauer classes. n in one variable by a Tschirnhaus transforma- tion and may be viewed as an algebraic variant of Cohomological Invariants Hilbert’s 13thproblem [1]. In his 1884“Lectures on Let G be an algebraic group over k. A cohomo- the Icosahedron”, F. Klein classified faithful finite logical invariant of G of degree n is a morphism 1 n n group actions on the projective line P and used of functors FG → H , where H (K) is the nth this classification to show that, in our terminology, Galois cohomology group. The coefficient mod- ed(S5) = 2. More generally, J. Buhler and I showed ule can be arbitrary; in the examples below it n (by a different method) that ed(Sn) ⌊n/2⌋ for will always be Z/2Z. (Observe that H (K) is a 1 any n, and ed(Sn) n − 3 for n 5; see [1]. group, whereas, in general, FG(K) = H (K, G) Using these inequalities one easily finds ed(Sn) has no group structure.) Serre noted that if k is for n 6. For larger n the only additional bit of algebraically closed and G admits a nontrivial co- insight we have is via birational classifications of fi- homological invariant of degree n, then ed(G) n. nite group actions on low-dimensional unirational Letting G be the On, so that FG is varieties (over an algebraically closed field), ex- the functor considered in Example 1, and applying tending Klein’s original approach. In dimension 2 the above inequality to the cohomological invari- n this yields ed(A6) = 3 (due to Serre) and in dimen- ant FG → H (K) which takes a quadratic form 2 2 sion 3, ed(A7) = ed(S7) = 4 (due to A. Duncan). q = a1x1 + + anxn to its nth Stiefel-Whitney n Serre’s argument is based on the Enriques-Manin- class (a1) (an) ∈ H (K), we obtain ed(On) n Iskovskikh classification of rational G-surfaces, and thus ed(On) = n (we showed that ed(On) n and Duncan’s is based on the recent work on ra- in Example 1). Another interesting example, also tionally connected G-threefolds by Yu. Prokhorov. due to Serre, is the degree 5 invariant of the excep- In higher this approach appears to be tional group F4, which gives rise to the inequality beyond the reach of Mori theory, at least for now. ed(F4) 5. The exact value of ed(Sn) remains open for every n 8. Nontoral Finite Abelian Subgroups Nontoral finite abelian subgroups first arose Projective Linear Groups in the foundational work of A. Borel on the The value of ed(PGLn) is intimately connected cohomology of classifying spaces for compact with the theory of central simple algebras. An Lie groups. Nontoral finite abelian subgroups of important open conjecture, due to A. A. Albert, algebraic groups were subsequently studied by is that every division algebra of prime degree p T. Springer, R. Steinberg, R. Griess, and many

November 2012 Notices of the AMS 1433 others. In a special group such as SLn, every finite THE HONG KONG UNIVERSITY OF abelian subgroup is contained in a torus. Ph. Gille, SCIENCE AND TECHNOLOGY B. Youssin, and I generalized this as follows: if A ⊂ G is a finite abelian subgroup, then ed(G) 0 (A) − rank(ZG(A) ). Here G is connected Department of Mathematics and reductive, rank(A) is the minimal number 0 of generators of A, ZG(A) is the connected Faculty (s) component of the centralizer of A in G, and the The Department of Mathematics invites applications for tenure-track rank of Z (A)0 is the dimension of its maximal faculty position(s) at the rank of Assistant Professor in all areas of G mathematics. Other things being equal, preference will be given to torus. (Note that the above inequality is of interest areas consistent with the Department’s strategic planning. only if A is nontoral; otherwise the right-hand side Applicants should have a PhD degree and strong experience in research is 0.) For example, the exceptional group G = E and teaching. Applicants with exceptionally strong qualifi cations and 8 9 experience may be considered for position(s) above the Assistant has a self-centralizing subgroup A ≃ (Z/2Z) ; Professor rank. hence ed(E8) 9. I am not aware of any other proof Starting rank and salary will depend on qualifi cations and experience. of the last inequality. In particular, no nontrivial Fringe benefi ts include medical/dental benefi ts and annual leave. Housing will also be provided where applicable. Initial appointment will cohomological invariants of E8 of degree 9 are be on a three-year contract, renewable subject to mutual agreement. known. A gratuity will be payable upon successful completion of the contract. Applications received on or before 31 December 2012 will be given full consideration for appointment in 2013. Applications received Spinor Groups afterwards will be considered subject to the availability of position(s). Applicants should send their curriculum vitae together with at least Over the past few years there has been rapid three research references and one teaching reference to the Human progress in the study of essential dimension, Resources Offi ce, HKUST, Clear Water Bay, Kowloon, Hong Kong. Applicants for position(s) above the Assistant Professor rank should based on an infusion of methods from the theories send curriculum vitae and the names of at least three research referees of algebraic stacks and algebraic cycles. These to the Human Resources Offi ce. developments are beyond the scope of the present More information about the University is available on the University’s homepage at http://www.ust.hk. note; please see the surveys [2, 3] for an overview (Information provided by applicants will be used for recruitment and other employment-related purposes.) and further references. I will, however, mention one unexpected result that has come up in my joint work with P. Brosnan and A. Vistoli: it turns out that the essential dimension of the spinor group Spinn increases exponentially with n. This AMERICAN MATHEMATICAL SOCIETY has led to surprising consequences in the theory

of quadratic forms. Note that FSpinn is closely related to n-dimensional quadratic forms with trivial discriminant and Hasse-Witt invariant, very much in the spirit of Example 1, and that there are no high rank finite abelian subgroups in Spinn to account for the exponentially high value of ed(Spinn). Are there high-degree cohomological invariants of Spinn? We do not know.

Further Reading [1] J. Buhler and Z. Reichstein, On the essential dimen- sion of a finite group, Compositio Math. 106 (1997), no. 2, 159–179. TheThhe AMSA o ers over 2,500 pure [2] A. Merkurjev, Essential dimension, in Quadratic Forms—Algebra, Arithmetic, and Geometry (R. Baeza, and applied mathematics titles W. K. Chan, D. W. Hoffmann, and R. Schulze-Pillot, in eBook form via Google Play. eds.), Contemporary Math., vol. 493, Amer. Math. Soc., Providence, RI, 2009, pp. 299–326. [3] Z. Reichstein, Essential dimension, Proceedings of Your favorite texts (even the the International Congress of Mathematicians, Vol- outofprint ones!) are now fully ume II, Hindustan Book Agency, New Delhi, 2010, pp. searchable and easily readable 162–188. on webbrowsing devices.

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1434 Notices of the AMS Volume 59, Number 10