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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 16 Выпуск 3 (2015)
UDK 511
EXTREMAL FORMS AND RIGIDITY IN ARITHMETIC GEOMETRY AND IN DYNAMICS
N. M. Glazunov (Kiev)
glanm@yahoo.com
Abstract
Ryshkov S. S. in his papers has investigated extremal forms and extremal lattices. Extremal forms and lattices are connected with hard or rigid (by M. Gromov and other) objects in mathematics. In their work with colleagues S. S. Ryshkov came also to the other hard (or rigid) objects, for instance, to rigidly connected chain.
Rigid and soft methods and results already evident in the study of the classical problems in number theory. Let us dwell briefly on the interpretation in terms of hard and soft methods of binary and ternary Goldbach problems.
Since the binary (respectively ternary) Goldbach problems in their present formulation there are about equalities of the type 2n = p1 + p2 (respectively 2n + 1 = p1 + p2 + p3), where n is a natural number greater than 1 (respectively n is a natural number greater than 2), p1,p2,p3 prime numbers, then these are hard (rigid) problems; the results of their studies are also hard.
However, the methods of their study include both rigid methods — the exact formula of the method of Hardy — Littlewood — Ramanujan and a combination of hard and soft methods under the investigation by the Vinogradov‘s method of trigonometric sums.
A number of problems of analytic number theory allow dynamic interpretation. We note in this regard that on connection of methods of analytic number theory and the theory of dynamical systems paid attention and has developed such analogies A. G. Postnikov.
The purpose of the paper is not to provide any sort of comprehensive introduction to rigidity in arithmetic and dynamics. Rather, we attempt to convey elementary methods, results and some main ideas of the theory, with a survey of some new results. We do not explore an exhaustive list of possible topics, nor do we go into details in proofs.
After giving an elementary number theoretic, algebraic and algebraic geometry introduction to rigid non-Archimedean spaces in the framework of local one dimensional complete regular rings, modules over rings, trees and formal schemes follow to I. R. Shafarevich, J.-P. Serre, J. Tate, D. Mumford, we review some novel results and methods on rigidity.
EXTREMAL FORMS AND RIGIDITY IN ARITHMETIC GEOMETRY... 125
These include (but not exhaust) methods and results by H. Furstenberg,
G. A. Margulis, G. D. Mostow, R. Zimmer, J. Bourgain, A. Furman, A. Linden-strauss, S. Mozes, J. James, T. Koberda, K. Lindsey, C. Silva, P. Speh, A. Io-ana, K. Kedlaya, J. Tuitman, and other.
I am grateful to V. M. Buchstaber for helpful comments during the discussion of my talk.
I thank the anonymous referee for his comments relatively the content and style of presentation and for providing suggestions for improvements.
Special thanks to N. M. Dobrovolskii for help and support under the preparation of the article for publication.
Keywords: rigid analytic space; Bruhat-Tits tree; formal scheme; rigid action; cocycle superrigidity; uniformly rigid ergodic action; superrigid action;
Bibliography: 51 titles.
ЭКСТРЕМАЛЬНЫЕ ФОРМЫ И ЖЕСТКОСТЬ
В АРИФМЕТИЧЕСКОЙ ГЕОМЕТРИИ И В
ДИНАМИКЕ
Н. М. Глазунов (г. Киев, Украина)
glanm@yahoo.com
Аннотация
С. С. Рышков в своих работах исследовал экстремальные формы и экстремальные решетки. Экстремальные формы и экстремальные решетки связаны с жесткими (в смысле М. Громова и других) математическими объектами. В своих работах, а также в работах с коллегами С. С. Рышков пришел и к другим жестким объектам.
Жесткие и мягкие задачи, методы и результаты проявляются уже при исследовании классических проблем теории чисел. Остановимся очень кратко на интерпретации с точки зрения жестких и мягких методов бинарной и тернарной проблем Гольдбаха, проблем гольдбахова типа и методов их исследования. Так как в бинарной (соответственно, тернарной) проблемах Гольдбаха в их современной постановке речь идет о равенствах типа 2n = pi + p2 (соответственно 2n + 1 = pi + p2 + p3), где n — натуральное число, большее 1 (соответственно n больше 2), pi,p2,p3 — простые числа, то в своей постановке это жесткие проблемы; результаты их исследования также являются жесткими.
Однако методы их исследования включают как жесткие методы — точная формула метода Харди — Литтлвуда — Рамануджана (Х-Л-Р), получаемая методами комплексного анализа, так и сочетание жестких и мягких (soft) методов исследования главного члена в форме Х-Л-Р и остаточного члена методом тригонометрических сумм Виноградова.
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Ряд задач аналитической теории чисел допускают динамическую интерпретацию. Отметим в связи с этим, что на связи методов аналитической теории чисел и теории динамических систем обращал внимание и развивал такие аналогии А. Г. Постников.
Целью предлагаемой работы не является исчерпывающее введение в жесткость в арифметике и в динамике. Скорее мы сделали попытку представить элементарные методы, результаты и некоторые основные идеи в этой области, вместе с обзором ряда новых результатов. Мы не даем исчерпывающего обзора возможных тем, а также не входим в детали доказательств.
После представления элементарного теоретико-числового, алгебраического и алгебро-геометрического введения в жесткие неархимедовы пространства на основе локальных одномерных полных регулярных колец, деревьев и формальных схем по И. Р. Шафаревичу, Ж.-П. Серру, Дж. Тэйту, Д. Мамфорду, мы даем обзор некоторых новых результатов и методов в направлении жесткости.
Изложение включает (но не исчерпывает) результаты и методы H. Fur-stenberg, G. A. Margulis, G. D. Mostow, R. Zimmer, J. Bourgain, A. Furman, A. Lindenstrauss, S. Mozes, J. James, T. Koberda, K. Lindsey, C. Silva, P. Speh, A. Ioana, K. Kedlaya, J. Tuitman, и других.
Я признателен В. М. Бухштаберу за полезные замечания в процессе обсуждения моего доклада.
Я благодарю рецензента за замечания относительно содержания и стиля изложения и за предложения по улучшению.
Особая признательность Н. М. Добровольскому за помощь и поддержку в процессе подготовки статьи к печати.
Ключевые слова: жесткое аналитическое пространство; дерево Брюа — Титса; формальная схема; жесткое действие; коцикленная супержесткость; равномерно жесткое эргодическое действие; супержесткое действие;
Библиография: 51 название.
1. Introduction
Ryshkov S. S. in his papers has investigated extremal forms and extremal lattices. Extremal forms and lattices are connected with hard or rigid (by M. Gromov and other) objects in mathematics. In their work with colleagues S. S. Ryshkov came also to the other hard (or rigid) objects, for instance, to rigidly connected chain.
Rigid and soft methods and results already evident in the study of the classical problems in number theory. Let us dwell briefly on the interpretation in terms of hard and soft methods of binary and ternary Goldbach problems[2, 3, 4, 5, 6]. Since the binary (respectively ternary) Goldbach problems in their present formulation there are about equalities of the type 2n = p\ + p2 (respectively 2n + 1 = p\ + p2 + p3), where n is a natural number greater than 1 (respectively n is a natural number
EXTREMAL FORMS AND RIGIDITY IN ARITHMETIC GEOMETRY... 127
greater than 2) pi,p2,p3 prime numbers, then these are hard (rigid) problems; the results of their studies are also hard. However, the methods of their study include both rigid methods — the exact formula of the method of Hardy — Littlewood — Ramanujan and a combination of hard and soft methods under the investigation by the Vinogradov‘s method of trigonometric sums.
A number of problems of analytic number theory allow dynamic interpretation. We note in this regard that on connection of methods of analytic number theory and the theory of dynamical systems paid attention and has developed such analogies
A. G. Postnikov [45]. An interesting approach to rigid and soft models is proposed by V. Arnold [8]. Special considerations need for application of the approach to problems of number theory and algebra of our paper.
The purpose of the paper is not to provide any sort of comprehensive introduction to rigidity in arithmetic and dynamics. Rather, we attempt to convey elementary methods, results and some main ideas of the theory, with a survey of some new results. We do not explore an exhaustive list of possible topics, nor do we go into details in proofs.
After giving an elementary number theoretic, algebraic and algebraic geometry introduction to rigid non-Archimedean spaces in the framework of local one dimensional complete regular rings, modules over rings, trees and formal schemes follow to I. R. Shafarevich, J.-P. Serre, J. Tate, D. Mumford, we review some novel results and methods on rigidity.
These include (but not exhaust) methods and results by S. S. Ryshkov [1], H. Furstenberg, G. A. Margulis, G. D. Mostow, M. Gromov, R. Zimmer, J. Bourgain,
A. Furman, A. Lindenstrauss, S. Mozes, J. James, T. Koberda, K. Lindsey, C. Silva, P. Speh, A. Ioana, K. Kedlaya, J. Tuitman, and other.
2. Quadratic modules over integers and sums of squares
Here we consider the partial case of quadratic modules [21] over integer numbers when the quadratic form is the sum of n squares. Let Л be a lattice [21] in та-dimensional real euclidean space that is defined by congruences. Davenport, Mordell, Cassels and others used the lattices and Minkowski‘s convex body theorem for proving results about existence of nontrivial solutions of some Diophantine equations.
We will give examples below. Recall the case of positive quadratic forms.
Let r be a complex number, Imr > 0, q = exp nir,
#з(т)
q
Ш=-Ж
the Jacobi function. Denote by Zn the h-dimensional integer lattice.
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Let rn(m) be the number of ways of writing m as a sum f (xf, ..., xn) = f of n squares. Put ©Zn = 93(r)n.
2.1. Sums of two squares
Let p = 1 (mod 4). In the case there is the integer l such that l2 + 1 = 0 (mod p). The lattice Л of pairs (a,b) of integer numbers is defined by congruences a = lb (mod p) and has determinant 9(Л) ^ p. From this and Minkowski's convex body theorem follow that every prime p = 1 (mod 4) is the sum of two squares.
Let у be the nontrivial Dirichlet character mod 4, integer m > 0. There is the well known
Proposition 1. The number of integer solutions of the equation xf + x2 = m is equal 4Ed]m x(d)-
In the framework of the function ©Zn we have
©Z2 = ^2 r2(m)qm.
m=0
2.2. Sums of three squares
In the case and in the case n = 4 it is possible to use quaternions (hermitions) but for simplicity we will formulate the well known result by ©Z3 and r3(m).
Proposition 2.
ж
©Z3 = ^2 ra(m)qm.
m=0
2.3. Sums of four squares
The quadratic form xf + xf + x3 + xf represents all positive numbers (Lagrange). The number of solutions of the equation xf + xf + x3 + x\ = m, where m is a positive integer is given by Jacobi.
Proposition 3. The number of integer solutions of the equation
2222 xf + x2 + x3 + x4 = m
is equal
8 V d
24 £d
d\m
if m = 2k + 1, and is equal
EXTREMAL FORMS AND RIGIDITY IN ARITHMETIC GEOMETRY... 129
if m = 2k.
In the framework of the function ©Zn we have
©Z4 = ^2 n(m)qm.
m=0
2.4. Sums of squares greater than four
Recall elements of Hardy-Littelwood-Kloosterman method in the case. This is valid also in the previous case n = 4. Consider a function of complex variable u, |u| < 1
Ж
uf
Xi^"Xn = — (X>
Then the number rn(m) of ways of writing m as a sum of n squares by Cauchy’s integral formula is given as
rn(m) = 2— £ i9(f,u)u-m-1du
where Г is the circle |u| = exp(—m). We omit here the very important step of the dividing Г into Farey-arcs.
4f,u)= V
3. Elements of history of rigidity
The history of rigidity is reflected in papers by A. Selberg, E. Calabi, E. Vesentini,
A. Weil, H. Furstenberg, G. Mostow, G. A. Margulis and their colleagues [22, 23, 24, 25, 26, 27]. There are interesting surveys by D. Fisher [28] and R. Spatzier [29]. Let G be a finitely generated group, D a topological group, and h : G ^ D a homomorphism. Follow to [28] recall
Definition 1. Given a homomorphism h : G ^ D, it is said that h is locally rigid if any other homomorphism h‘ which is close to h is conjugate to h by a small element of D.
Recall follow to [23, 24] in framework of [29] the Local Rigidity Theorem.
Theorem 1. Cocompact discrete subgroups H in semisimple Lie groups without compact nor SL(2, R) nor SL(2, C) local factors is deformation rigid.
The notion of uniform rigidity was introduced as a topological version of rigidity by S. Glasner and D. Maon [30].
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4. Rigid non-Archimedean spaces and Formal Groups
At first we formulate very briefly some elementary (and probably well known) results on connections among local one dimensional complete regular rings, trees and formal schemes. We follow to [9, 10, 52].
Let A be a local one dimensional complete regular ring with maximal ideal n, K its field of fractions with the multiplicative group K*, V a two dimensional vector space over K, M a module of the rank 2 over A (a two-dimensional lattice in the space V). Denote by S(M) the symmetric algebra of the module M. The main example is the case of the ring A = Zp of integer p— adic numbers, K = Qp the field of p— adic numbers, n = p the prime number, M a module of the rankZpM = 2 over Zp.
Definition 2. Let K be a locally compact non-Archimedean field, A its valuation ring, m the maximal ideal of A. A free module of rank n over A is called a lattice in K n.
Two modules M and M‘ of the rank 2 over A are called similar if M‘ = xM, x E K*. Denote by T the set of classes of similar modules.
Definition 3. Let X be the graph whose vertices are equivalence classes [M] of similar modules M of the rank 2 over A in V , where two vertices x and y are joint by an edge if x = [M] and y = [M‘] with M‘ C M, M‘ C nM, M/M‘ ~ A/nA.
Two modules are called adjacent if the length l(M/M‘) = 1, i.e. M/M‘ ~ A/nA.
Theorem 2. The graph X is a homogeneous or a regular tree. We will denote the tree by T.
By dT denote the set of ends of T and by P*(A) denote the one-dimensional projective space over A.
Theorem 3. dT ~ P:(A).
Recall that a group G acts on a set X if there is a map G x X ^ X, (g; x) gx such that the following are true: (i) For e the identity of G, ex = x; (ii) For h; g E G,x E X,h(gx) = (hg)x. On the space V act the projective linear group PGL2(K) and its subgroups. This action extends to the action on the tree T.
Theorem 4. Let a group G acts on a tree T without fixed points and without inversions. Then G is the free group.
Let a group G C PGL2(K) acts on T discretely and freely. Follow to Mumford it is possible to construct a subtree Tg of T.
Theorem 5. If the group G has finite number of generators then Tg/G is finite.
EXTREMAL FORMS AND RIGIDITY IN ARITHMETIC GEOMETRY... 131
For the above mentioned symmetric algebra S(M) of the module M define the corresponding scheme P(M) by the formula P(M) = Proj S(M). For each module M ^ V there is the birational isomorphism P(M) = P^A) K ——% P^. Now
let S be a finite subtree of T. It is possible to construct many formal schemes from these data. We indicate here the formal scheme P that is the formal completion (P(S)0) of the scheme P(S) along its closed fibre P(S)0 only. Recall that the generic fiber of P^A) is the one-dimensional projective space P^ over K.
5. Formal groups and formal stacks
Here we present results on two-dimensional commutative formal groups and on formal stacks
5.1. On two-dimensional commutative formal groups
Let F be a commutative formal group low of n variables over commutative ring R with unit. In the case n =1, following to the known results by M. Lazard, there is only one 1— bud of the form x + y + axy.
Proposition 4. Let n = 2, A = Zp[a,e] be the ring of polynomials with integer p-adic coefficients from a, в. 1— buds are
F(x, y)
Fa(x,y)
Fb(x,y)
f xi + yi + axiyi
[ x2 + y2 + ex2y2,
f xi + yi + axiyi
[ x2 + y2 + exiyi,
f xi + yi + ax2y2
[ x2 + y2 + ex2y2,
Fc (x,y)
{
xi + yi + a(xi + x2)(yi + y2)
x2 + y2 + в (xi + xf)(yi + y2),
Remark 1. 1— buds given in Proposition 1 are also two-dimensional formal group lows, whose coefficients under terms of degrees ^ 3 are zeros.
Remark 2. These group lows define classes of group lows. In particular, the class Fa contains under values of parameters a = 0,в = —1, the Witt group, that corresponds to prime number p = 2.
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5.2. Formal stacks
Let now the ring R is the field k. Recall, that formal k— scheme is formal k— functor, that is the limit of directed inductive system of finite k— schemes, and a formal group is a group object in the category of formal k— schemes. The notion of a stack, as one of category theory variants of moduli space is defined by P. Deligne and D. Mumford.
Proposition 5. There exist formal stacks, that are categories that are bundled on formal groupoids and that satisfy axioms of decent theory.
6. Uniformly rigid and measurable weak mixing
Authors of the paper [14] investigate properties of uniformly rigid transformations and analyze the compatibility of uniform rigidity and measurable weak mixing along with some of their asymptotic convergence properties.
This interesting survey includes some resent results on genericity of rigid and multiply recurrent infinite measure preserving and nonsingular transformations by
O. Ageev and C. Silva [31] and on measurable sensitivity by J. James, T. Koberda, K. Lendsey, C. Silva, P. Speh [32].
All spaces of the paper [14] are considered simultaneously as topological spaces and as measure spaces. Presented results concern either the measurable dynamics on the spaces or the interplay between the measurable and topological dynamics.
After some introductory section, second section of the paper[14] considers functional analytic properties of uniform rigidity that is similar to the properties of rigidity. Authors formulate and prove
Theorem 6. (Theorem 1.) Every totally ergodic finite measure-preserving transformation on a Lebesgue space has a representation that is not uniformly rigid, except in the case where the space consists of a single atom.
The proof of the theorem connects with results of authors of the paper [14] that uniform rigidity and weak mixing are mutually exclusive notions on a Cantor set, and follows from the Jewett-Krieger Theorem by [33].
Third section concerns with uniform rigidity and measurable weak mixing.
Author motivation for this topic is that a (nontrivial) measure-preserving weakly mixing transformation that is uniformly rigid would yield an example of a measurable sensitive transformation that is not strongly measurably sensitive. For a subset Y of a metric space X and a measurable transformation of X authors of the paper [14] define the notion of uniformly rigid transformation on Y and prove Theorem 3.4 that is reminiscent of Egorov Theorem by P. Halmos [34]. In forth section authors present asymptotic convergence behavior. Let X be a compact metric space and let T be a finite measure-preserving ergodic transformation. Authors prove:
EXTREMAL FORMS AND RIGIDITY IN ARITHMETIC GEOMETRY... 133
Proposition 6. If T is uniformly rigid, then the uniform rigidity sequence has zero density.
The aim of section five is to study group action and generalized uniform rigidity. Let G be a countable group endowed with the discrete topology acting faithfully on a finite measure space by measure-preserving transformations. Following authors of the paper [14] the action of G is uniformly rigid if there exists a sequence {gi} of group elements that leaves every compact K C G, denoted gi ^ x>, such that d(x,gi ■ x) 0 uniformly. The main result of the section is Theorem 5.3:
Theorem 7. Let X admit a weakly mixing group action and a uniformly rigid action by nontrivial subgroups of a fixed group of automorphisms G. Then there exists a G—action on X that is simultaneously weakly mixing and uniformly rigid.
Authors formulate several interesting questions that arise under investigations of weak mixing and uniform rigidity.
Some results and methods that are connected with topics of this and next section are considered in the paper [51].
7. Actions of groups and semigroups
Furstenberg and Berent investigate the action of abelian semigroups on the torus Td for d =1 and d > 1 respectively. The authors of the paper [12] extend to the noncommutative case some results of Furstenberg and Berent. Author‘s results answer problems raising by H. Furstenberg [35] and by Y. Guivarc’h [private communication to authors of the paper [12]].
Let v be a probability measure on SLd(Z) satisfying the moment condition
Ev(| g ||e) < »
for some e. The authors of the paper [12] show that if the group generated by the support of v is large enough, in particular if this group is Zariski dense in SLd, for any irrational x E Td the probability measures v*n * 5x tend to the uniform measure on Td. If in addition x is Diophantine generic, authors show this convergence is exponentially fast.
This interesting survey includes resent results on rigidity theory by M. Einsiedler,
E. Lindenstrauss [36] and by G.A. Margulis [37], convolution of measures, on v—stiff action, on Fourier coefficients of measures and on notions of coarse dimension.
Let the action of semigroup Г on Td satisfy the following three conditions: (Г—0) Г < SLd(R), (Г— 1) Г acts strongly irreducibly on Rd, (Г—2) Г contains a proximal element: there is some g E Г with a dominant eigenvalue which is a simple root of its characteristic polynomial.
In Section 1 authors formulate main result of the paper.
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Theorem 8. Let Г < SLd(R) satisfy (T — 1) and (Г — 2) above, and let v be a probability measure supported on a set of generators of Г satisfying
^ v(g) I g 1Г< ж
ger
for some t > 0. Then for any 0 < A < Ai(v) there is a constant C = C(v, A) so that if for a point x E Td the measure pn = v*n * 5x satisfies that for some a E Zd \ {0} \ jan(a) \> t > 0, with n > C • log(“j1), then x admits a rational approximation p/q for p E Zd and q E Z+ satisfying || x — - ||< exp-An and
\ q \< (A1 )c.
Authors of [12] denote the theorem as Theorem A.
Section 2 is devoted to the deduction of two corollaries from Theorem A. Let in the corollaries Г and v be as in theorem A.
Corollary 1. Let x E Td \ (Q/Z)d. Then the measures pn = v*n * 5x converge to the Haar measure m on Td in weak-* topology.
This is authors [12] Corollary B. Next corollery is the authors [12] Corollary C:
Corollary 2. Let x E Td and pn = v*n*6x. Then there are c1, c2 depending only on v so that the following holds: (1) Assume x is Diophantine generic in the sense that for some M and Q || x — - ||> q-M for all integers q h Q and p E Zd. Then for
n > c1 log Q maxbeZdm<B \ jan(b) \< B exp-C2n/M. (2) Assume x E Q/Z)d. Then there is a sequence щ ^ ж along which maxbeZd)0<||b||<expc2"i \ jan(b) \< exp-C2ni.
Section 3 gives the deduction of authors' solution of Furstenberg problem from the authors [12] Proposition 3.1:
Proposition 7. Let Г and v be as in theorem A, 0 < A < A1(v). Then for some constant C depending on v, A the following holds: for any probability measure p0 on Zd, if pn = v*n * p0 has a nontrivial Fourier coefficient a E Zd \ {0} \ pn(a) \> t, with n > C • log(21j1), then p0(Wqexp-\n) > (|)C where Q = (-j1 )C.
Theorem A follows from Proposition 3.1.
Section 4 is devoted to random matrix products. It includes estimates of the metric on Pd-1 and random walks. In Section 5 two notions of coarse dimension are discussed. Section 6 describes the structure of the set of t—large Fourier coefficients. The last Section "Granulated measures"gives the prove of Proposition 3.1.
The results of the paper[12] will be of use to specialists interested in Diophantine approximation, measure theory and algebraic dynamics.
8. Rigid cohomology
At first we review the necessary results on the connection and on the Gauss-Manin connection. We follow the ideas and results by Grothendieck, Griffiths, Manin, Katz, Deligne and others.
EXTREMAL FORMS AND RIGIDITY IN ARITHMETIC GEOMETRY... 135
8.1. Connection
Let S/k be the smooth scheme over field k, U an element of open covering of S, OS the structure sheaf on S, r(U, OS) the sections of OS on U. Let (lS/k be the sheaf of germs of 1—dimension differentials, F a coherent sheaf on S. The connection on the sheaf F is the sheaf homomorphism
V : F —— (S/k О F,
such that, if f E r(U, OS), g E r(U, F) then
V(fg) = f V(g) + df О g.
There is the dual definition. Let F be the locally free sheaf, &S/k the dual to sheaf (S/k, d E r(U, @S/k). The connection is the homomorphism
p : ®S/k — EvAos (F, F),
p(d )(fg) = d(f )g + fp(d)•
8.2. Integration of connection
Let (iS/k be the sheaf of germs of «—differentials,
Vl(a О f ) = da О f + (— 1)га Л V(f).
Then V, Vг define the sequence of homomorphisms:
F — (S/k oF — (S/k oF —■■■ ,- (1)
The map
K = VoV1 : F — (S/k О F
is called the curvature of the connection V.
The cochain complex
(K \d) = [K° — K1 — K2 — ■■■}
is the sequence of abelian groups and differentials d : Kp — Kp+1 with the condition d o d = 0.
Let A be an abelian category, K(A) the category of complexes over A. Furthermore, there are various full subcategories of K(A) whose respective objects are the complexes which are bounded below, bounded above, bounded in both sides.
A connection is integrable if (1) is a complex.
Proposition 8. The statements a), b), c) are equivalent:
a) the connection V is integrable;
b) K = VoV1 = 0;
c) p is the Lie-algebra homomorphism of sheaves of Lie algebras.
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More generally, let X, S be smooth schemas over k, f : X -— S a smooth morphism,
OX/S : OX — OX/S — OX/S ^ ■ ,
the de Rham complex of relative differentials. There is the exact sequence
0 — f *(OS) — ox — oX/s — 0
In the case there is an integrable connection, the Gauss-Manin connection.
By an abelian sheaf we mean a sheaf of abelian groups. Let R0 f be the functor from the category of complexes of abelian sheaves on X to the category of abelian sheaves on S. Denote by WDR(X/S) the sheaf of de Rham cohomologies such that
hDr (x/s) = Rf&X/S.
Recall that
H = HDr = Rf&x/S
is called the Gauss-Manin bundle.
Here Rlf(,i > 0, are the hyperderved functor of R°f^.
For each i ^ 0, HlDR(X/S) is a locally free coherent algebraic sheaf on S, whose fiber at each point s E S is the C—vector space Hl(Xs, C) and has the Gauss-Manin connection. HlDR(X/S) has the main interpretation as the Picard-Fuchs equations and Hl(Xs, C) in the interpretation is the local system of germs of solutions of the equations.
There is the canonical filtration of OX/S by locally free subsheaves
OX/s = F (Ox/s) D F1 (oX/s) D • • •
given by
Let
Fi(Qnx) := Im(f*(OS) 0 OX- — OX). gri = Fl/Fl+\ i = 0,1,...
Example 1. Let S = SpecB,X = Spec A be schemas over k of algebraic dimensions dim B =1 (for instance, B = k[x], ) dim A = 2,
HTDr(X/S) = HDr(A/B ).
Let OB = Bdt, r = 1,
0 — gr1 -— F0/F2 — gr° — 0
where F0 = OX- The exect sequence has the form,
0 —— OB 0 A 0 OX —— Ox —— A 0 OX/b —— 0.
EXTREMAL FORMS AND RIGIDITY IN ARITHMETIC GEOMETRY... 137
Consider the simplest case of elliptic curve
y2 = x3 + t.
In the case
B = k[t, t ],A = В[x,y\/(y - x + t), and ш = dj- £ 0У Finally, we obtain a partial case of Fuchs equation:
du 1
_:— + -Ш = 0.
dt 6t
8.3. Rigidity
Let p be a prime, n a positive integer, and Fq the finite field with q = pn elements. Let Qq denote the unique unramified extension of degree n of the field of p-adic numbers. Let U be an open dense subscheme of the projective space PQ^ with nonempty complement Z. Let V be the rigid analytic subspace of PQq which is the complement of the union of the open disks of radius 1 around the points of Z. A Frobenius structure on E with respect to a is an isomorphism F : a*£ ~ E of vector bundles with connection defined on some strict neighborhood of V.
A meromorphic connection on P1 over a p-adic field admits a Frobenius structure defined over a suitable rigid analytic subspace. Authors of the paper[11] give an effective convergence bound for this Frobenius structure by studying the effect of changing the Frobenius lift. They also give an example indicating that their bound is optimal.
The techniques used are computational. This is a good place to see the interplay between matrix representation of a Frobenius structure and a Gauss-Manin connection.
The theory of rigid p-adic cohomology are developed by Berthelot [38] and others. Rigid cohomology in some sense extends crystalline cohomology. Review of some novel results and applications of crystalline cohomology is given in paper [52].
9. Superrigidity
The notion of property (T) for locally compact groups was defined by D. Kazhdan [39] and the notion of relative property (T) for inclusion of countable groups Г0 С Г was defined by G. Margulis [40].
The concept of superrigidity was introduced by G. D. Mostow [41] and by
G. A. Margulis [42] in the context of studying the structure of lattices in rank one and higher rank Lie groups respectively. The first result on orbit equivalent (OE) superrigid actions was obtained by A. Furman [43], who combined the cocycle superrigidity by R. Zimmer [44] with ideas from geometric group theory to show
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that the actions SLn(Z) ^ Tn(n ^ 3) are OE superrigid. The deformable actions of rigid groups are OE superrigid by S. Popa [45].
The paper [13] presents a new class of orbit equivalent superrigid actions. The main result of the paper [13] is the Theorem A on orbit equivalence (OE) superrigidity. As a consequence of Theorem A the author can constructs uncountable many non-OE profinite actions for the arithmetic groups SLn(Z)(n ^ 3), as well as for their finite subgroups, and for the groups that are semi direct products of groups SLm(Z) and Zm( m ^ 2). The author deduces Theorem A as a consequence of the Theorem B on cocycle superrigidity.
Let Г ^ X be a free ergodic measure-preserving profinite action (i.e., an inverse limit of actions Г ^ Xn with Xn finite) of a countable property (T) group Г (more generally, of a group Г which admits an infinite normal subgroup Г0 such that the inclusion Г0 С Г has relative property (T) and Г/Г0 is finitely generated) on a standard probability space X. The author prove that if ш : Г x X ^ Л is a measurable cocycle with values in a countable group Л, then ш is a cohomologous to a cocycle ш which factors through the map Г x X ^ Г x Xn, for some n. As a corollary, he shows that any free ergodic measure-preserving action Л ^ Y comes from a (virtual) conjugancy of actions.
10. Newton strata in the loop group of a reductive group
Let G be a split connected reductive group over Fp, let T be a split maximal torus of G and let LG be the loop group of G by Faltings [46].
Let R be a Fg-algebra and K be the sub-group scheme of LG with K(R) = G(R[[z]]). Let a be the Frobenius of k over Fg and also of k((z)) over Fg((z)). For algebraically closed k, the set of a-conjugacy classes [6] = [g-1ba(g)\g E G(k((z)))} of elements b E LG(k) is classified by two invariants, the Kottwitz point Kc(b) and the Newton point v.
The author of the paper [47] proves the following two main results.
Theorem 9. Let S be an integral locally noetherian scheme and let b E LG(S). Let j E J(v) be a break point of the Newton point v of b at the generic point of S. Let Uj be the open subscheme of S defined by the condition that a point x of S lies in Uj if and only if prj)(vb(x)) = pr j)(v). Then Uj is an affine S-scheme.
Theorem 10. Let p1 Y p2 E X*(T) be dominant coweights. Let
S^ = U Kz» K.
Let [b] be a a-conjugacy class with Kc(b) = R1 = p2 as elements of n1(G) and with vb Y p2. Then the Newton stratum Nb = [b] П S^^2 is non-empty and pure of
EXTREMAL FORMS AND RIGIDITY IN ARITHMETIC GEOMETRY... 139
codimension (p,^2 — vb) + 2def (b) in S^1>^2. The closure of Nb is the union of all Nb‘ for [b‘] with Kc(b‘) = Щ and vb‘ < vb.
Here p is the half-sum of the positive roots of G and the defect def (b) is defined as rkG — rkFq Jb where Jb is the reductive group over Fg with Jb(k((z))) = {g E LG(k)\gb = ba(g)} for every field k containing Fg and with algebraically closed k.
The proof of Theorem 9 is based on a generalization of some results by Vasiu [48]. An interesting feature of E. Viehmann method in the prove of Theorem10 is the using of various results on the Newton stratification on loop groups as Theorem 9 and the dimension formula for affine Deligne-Lusztig varieties by G’ortz, Haines, Kottwitz, Reuman [49] together with results on lengths of chains of Newton points by Chai [50].
11. Conclusion
Rigid problems, methods and results in arithmetic algebraic geometry and in dynamics have presented. Diverse notions of rigidity and respective novel results are reviewed.
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National Aviation University Received 31.07.2015
