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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 16 Выпуск 3 (2015)
ONE CONSTRUCTION OF INTEGRAL
REPRESENTATIONS OF p-GROUPS AND SOME
APPLICATIONS
Dmitry Malinin (Kingston, Jamaica)
Department of Mathematics, University of the West Indies,
Mona, Kingston 7, Jamaica
Abstract
Some well-known classical results related to the description of integral representations of finite groups over Dedekind rings R, especially for the rings of integers Z and p-adic integers Zp and maximal orders of local fields and fields of algebraic numbers go back to classical papers by S. S. Ryshkov,
P. M. Gudivok, A. V. Roiter, A. V. Yakovlev, W. Plesken. For giving an explicit description it is important to find matrix realizations of the representations, and one of the possible approaches is to describe maximal finite subgroups of GLn(R) over Dedekind rings R for a fixed positive integer n.
The basic idea underlying a geometric approach was given in Ryshkov's papers on the computation of the finite subgroups of GLn(Z) and further works by W. Plesken and M. Pohst. However, it was not clear, what happens under the extension of the Dedekind rings R in general, and in what way the representations of arbitrary p-groups, supersolvable groups or groups of a given nilpotency class can be approached.
In the present paper the above classes of groups are treated, in particular, it is proven that for a fixed n and any given nonabelian p-group G there is an infinite number of pairwise non-isomorphic absolutely irreducible representations of the group G. A combinatorial construction of the series of these representations is given explicitly.
In the present paper an infinite series of integral pairwise inequivalent absolutely irreducible representations of finite p-groups with the extra congruence conditions is constructed.
We consider certain related questions including the embedding problem in Galois theory for local faithful primitive representations of supersolvable groups and integral representations arising from elliptic curves.
Keywords: finite nilpotent groups, integral domain, Dedekind ring, elliptic curves.
Bibliography: 27 titles.
ONE CONSTRUCTION OF INTEGRAL REPRESENTATIONS OF ... 323
ОБ ОДНОЙ КОНСТРУКЦИИ ЦЕЛОЧИСЛЕННЫХ ПРЕДСТАВЛЕНИЙ р-ГРУПП И ЕЁ ПРИЛОЖЕНИЯ
Д. А. Малинин (Кингстон, Ямайка)
Аннотация
Некоторые хорошо известные классические результаты, относящиеся к описанию целочисленных представлений конечных групп над дедекин-довыми кольцами R, в частности, для колец целых чисел Z и р-адических чисел Zp и максимальных порядков локальных полей и полей алгебраических чисел берут начало в классических работах С. С. Рышкова, П. М. Гу-дивка, А. В. Ройтера, А. В. Яковлева, В. Плескена. Для их явного описания важно найти матричные реализаций представлений, и один из возможных подходов состоит в описании максимальных конечных подгрупп GLn(R) над дедекиндовым кольцом R при фиксированном натуральном п.
Основная идея, лежащая в основе геометрического подхода, была приведена в работах С. С. Рышкова по вычислению конечных подгрупп из GLn(Z) и дальнейших работах М. Поста и В. Плескена. Тем не менее, было неясно, что происходит при расширении дедекиндова кольца R в общем случае, и в случаях представлений произвольных р-групп, сверхразрешимых групп или групп заданного класса нильпотентности.
В настоящей работе изучаются представления вышеуказанных классов групп, в частности, доказано, что при фиксированном п и любой заданной неабелевой р-группы G существует бесконечное число попарно неизоморфных абсолютно неприводимых представлений группы G. Комбинаторная конструкция серии этих представлений получена в явном виде.
В настоящей работе построена бесконечная цепочка целочисленных попарно неэквивалентных абсолютно неприводимых представлений конечных р-групп с дополнительными условиями сравнимости по модулю дивизоров простого числа р.
Мы рассматриваем некоторые связанные нашей конструкцией вопросы, включая задачи погружения в теории Галуа для локальных точных примитивных представлений сверхразрешимых групп и целочисленные представления, возникающие из эллиптических кривых.
Ключевые слова: конечные нильпотентные группы, целые области, Де-декиндовые кольца, эллиптические кривые.
Библиография: 27 наименований.
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1. Introduction
Let K be a finite extension of the p-adic field Qp, and let OK be its ring of integers. If K is fixed, the number of irreducible pairwise inequivalent representations of a given finite group group over OK is finite. In this paper we do not fix K, we allow K to be extended via adjoining certain roots of 1, we construct an infinite number of absolutely irreducible pairwise inequivalent representations of a given p-group over OK for different K, and we consider the possible realization fields of these representations.
We construct some infinite series of integral pairwise inequivalent absolutely irreducible representations of finite p-groups over the rings of integers of number fields and local fields, and we apply this construction to representations having the minimal possible degrees. We also prove the extra condition that the matrices of these representations are contained in the kernel of reduction modulo a prime divisor of p. By giving a complete combinatoric description of all irreducible representations of a finite p-group of class 2 we show that a nonabelian p-group possesses infinitely many absolutely irreducible integral representations which are not equivalent over the ring of integers.
Remark that the class of groups considered in the first section below can be extended to classification of absolutely irreducible primitive representations of some supersolvable groups (see [16]), and this can be applied to the classification of the primitive representations of the Galois groups of local fields.
2. Notation
We denote C, Q and Qp the fields of complex, real, rational and rational p-adic numbers. Z and Zp are the rings of rational and p-adic integers. NE/F(a) denotes the norm of a E E in the field extension E/F.
We denote GLn(R) the general linear group over a ring R, SLn(R) denotes the special linear group.
[E : F] denotes the degree of the field extension E/F.
Mn(R) is the full matrix algebra over a ring R.
Finite groups are usually denoted by capital letters G, H, and their elements by small letters, e.g. g E G, h E H, (a,b...) denotes a group generated by a,b,..., Z = Z(G) is the center of G, [a,b] = aba-1b-1 denotes the commutator of a,b.
We write (t for a primitive t-root of 1. Diagonal matrices are denoted by diag(d1,
,dn), I (and Im) stands for a unit
(m x m-matrix).
Binomial coefficients are denoted by (m) ■
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3. The construction
Let us consider a nonabelian group G0 generated by two elements a and b of order t = pm, at=bt=1 such that the commutator c = [a, b] = 1 is contained in the center of G0, and ct=l, t is the minimal positive integer having this property. Let Z be a primitive root of 1 of degree t. The following representation of Go is faithful and absolutely irreducible.
A = A(a)
(0 1 • • .0
0 0 •• • 1
1 0 •• .0
B = A(b) = drng(1, t 1
Indeed, all matrices of this representation are unitary, and any matrix in GLt(C) commuting with all matrices of this representation is a scalar matrix; it follows from [8], p. 8 that A(G0) is absolutely irreducible.
Proposition 1. (see e.g. theorem (2.32), p. 29 [11]. A p-group has a faithful irreducible representation if and only if its center is cyclic.
For the n x n-matrices e ij having precisely one nonzero entry in the position (i,j)
equal to 1 we can define a n x n-matrix using the binomial coefficients
in the case i = j = n we replace the above coefficients with 1. Let us consider n = t and the following triangular matrices:
n - j i - j
C
E (-uj n - j)
-j
n>i>j>1
C1
E (I - j)eij •
n>i>j>1 V 7
Let X = diag(1,x,x2,... ,xt 1), then CiXC--
N (n - j)xj-1(1 - xy--e.ij,
i j 1 -
n>i>j>1
and if we take x = 1, this will imply C 1 = C1. If we take x = Z, we will obtain:
A'(b) = C1A(b)C = C\BC = N ("i _ j ) Zj-1(1 - Z)i-jeij,
n>i>j>1 ^ j 7
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D. MALININ
We can see that all matrix entries below the main diagonal are divisible by powers of ( — 1, and the exponents of the powers are growing proportionally to the distance from the main diagonal. An elementary computation shows that
< 1 — l 1 0 ... 0 0
—(2) 1 1 ... 0 0
A (a) = C _ 1AC =
—a о 0 ... 1 1
V о о 0 ... 0 1
Let K be (as earlier in Introduction) a finite extension of the p-adic field Qp. Let us assume that K contains Z. For a positive integer h let us consider a finite extension Lh C K((pr) of degree h over Qp for an appropriate integer r and a primitive pr-root of 1 (pr; its maximal order OLh, a prime divisor P of p and its prime element nh, this prime element may be chosen as (pr — 1 in the case if Lh is a cyclotomic field Qp((pr). Let Dh = diag(1,nh,nh,... ,nfh_l), then
f
Ah = Ah(a) = D- A'(a)Dh
nh 0 .. .0 0
( 1) — t —(2) 1 nh . .0 0
i to • 1 • 0 0 .. .1 nh
0 0 0 .. .0 1
Bh = Ah(b) = D_1A(b)Dh
E
n>i>j>l
(:—
— Z )i-j nh-ie
ij
In the field Qp the prime p factorizes as (1 — ZУ™ 1 (p 1 ^ The entries of the first column of A'(a) (except the first one) are divisible by p, all of them lower than
pm 1 (p_ 1)
are even divisible by p2, and nh divides (1 — Z), therefore, all subdiagonal
entries of Ah(a) (except the first one) are divisible by nh_j. The same is true for the matrix A'(b), and the representations Ah are integral in OK, and they are contained in the kernel of reduction (modnh). Moreover, the matrices Ah(a) = It(modnh), but Ah(a) = It(modnh). Thus Ah and Ar are not equivalent over OK if h = r. This gives an explicit construction of an infinite series of pairwise integrally inequivalent representations over OK. Thus we have the following theorem:
Theorem 1. Let L denote Qp(Zp~), the extension of Qp obtained by adjoining all roots Zpi ,i = 1, 2, 3,... of p-prim,ary orders of 1. Let G be a finite nonabelian two-generator p-group admitting representations by matrices A(a), A(b) above, and let OL be the ring of integers of L. Then there is an infinite number of integral pairwise inequivalent absolutely irreducible representations of G over OL.
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We shall extend the construction of Ah from theorem 1 above first to the case of a nonabelian p-group G1 generated by its center Z and elements a, b with central nontrivial commutator c = [a,b] = aba-1 b-1. Let x be a character of Z such that x(c) = Z = 1 for a primitive root of 1 of some degree t = pm. There is an absolutely irreducible representation of G1 extending x; the central elements z correspond to scalar matrices x(z)I, and let us denote by Cx the kernel of x. Let us denote by Cx the kernel of x. Then we have a representation of the factorgroup Gx = G1 /Cx. Let Zx C G1 be the preimage of the center of Gx; it consists of the elements x E G1 such that x(xax-la-1) = x(xax-la-1) = 1. It is also clear that a* E Zx and b E Zx, since x(ataa-ta-1) = x(xax-1a-1) = 1,x(atba-tb-1) = x(c) = 1 and x(btbb-tb-1) = x(btab-ta-1) = 1. The same computation shows that powers of a and b lower than tth powers are not contained in Zx. Further, Zx/Cx is an abelian group containing Z/Cx, and we can extend the character x of Z/Cx to Zx/Cx, and thereby to a linear character of Zx. In the absolutely irreducible representation A of G1 such that A(z) = x(z)I, the elements x correspond to scalar matrices x1(z)I for the extension x1 of the character x(z) to Zx.
Denote Z1 = ^x1(at) and Z2 = ^xJb). The matrices Ah(z) = x(z)I, for z E Zx, together with Ah(a) = Z1Ah and Ah(b) = Z2Bh determine a representation of G1,
and since Z1 = Z2(m°dnh) for sufficiently large n, for large enough distinct n and n' the integral representations Ah and Ah are integrally inequivalent.
Now let us consider an arbitrary p-group G of the nilpotency class 2 having the center Z. For every character x of the center let us denote its kernel by Cx. Denote by Zx C G be the preimage of the center of G/Cx. Then Zx is the set of of the elements x E G such that x(xyx-1y-1) = 1 for all y E G. Let x1 be an extension of x from Z/Cx to Zx/Cx. In the absolutely irreducible representation of G extending the representation x(z)In of the center, the elements y E Zx correspond to scalar matrices x(y)In. Let us assume that the commutator subgroup G of G is not 1, and
G = Zx.
Let us define an "inner product"(x,y) = x([x,y]), where x,y E G and [x,y] = x-1y-1xy.
The following lemma is well known in the theory of nilpotent groups and can be checked by a direct calculation of commutators [x,y].
Lemma 1. Suppose G is a nilpotent group of nilpotency class is two. Then, for any element x E G, the map
y ^ [U y]
is an endomorphism of G.
The image of these endomorphism lie in the commutator subgroup G of G, hence in the center of G, so it is abelian. The kernel of this endomorphism contains the center of the group, more specifically, it is the centralizer of x in G.
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PROOF. Consider an element x E G. Since G has nilpotency class two, the commutator [x,y] = xyx-1y-1 is in the center Z of G, and hence it commutes with any y E G.
Let y1,y2 E G. Since [x,y2] E Z, we have [x,yi][x,y2]
[x,yi]yi[x,y2]yi 1
xyix 1y-1yi xy2x 1y- 1y 1 1
xy1y2x 1y11y1 1 = [x,y1y2]. This completes the proof of lemma 1. □
The above inner product (x,y) = x([x,y]) is multiplicative in both arguments and antisymmetric, since (x,x) = 1. The value of (x,y) depends only on cosets containing x and y modulo Zx. Thus we can view (x,y) as being defined on Gx = G/Zx. The product (x,y) is nondegenerate on this group by the definition of Zx. Now Gx is an abelian p-group. Let a, b,... be the generators of its cyclic direct factors. The values of (x, y) are roots of 1 of degrees that are powers of p. They are generated by the values of the symbol (x,y) on the generators. Therefore, there is a pair of generators on which the value of the symbol is a root of 1 of the highest possible degree t = pm. Let a and b be such generators, and let (a,b) = ( = f/l. Thus at = bt
in Gx.
Lemma 2. Gx is the direct product of the group H generated by 2 elements a and b and its orthogonal product H±. In particular, the number of generators is even, and they are divided to pairs ai, bi such that the generators from different pairs are orthogonal, the orders of ai and bi are equal, and the degrees of the roots (ai,bi) of 1 are equal.
PROOF. Suppose that x E Gx. Then (x,a) = (kl and (x,b) = (k2. for some integers k1,k2. Then we have (x • a1 k2 bkl, b) = 1, thus x • a 1 k2 bkl E H±, and H • H± = Gx. Any element u E Hf]H± is orthogonal to both H and Hx, and thus to all Gx = H • H± and since the symbol (x,y) is nondegenerate, u =1. This argument implies that Gx is a direct product of H and H±. We can apply the same argument to H± and use induction on the number of generators G. Finally we find that G can be expressed as a direct product of pairwise orthogonal two-generator subgroups.
Let Ai and Bi be representatives in G of the classes of ai and bi from the constituents of G/Zx. Then both Af = В*1 are contained in Zx, and
X(aibia-1b- 1)(ti = Vl,
and the values of x on commutators of elements from different pairs are all equal 1.
The representation of G extending the character X1 of Zx is also a representation of an algebra over K (K is Qp(()) with generators u1,v1,... ,us,vs with multiplication given by vfti = x1(Ati),vti = X'lBf ),uivi = viui(ti, and the generators from different pairs commute. This algebra is a tensor product over K of the algebras generated by the pairs ui,vi, and representations of these algebras are representations of groups of type G1 as considered above. These algebras determine symbols K[u,v] satisfying the properties of Hilbert symbol which can be identified with an element
ONE CONSTRUCTION OF INTEGRAL REPRESENTATIONS OF ... 329
of the Brauer group (see [12], theorem A.2.3, p. 142). Note that the degree of the irreducible representation of each two-generator group (ui,Vi) described above is equal to p*1. Compare section 2 in [10].
Lemma 3. Let G be a two-generator group (u,v) as above with cyclic center Z = (c) of order pn, and the order of (u,v) = d is p*. For p = 2 we can find the generators u,v of the group G above in such a way that either up =1 or vp =1 for p =2.
PROOF. We can use our previous remarks and replace the generators in the following way: if neither of the conditions up =1 or vp =1 is true, and the order of u does not exceed the order of v, then we can change v: consider v0 = urv, then [u,v] = S is in Z, the order of [u,v] is d, and computations of the commutators
show that v0 = (urv)p = urptvp*S-^^2 , and v0 = 1 for an appropriate choice
of an integer r if p = 2, and we can take the generators u,v0 instead of u,v; this replacement will not change the group (u,v). □
This implies that the number of OK—inequivalent representations of G is infinite.
The constructed representations are contained in the kernel of reduction modulo some prime divisor P of p.
Further let us formulate the following propositions based on results. Note that there are some general results on the classification of two-generator p-groups G of the nilpotency class 2, see [2], [25], see also earlier papers: [1], theorem 2.6, [14], theorem 2.5.
Proposition 2. [21] or [22], Satz 6.1, p. 291. Let G be a minimal nonabelian p-group. Then G = (a, b) and one of the following holds:
(a) Apm = Bpn = Cp = 1, [A,B] = C,B2h = A-m, [A,C] = [B,C] = 1 is not
m,etacyclic. Furthermore, this group is not metacyclic and in the case p = 2, we have m > n; m > 2.
Also, \G\ = pm+n+1; G' = (C) and Z(G) = (Ap) x (Bp) x (C);
(b) G = Q8 is the group of quaternions of order 8;
(c) G = (A,B\Apm = Bpn = Cp = 1, [A,B] = Apm-1) is metacyclic.
Proposition 3. [3]. Let G be a 2-generated finite 2-group and \G'\ = 2. Then G is minimal nonabelian.
Now we can extend theorem 1 to representations (which are not always faithful) of an arbitrary p-group G. First, let us observe that among the representations of G there occur the absolutely irreducible representations of the factor-group by the third term of the lower central series, and this is nilpotent of class 2. We can also observe that the absolutely irreducible representations of the factor-group by the
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D. MALININ
orthogonal complement H± to any two-generator subgroup H = (u,v) C G are also the representations of G. If H is not abelian, any its faithful absolutely irreducible representation will be an absolutely irreducible representation of G. Using lemma 3, we can start from the representation determined by matrices A and B together with scalar pk x pk-matrices composing the centre Z = (c) of G:
( 0 l .. .0
A' = A(a) =
0 0 .. . l
l 0 .. .0
B = A(b) = t-1)
к
Let the order of Z be pm, and let t be a primitive pm-root of 1. Let up = l, vpk = cf, ( = tpm k, 9 = Then the representation Ah of H determined by
u ^ A, v ^ 9B' = B, c ^ tI' = C is absolutely irreducible and faithful. As earlier, we can obtain the integral representation by matrices congruent to the unit matrix I (mod P):
/
Ah = Ah(a) = D-1C-1A'CD
l -1 nh 0 .. .0 0
- Q n-1 1 nh . .0 0
i to • 1 • 0 0 .. .1 nh
0 0 0 .. .0 1
Bh = Ah(b) = D-1C ~lB'CD,
£ 0-
-Ic-ibcd = I t j \(j-\l - Z)i-3П~гej.
n>i>j>l
In the case p = 2 we can consider a character у of Z C G = (u,v) as earlier, and for the subgroup ([u,v]) = C C Z generated by the commutator c = [u,v]. Let x(c) = Y = l be an element of of order 2x; then and for у2 the image of the commutator subgroup G C G is a group of order 2, the image у2 (G) is
nonabelian, we can apply Propositions 2 and 3 together with examples 1) - 3) for constructing an infinite series of pairwise inequivalent representations of G of the minimal degree. Alternatively, we can use our construction from theorem 1.
Thus we have the following
Theorem 2. Let L denote Qp((p~), the extension of Qp obtained by adjoining all roots Zpi,i = l, 2, 3,... of p-prim,ary orders of 1. Let us fix the degree t of matrix representations. Let G be any finite nonabelian p-group, and let OL be the ring of
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integers of L. Then there is an infinite number of integral pairwise inequivalent absolutely irreducible representations of finite groups G in GLn(OL).
The constructed representations are contained in the kernel of reduction It (mod P) modulo some prime divisor P of p.
Remark. Our construction applied to two-generator nonabelian subgroups gives the representations of G having minimal possible degrees.
The results above preceding theorem 2 (in particular, lemma 2) can be reformulated for some supersolvable groups and used for classification of absolutely irreducible primitive representations of the absolute Galois groups of local fields, see (see [16], theorem 2.2), see also [26], [27] and [23].
4. Some related topics and applications
Proposition 4. Let G be a finite group, H - its normal p-subgroup, let G/H be supersolvable, p : G 4 GLn(C) - a faithful primitive representation. Then:
• n = pd. The center Z = Z(H) is cyclic of order pz, and for c E Z of order p there are elements u1,v1,... ,ud,vd which together with Z generate H and satisfy the generating relations: [ui,Uj] = [vi,Vj] = 1, [ui,vj] = cSij, where 8itj is the Kronecker’s delta, (i,j = 1,...,d), and the generators from different pairs commute.
• There are 2 possibilities:
1) up = vp =1 for p = 2
2) up = vp = c (quaternion type), or up = vp =1 (dihedral type) for p = 2.
• H/Z is p-elementary abelian of order p2d
• H has (p — 1)pz-1 inequivalent faithful absolutely irreducible representations
This result is closely related to the embedding problem with a nonabelian kernel for local fields which has been studied in [12] and [10].
Let
1 ^ B -a G 4 F -a 1
be an exact sequence of p-groups, K/k be a Galois extension of a local field with the Galois group F, and p > 2 be the characteristic of the residue field Qp of k. The embedding problem consists in constructing an extension L of K having the Galois group G over k, such that the automorphisms g E G, being restricted on K, coincide with p(g). The associated abelian problem is a similar problem for the sequence 1
1 4 B/B 4 G/B 4 F 4 1
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D. MALININ
where B is the commutator subgroup of B; the solution of the abelian embedding problem is well known. Let F be the Demushkin group of k, that is, the Galois group of the maximal p-extension of k. The number d(F) of generators of the group F is equal to [k: Qp] + 2. Let d(F) be the number of generators of F. In [10] the authors prove that if d(F) ^ d(F) + 3 then the embedding problem is equivalent to the associated abelian problem. In the proof they used the generalized Hilbert symbol and orthogonality of elements of k*/k*p for an option of a basis k*/k*p and abelian radical extensions of k and for the fulfillment of the Faddeev-Hasse compatibility conditions. In our argument above we used similar techniques.
In his recent publication [24] J.-P. Serre emphasized remarkable connections between integral irreducible representations of the group of quaternions and genus theory of Gauss and Hilbert, and the theory of Hilbert’s symbol. This was also considered in our recent paper [17] as an application to the description of globally irreducible representations over arithmetic rings which was earlier introduced by F. Van Oystaeyen and A. E. Zalesskii, see [20].
Let p : G ^ GLn(K) be a linear representation of a finite group G over a number field K. Is it possible to realize p over OK, the ring of integers of K, i. e. is p conjugate to a homomorphism of G into GLn(OK) ?
Another approach to generalization of integral representations of finite groups was proposed by D. K. Faddeev in [8] (see also [9]) where a generalization of the theory of Steinitz and Chevalley has been suggested.
Remark that in the paper by Serre [24] only imaginary quadratic fields \/QFd), d > 0, were considered as realization fields for representations of the group G of quaternions.
It would be interesting to find the conditions for realizations of G C GL2(OK) for any algebraic number field. The necessary condition is that K should be a splitting field of G, or in the terms of Hilbert symbol,
Proposition 5. (1) An algebraic number field K is a splitting field for group
G of quaternions if and only if K is totally imaginary and for all localizations Kv for all prime divisors v of 2 the local degree [Kv : Q2] is even.
(2) If (1) is true, then [K : Q] is even.
(3) If (1) is true and K/Q is abelian, then K has a quadratic subfield Q(Vd).
Proof. By Hasse-Brauer-Noether theorem, K is a splitting field for (G)q = QG, Q-span of G, if and only if Kv is a splitting field for (G)qp = QpG locally for all prime divisors v of p. Since the quaternion algebra has invariants 1/2 at 2 and to in the Brauer group, and 0 at all other primes p, K is a splitting field for G if and only if K is totally imaginary and for all localizations Kv for all prime divisors v of 2 the local degree [Kv : Q2] is even [6], Satz 2, ch. VII, sect. 5.
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Since [K : Q] is the sum of [Kv : Q2], it must be even, and this implies (2).
If K/Q is abelian, its degree is even, and its Galois group has a subgroup of index 2, therefore, the fixed subfield of this subgroup is a quadratic extension of Q.
This completes the proof of Proposition 5. □
Let us consider two examples.
1) We can use our construction in the case of the generalized quaternion group
G generated by Ah and Bh, A2hm = 1, [Ah,Bh] = BhAhB'-lA-l = A-2,Bh = A-m
we can use the following construction of an infinite series of pairwise integrally inequivalent over OK representations in GL2(Ok):
Ah = ЫЬ) = (
where Z is a primitive 2m-root of 1,
Bh = Ah(a) = | z2-l V (nh
2) For the following finite extension K/Qp of local fields obtained via adjoining torsion points of elliptic curves, let OK be the ring of integers of K with the maximal ideal P. Consider an elliptic curve E over Zp with supersingular good reduction (see [24], sect. 1.11). Let K/Qp be the field extension obtained by adjoining p-torsion points of E, then the formal group associated to E has height 2, its Hopf algebra Oa is a free module of rank p2 over Zp, and for the kernel Ep of multiplication by p \Ep\ = p2 (see [5], 1.3 and sect. 2). Note that for some E the ramification index e = e(K/Qp) = p2 — 1 ([24], p. 275, Proposition 12).
We can consider the group G of p-torsion points as Zp-algebra homomorphisms from the Hopf algebra OA to the Zp-algebra OK, then
G = HomZp(Oa,Ok), and the algebra OA is isomorphic to Zp[X]/(ciX + c2X2 + ... + Xp2), see [5], sect.2 and [15]. So there is a representation v : G ^ GLp2 (OK), and since E is supersingular, the image of v is contained in the kernel of reduction modulo P.
nh
Z
,
-2nhZ
z2-
-1
b
5. Conclusion
There are many classical results related to the description of integral representations of finite groups over Dedekind rings R, especially for the rings of integers Z or p-adic integers Zp and maximal orders of local fields or fields of algebraic numbers. Some of them given by P. M. Gudivok, A. V. Roiter, A. V. Yakovlev, W. Plesken go back to the classification of irreducible and indecomposable representations that can give an explicit description only for certain classes of groups and rings R. There are classification results for finite, wild and tame representation types, including the classification of arbitrary commutative R-rings having finitely many non-isomorphic
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indecomposable integral representations. For an explicit description it is important to find matrix realizations of the representations, and one of possible approaches is to describe maximal finite subgroups of GLn(R) over Dedekind rings R for a fixed positive integer n. The basic idea underlying a geometric approach was given in Ryshkov’s papers on the computation of the finite subgroups of GLn(Z) and further papers by W. Plesken and M. Pohst. However, it was not clear, what happens under the extension of the Dedekind rings R in general, and in what way the representations of arbitrary p-groups, supersolvable groups or groups of a given nilpotency class can be approached. In this paper the above classes of groups are treated, in particular, it is proven that for a fixed n and any given nonabelian p-group G there is an infinite number of pairwise non-isomorphic absolutely irreducible representations of G. The series of these representations is constructed explicitly. We study group representations with extra properties of congruences, and we give some links to representations arising from elliptic curves. The integral representations in question are very sensitive to changing the ground ring and the ramification index. Besides, the group of units of the Dedekind rings R, especially its torsion subgroup, plays an important role.
There are some applications to the embedding problem in Galois theory, globally irreducible representations and Schur rings which are discussed in proposition 4 and section 2 of the paper. Throughout the paper we give examples of particular representations. There are some more applications, which can be considered for the group representations with extra properties of congruences in our construction, arising from the class of Galois stable subgroups of GLn(R) and considered earlier in [18]. Besides a score of generalizations, finite groups that are stable under Galois action have some interaction with seemingly unrelated results in the theory of definite quadratic forms and Galois cohomologies of certain arithmetic groups.
Acknowledgement: The author is grateful to the referees for many useful remarks and suggestions which improved the paper essentially.
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СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
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3. V. Cepulic, O. S. Pyliavska A class of nonabelian nonmetacyclic finite 2-groups // Glasnik matematicki. 2006. Vol. 41(61). P. 65-70.
4. Ch. Curtis, I. Reiner Representation theory of finite groups and associative algebras. Reprint of the 1962 original. AMS Chelsea Publishing, Providence, RI, 2006. xiv+689 pp. ISBN: 0-8218-4066-5
5. F. Destrempes Deformations of Galois representations: the flat case. Seminar on Fermat’s Last Theorem (Toronto, ON, 1993-1994), p. 209-231, Canad. Math. Soc. Conf. Proc., Amer. Math. Soc., Providence, RI, 1995. Vol. 17. P. 209-231.
6. Deuring, Max Algebren. (German) Zweite, korrigierte auflage. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 41 Springer-Verlag, Berlin-New York 1968. viii+143 pp.
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Department of Mathematics, University of the West Indies, Mona, Kingston 7, Jamaica.
Received 10.07.2015
