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Symplectic Representations of Inertia Groups
Journal of Algebra 238, 400᎐410 Ž2001. doi:10.1006rjabr.2000.8638, available online at http:rrwww.idealibrary.com on
Symplectic Representations of Inertia Groups1 A. Silverberg Mathematics Department, Ohio State Uni¨ ersity, 231 W. 18 A¨ enue, Columbus, Ohio 43210 E-mail: [email protected]
and Yu. G. Zarhin Mathematics Department, Pennsyl¨ ania State Uni¨ ersity, Uni¨ ersity Park, Pennsyl¨ ania 16802, and Institute for Mathematical Problems in Biology, Russian Academy of Sciences, Pushchino, Moscow Region, 142292, Russia E-mail: [email protected] Communicated by Walter Feit Received August 31, 2000
1. INTRODUCTION AND NOTATION The aim of this paper is to study finite inertia subgroups of symplectic groups over the field Q l of l-adic numbers. A finite group is called an inertia group with respect to a given prime p / l if it is a semi-direct product of a finite normal p-subgroup and a cyclic p⬘-group. These groups are exactly the inertia groups of finite Galois extensions of discrete valuation fields of residue characteristic p. The study of semistable reduction of abelian varieties over such fields leads naturally to certain finite inertia subgroups of the symplectic group Sp2 g ŽQ l . w6x. In w7x we constructed examples for every odd prime l of inertia subgroups of Sp2 g ŽQ l ., which are not conjugate, even in GL 2 g ŽQ l ., to a subgroup of Sp2 g ŽZ l .. However, it turns out Žand this is the main result of this paper. that every 1 Silverberg thanks NSA and the Alexander-von-Humboldt Stiftung for financial support, and H. Lange and the Mathematics Institute of the University of Erlangen for their hospitality. Zarhin thanks NSF for financial support. We thank W. Feit for suggesting that we include a discussion of the case l s 3.
400 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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finite inertia subgroup of Sp2 g ŽQ l . is isomorphic to a subgroup of Sp2 g ŽZ l ., if l ) 3. See w3, 4x for a study of representations of inertia groups in characteristic 0. Throughout this paper l is an odd prime, d is a positive integer, K is a field that is an unramified finite extension of Q l , and G is a finite group that is a semi-direct product G s HL of a normal l⬘-subgroup H and a cyclic l-group L. If G is a finite inertia group for some prime p, then G is of this form for every prime l / p Žsee Lemmas 3.2 and 3.3 of w7x.. We write Sp2 d Ž R . for the group of 2 d = 2 d symplectic matrices over a ring R. If E is a field that is a finite extension of Q l , let OE denote the ring of integers. Let n denote a primitive nth root of unity in K. The following theorem is the main result of this paper. THEOREM 1.1 ŽEmbedding Theorem.. If there is an embedding G ¨ Sp2 d Ž K ., and l G 5, then there is an embedding G ¨ Sp2 d Ž OK .. Let m denote the least common multiple of the orders of the elements of H. For l s 3, we can show: THEOREM 1.2 ŽEmbedding Theorem Bis.. embedding G ¨ Sp2 d Ž K .. If
Suppose l s 3, and there is an
Ža. d F 5, or Žb. 9 ¦ 噛G, or Žc. 3 ¦ d and the natural representation of G on K 2 d is irreducible, or Žd. 3 ¦ w K Ž m . : K x Ž e. g., K s K Ž 噛H .., then there is an embedding G ¨ Sp2 d Ž OK .. We will make use of the following result, which we prove in Section 2. Our proof was inspired by Sect. 17.6 in w5x and the proof of Lemma 1.1 in Sect. 1 of Chap. X in w2x. THEOREM 1.3 ŽExtension Theorem.. Suppose that W is a finite dimensional K-¨ ector space, f : W = W ª K is a non-degenerate alternating Ž resp., symmetric. K-bilinear form, and : H ª AutŽW, f . is a group homomorphism that makes W into a simple K w H x-module. Assume that, for e¨ ery g g G, the representation of H
g : H ª Aut Ž W . ,
h ¬ Ž ghgy1 .
is isomorphic to . Suppose T is a H-stable OK -lattice in W. Then
: H ª Aut Ž T , f . ; Aut Ž W , f . can be extended to a homomorphism
G : G ª Aut Ž T , f . ; Aut Ž W , f . .
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We will say that two bilinear forms have the same parity if they are either both symmetric or both alternating. Let denote a uniformizer of OK . 2. PROOF OF THE EXTENSION THEOREM Since the H-module W is simple and H is an l⬘-group, the H-module Tr T is also simple. Replacing f by i f if necessary, we may assume that f ŽT, T . s OK and f : T = T ª OK is perfect Žsince the simplicity of Tr T implies that the induced non-zero pairing on Tr T is non-degenerate.. Let E denote the centralizer of H in End K ŽW .. By Cor. 1 on p. 179 of w3x, K w H x is decomposable Ži.e., splits into a direct sum of matrix algebras over fields.. Therefore, since W is a simple K w H x-module, E is a field that is a finite extension of K. It follows from Theorem 74.5 Žespecially a description of the center K in Žii.. of w1x that E is the field of definition of a certain character of H; in particular, E : K Ž m . : K Ž 噛H . .
Ž 1.
Thus ErK is unramified, since 噛H is prime to l. Therefore ErQ l is also unramified, since KrQ l is unramified. The simplicity of the H-module Tr T implies that the H-stable OK sublattice OE T of W is of the form aT with a g K * Žsee Exercise 15.2 of w5x.. Thus T s ay1 Ž OE T . is OE -stable; i.e., T is an OE -lattice in the E-vector space V. It follows from the Jacobson density theorem that the image of K w H x in End K ŽW . is End E ŽW .. In other words, End E ŽW . is the K-vector subspace of End K ŽW . generated by the Ž h. for h g H. By the non-degeneracy of f, there is an involution u ¬ u⬘ of End K W . characterized by f Ž ux, y . s f Ž x, u⬘ y .
᭙ x, y g W . y1 .
By the H-invariance of f, Ž h.⬘ s Ž h ᭙h g H. Thus the involution u ¬ u⬘ sends End E ŽW . into itself, and therefore sends E, the center of End E ŽW ., into itself. Let E0 s u g E N u⬘ s u4 . Then K : E0 : E. Either E s E0 or ErE0 is a quadratic extension. Let c be a generator of L. The homomorphisms
: H ª Aut O K Ž T . ,
c : H ª Aut O K Ž T .
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define OK w H x-module structures on T in such a way that the corresponding K w H x-modules are isomorphic. Since H is an l⬘-group, the corresponding OK w H x-modules are isomorphic Žsee Sects. 14.4 and 15.5 of w5x.; i.e., there exists A g Aut O KŽT . such that
Ž chcy1 . s A Ž h . Ay1
᭙h g H.
Then
Ž c i hcyi . s Ai Ž h . Ayi
᭙h g H , i g Z.
Since c 噛L s 1, we have A噛L g OEU . Further, A End E ŽW . Ay1 s End E ŽW .. Since E is the center of End E ŽW ., AEAy1 s E. In other words, defining Ž c .Ž u. s AuAy1 for u g E induces a homomorphism
: L ª Gal Ž ErK . . Since w E : E0 x divides 2 and l is odd, the composition L ª GalŽ ErK . ¸ GalŽ E0rK . has the same kernel as . Thus 噛 Ž L . divides w E0 : K x .
Ž 2.
Let f AŽ x, y . s f Ž Ax, Ay . for x, y g T. For all h g H, f A Ž Ž h . x, Ž h . y . s f Ž A Ž h . x, A Ž h . y . s f Ž A Ž h . Ay1Ax, A Ž h . Ay1Ay . s f Ž Ž chcy1 . Ax, Ž chcy1 . Ay . s f A Ž x, y . , since chcy1 g H and f is H-invariant. Thus f A is H-invariant and of the same parity as f. Therefore there exists a g E0U such that f Ž Ax, Ay . s f Ž ax, y . s f Ž x, ay . ᭙ x, y g W . Since A g AutŽT . and f : T = T ª OK is perfect, we have a g AutŽT .. This implies easily that a g OEU0 . Let [ Ž c . g GalŽ ErK .. Then aA s A y1 Ž a.. There exists a1 g OEU0 such that a y1 Ž a . s a12 . ŽIndeed, let be a uniformizer for OE . Then y1 Ž a. ' a l Žmod . for 0 some non-negative integer j. Since l is odd, l j q 1 is even, so a y1 Ž a. is a square modulo . Thus a y1 Ž a. is a square, since all elements of OE 0 congruent to 1 modulo are squares.. j
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For all x, y g T, f Ž Ž A2 a1y1 . x, Ž A2 a1y1 . y . s f Ž aAa1y1 x, Aa1y1 y . s f Ž A y1 Ž a . a1y1 x, Aa1y1 y . s f Ž a y1 Ž a . a1y1 x, a1y1 y . s f Ž x, y . . Let A1 s A2 a1y1 g Aut O KŽT .. Then f is A1-invariant, detŽ A1 . s 1, and conjugation by A1 coincides with conjugation by c 2 . Thus A噛L 1 s bI is a scalar operator of determinant 1 on the E0-vector space W. Therefore b g E0U is a root of unity, i.e., b s 1, where is the number of roots of unity in E0 . Since E0 : E, the extension E0rQ l is unramified, so l does not divide . Letting B [ A1, then conjugation by B coincides with conjugation by b1 [ c 2 , and B 噛L s I. Since b1 is a generator of L, sending b1 to B defines the desired extension G of . 3. LEMMAS FOR THE EMBEDDING THEOREMS Assume from now on that we are in the setting of the embedding theorems. Therefore there exist a 2 d-dimensional K-vector space V, a non-degenerate alternating K-bilinear form e: V = V ª K , and a faithful symplectic representation
: G ¨ Aut Ž V , e . . Clearly, to prove the embedding theorems it suffices to show the existence of an embedding G ¨ Sp2 d⬘Ž OK . for some d⬘ F d. PROPOSITION 3.1. Suppose that V is simple as a G-module but not as an H-module. Then either Ži. the H-module V is isomorphic to W r for some simple H-module W and some r ) 1, or Žii. there exist a normal subgroup G 1 of G, and a simple symplectic G 1-module V1 which is a K-¨ ector space of e¨ en dimension 2 d1 s 2 drw G : G 1 x, such that H : G 1 / G and such that if g 1 is a non-identity element of G 1 , then there exists g g G such that gg 1 gy1 is not in the kernel of G 1 ª AutŽ V1 .. Further, d is di¨ isible by l and d1 F drl. Moreo¨ er, if l 2 does not di¨ ide 噛G then G 1 s H. n Proof. We follow the proof of Prop. 24 of w5x. Let V s [is1 Vi be the canonical decomposition of the restriction of to H into a direct sum of
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isotypic representations. Since the G-module V is simple, G permutes the Vi transitively. If V is some Vi , then the H-module V is isotypic and Ži. holds. Assume from now on that Ži. does not hold. Let G 1 s s g G < s Ž V1 . s V1 4 ; G. Then H : G 1 / G. Since G 1 contains H, it is normal in G. Thus for every Vi , G 1 s s g G < s Ž Vi . s Vi 4 ; G. Every Vi is a simple G 1-module, because V is a simple G-module. The kernels of the natural maps G 1 ª AutŽ Vi . have trivial intersection and are conjugate in G. It follows that if g 1 is a non-identity element of G 1 , then it has a conjugate which does not lie in the kernel of G 1 ª AutŽ V1 .. Since w G : G 1 x divides w G : H x, it is an l-power, and therefore odd. Thus n is odd, so at least one of the simple G 1-modules Vi is self-dual. This implies easily that all the Vi are self-dual. Suppose V1 is not symplectic. Then none of the Vi are symplectic. Since Vi is a simple G 1-module, every G 1-invariant alternating bilinear form on Vi is zero. Since the Vi are mutually non-isomorphic simple G 1-modules, every G 1-invariant bilinear pairing between Vi and Vj for i / j induces the zero map Vi ª VjU Žs Vj ., and therefore is zero. Therefore, every G 1-invariant alternating bilinear n form on V s [is1 Vi is zero, contradicting that V is symplectic. Thus V1 is symplectic. Since 2 d is divisible by w G : G 1 x, therefore d is divisible by the odd prime l. If l 2 does not divide 噛G then w G : H x s l so G 1 s H. We leave the next lemma as an exercise. LEMMA 3.2. If G 0 is a finite group, V0 is a finite-dimensional K-¨ ector space which is also a faithful K w G 0 x-module, and T0 is a G 0-stable OK -lattice in V0 , then e0 Ž Ž x, f . , Ž y, g . . s g Ž x . y f Ž y . for x, y g T0 and f , g g T0U [ Hom O K Ž T0 , OK . defines a perfect alternating G 0-in¨ ariant form on T0 [ T0U , and induces a natural embedding G 0 ¨ AutŽT0 [ T0U , e0 .. LEMMA 3.3. If Theorem 1.1 Ž resp., Theorem 1.2. holds for all irreducible faithful symplectic representations of dimension F 2 d o¨ er K of all quotients of G, then it holds for all faithful symplectic representations of G o¨ er K of dimension F 2 d.
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Proof. The G-module V splits into a direct sum of G-modules V ⬘ such that the restriction of e to V⬘ is non-degenerate, and either V ⬘ is simple or V⬘ s V0 [ V0U , where V0 is simple. In the latter case choose a G-stable OK -lattice T0 in V0 and apply Lemma 3.2. LEMMA 3.4. Suppose V is simple as both a G-module and an H-module. Suppose T is a G-stable OK -lattice in V, and choose i g Z so that i eŽT, T . s OK . Then i e: T = T ª OK is a perfect G-in¨ ariant alternating bilinear form and induces an embedding G ¨ Aut Ž T , i e . ( Sp2 d Ž OK . . Proof. Let e: Tr T = Tr T ª OK r OK be the non-zero pairing induced by i e. Its kernel is an H-submodule of Tr T, so is zero by the H-simplicity of Tr T ; i.e., e is non-degenerate. By Nakayama’s lemma, i e is perfect. LEMMA 3.5. Suppose G 0 is a finite group and G 1 is a normal subgroup of G 0 . Suppose there exist a free OK -module T1 of rank 2 d1 , an alternating perfect form e1: T1 = T1 ª OK , and a homomorphism f : G 1 ª AutŽT1 , e1 . such that whene¨ er g 1 is a non-identity element of G 1 there exists g g G such that gg 1 gy1 f kerŽ f .. Then there exist a free OK -module T of rank 2 d1w G 0 : G 1 x, an alternating perfect form e: T = T ª OK , and an injecti¨ e homomorphism : G 0 ¨ AutŽT, e .. Proof. Let T s u: G 0 ª T1 N u Ž xs . s sy1 u Ž x .
᭙s g G 1 , ᭙ x g G 0 4 ,
choose a section p: G 0rG 1 ª G 0 , and let e Ž u, ¨ . s
Ý
␥gG 0rG 1
e1 Ž u Ž p Ž ␥ . . , ¨ Ž p Ž ␥ . . .
for u, ¨ g T .
Note that e is independent of the choice of section p. Define a homomorphism : G 0 ª AutŽT, e . by Ž g .Ž u.Ž x . s uŽ gy1 x . for g g G 0 , u g T, x g G 0 . Then the desired conditions are all satisfied. COROLLARY 3.6. Suppose G 0 is a finite group and G 1 is a normal subgroup of G 0 . Suppose there exists an injecti¨ e homomorphism G 1 ¨ Sp 2 d 1Ž OK .. Then there exists an injecti¨ e homomorphism G 0 ¨ Sp2 d 1wG 0 : G 1 xŽ OK .. LEMMA 3.7. If ⌳ is a finite cyclic group of order l m , then there exists an injecti¨ e homomorphism ⌳ = "14 ¨ Sp Ž l m .ŽZ l ..
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Proof. Let C denote the subgroup of ⌳ of order l. By Lemma 3.7 of w7x there is an injective homomorphism C = "14 ¨ Sp ly1ŽZ l .. Now apply Corollary 3.6 to G 1 s C = "14 : ⌳ = "14 s G 0 . LEMMA 3.8. Suppose S is a normal l-subgroup of G. Then S : L, and S is central in G. If 噛S ) 1, then e¨ ery element of order l in G is contained in S. Proof. By the conjugacy of Sylow-l-subgroups, S : L. The natural map  : L ¨ G ¸ GrH ( L is the identity map. Suppose x g S : L and g g G. Then gxgy1 g S : L. Since L is commutative,  Ž x . s  Ž gxgy1 .. Thus x s gxgy1 , so S is central in G. Suppose y g G has order l. Then y is conjugate to some w g L of order l. Suppose 噛S ) 1. Since L is a cyclic l-group, w g S. Since S is normal, y g S.
4. PROOF OF THE EMBEDDING THEOREMS Note that the embedding theorems are trivial for d s 1. By Lemmas 3.3 and 3.4, we may assume from now on that the G-module V is simple and the H-module V is not simple. If Žii. of Proposition 3.1 holds, then we may induct on d, applying Lemma 3.5. Note that if l s 3, then Ža. d1 F dr3 and therefore d1 s 1 if d F 5; Žb. if 噛G is not divisible by 9 then G 1 s H; Žc. 3 divides d. By Proposition 3.1, we may now assume that V ( W r for some simple H-module W, where r ) 1. It follows that W is self-dual; i.e., there is an H-invariant non-degenerate alternating or symmetric K-bilinear form f : W = W ª K. We may choose an H-stable lattice T in W and Žreplacing f by i f for suitable i, if necessary . we may assume that f : T = T ª OK is perfect. Let w s dim K Ž W . . Assume first that w s 1. Then H ; Aut K ŽW . s K *. Since V s W r and H : G : SpŽ V ., we have H : "14 , and Lemma 3.7 gives the desired embedding. Assume from now on that w G 2. Let
: H ¨ Aut Ž W , f . ; Aut Ž W . be the injective homomorphism defining the H-module structure on W. Since H is normal in G, the subspace gW ; V is an H-submodule of V
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for every g g G. The natural representation H ª AutŽ gW . is isomorphic to the representation
g :
H ª Aut Ž W . ,
h ¨ Ž ghgy1 . .
On the other hand, since V ( W r as H-modules, therefore gW ( W as H-modules. ŽIndeed, every H-submodule in V is isomorphic to a direct sum of copies of W, and w s dimŽ gW ... Now apply the extension theorem and extend to a homomorphism
G : G ª Aut Ž T , f . : Aut Ž W , f . . Since f is perfect, we have T * s T. Lemma 3.2 gives a perfect alternating G-invariant form e0 on T [ T, and we let
0 : G ª Aut Ž T [ T , e0 . ( Sp2 w Ž OK . be the direct sum of two copies of G . Suppose G is injective. If f is alternating Žresp., symmetric., then G Žresp., 0 . gives the desired embedding. Now assume G is not injective. Then kerŽ G . l H is the identity, so kerŽ G . is a normal l-subgroup of G. By Lemma 3.8, kerŽ G . : L. Now retain the notation from the proof of the extension theorem. Suppose dim E 0ŽW . s 1. Then H ; E0U , and for h g H, f Ž x, y . s f Ž hx, hy . s f Ž h 2 x, y . ᭙ x, y g W. Thus h 2 s 1, i.e., H : 1, y14 , and we are done by Lemma 3.7. Now assume that dim E 0ŽW . G 2. Then
w E0 : K x F wr2.
Ž 3.
Let L0 s ker Ž . s x g L: G Ž x . y s y G Ž x . ᭙ y g E 4 . Then kerŽ G . : L0 : L. Let G 0 s HL0 , a normal subgroup of G. Since V s W r , the restriction of g to G 0 can be viewed as a homomorphism
G : G 0 ª Aut E Ž W . ; Aut E Ž V . . Then s G on H ; G 0 , and
Ž z . Ž h. Ž z .
y1
s G Ž z . G Ž h . G Ž z .
y1
s G Ž z . Ž h . G Ž z .
for every z g L0 , h g H. Thus
: L0 ª End H Ž V . * s GL r Ž E . ,
z ¬ G Ž z . Ž z .
y1
y1
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is a homomorphism. If ␥ is an element of order l in the cyclic l-group L0 , then a homomorphism from L0 is injective if and only if it does not kill ␥ . It follows that is injective, since G is not injective on L0 but is. Write 噛L0 s l t. Then r G Ž l t . s Ž l y 1 . l ty1 G l y 1.
Ž 4.
Since 噛L s 噛kerŽ .噛 Ž L. s l t噛 Ž L., we have, by Ž2., Ž3., and Ž4.,
Ž 噛L . s Ž l t . 噛 Ž L . F rwr2.
Ž 5.
Let
: G ¸ GrH ( L ¨ Sp Ž噛L. Ž OK . be the map from Lemma 3.7. If f is alternating, we are done by taking G [ . Suppose from now on that f is symmetric. In particular, G is an orthogonal representation. The direct sum of 0 and will give the desired embedding if 2 w q Ž 噛L . F 2 d
Ž s dim K Ž V . s rw . .
If l G 5, then by Ž4. and Ž5. we have Ž噛L. F Ž r y 2. w, and we are done. Now suppose l s 3. We will prove that either the desired embedding exists, or d G 6, 9 <噛G, 3 < d, and 3
G m : G ª Aut Ž T , f . = Sp Ž噛L. Ž OK . ª Aut Ž T Ž噛L. , h . . ŽTo see it is injective, suppose x g G has prime order q. If q / l, then x g H; so G Ž x . / I and Ž x . s I, where I denotes the identity. If q s l, then x g kerŽ G . : L0 s L by Lemma 3.8, and x f H s kerŽ .; so G Ž x . s I but Ž x . / I. Thus in both cases, x f kerŽ G m .; so kerŽ G m . is trivial.. Thus we may assume that L / L0 s kerŽ .. So 3 divides 噛 Ž L., which by Ž2. divides w E0 : K x, which divides w s 2 drr. Thus 3 divides d, and w G w E0 : K x G 3. Therefore w s w E0 : K xdim E 0ŽW . G 3 ⭈ 2 s 6 and 2 d s rw G 2 w G 12, so d G 6. Since 1 / kerŽ G . : L0 « L, therefore 噛L cannot be 1 or 3. Thus 9 <噛G. If w K Ž m . : K x is not divisible by 3, then by Ž1., neither is w E0 : K x. By Ž2. we have L s L0 , and we are done as before.
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REFERENCES 1. C. Curtis and I. Reiner, ‘‘Methods of Representation Theory,’’ Vol. II, Wiley, New York, 1987. 2. W. Feit, ‘‘The Representation Theory of Finite Groups,’’ North-Holland, AmsterdamrNew York, 1982. ´ 3. J.-M. Fontaine, Sur la decomposition des algebres de groupes, Ann. Sci. Ecole Norm. Sup. ´ ` 4 Ž1971., 121᎐180. 4. J.-P. Serre, Sur la rationalite d’Artin, Ann. of Math. 72 Ž1960., ´ des representations ´ 405᎐420. 5. J.-P. Serre, ‘‘Linear Representations of Finite Groups,’’ Springer-Verlag, New YorkrHeidelberg, 1977. ŽTranslation of second French edition.. 6. A. Silverberg and Yu. G. Zarhin, Subgroups of inertia groups arising from abelian varieties, J. Algebra 209 Ž1998., no. 1, 94᎐107. 7. A. Silverberg and Yu. G. Zarhin, Polarizations on abelian varieties and self-dual l-adic representations of inertia groups, available at http:rrwww.math.uiuc.edurAlgebraicNumber-Theoryr0171. ŽTo appear in Comp. Math..
Symplectic Representations of Inertia Groups1 A. Silverberg Mathematics Department, Ohio State Uni¨ ersity, 231 W. 18 A¨ enue, Columbus, Ohio 43210 E-mail: [email protected]
and Yu. G. Zarhin Mathematics Department, Pennsyl¨ ania State Uni¨ ersity, Uni¨ ersity Park, Pennsyl¨ ania 16802, and Institute for Mathematical Problems in Biology, Russian Academy of Sciences, Pushchino, Moscow Region, 142292, Russia E-mail: [email protected] Communicated by Walter Feit Received August 31, 2000
1. INTRODUCTION AND NOTATION The aim of this paper is to study finite inertia subgroups of symplectic groups over the field Q l of l-adic numbers. A finite group is called an inertia group with respect to a given prime p / l if it is a semi-direct product of a finite normal p-subgroup and a cyclic p⬘-group. These groups are exactly the inertia groups of finite Galois extensions of discrete valuation fields of residue characteristic p. The study of semistable reduction of abelian varieties over such fields leads naturally to certain finite inertia subgroups of the symplectic group Sp2 g ŽQ l . w6x. In w7x we constructed examples for every odd prime l of inertia subgroups of Sp2 g ŽQ l ., which are not conjugate, even in GL 2 g ŽQ l ., to a subgroup of Sp2 g ŽZ l .. However, it turns out Žand this is the main result of this paper. that every 1 Silverberg thanks NSA and the Alexander-von-Humboldt Stiftung for financial support, and H. Lange and the Mathematics Institute of the University of Erlangen for their hospitality. Zarhin thanks NSF for financial support. We thank W. Feit for suggesting that we include a discussion of the case l s 3.
400 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
SYMPLECTIC REPRESENTATIONS OF INERTIA GROUPS
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finite inertia subgroup of Sp2 g ŽQ l . is isomorphic to a subgroup of Sp2 g ŽZ l ., if l ) 3. See w3, 4x for a study of representations of inertia groups in characteristic 0. Throughout this paper l is an odd prime, d is a positive integer, K is a field that is an unramified finite extension of Q l , and G is a finite group that is a semi-direct product G s HL of a normal l⬘-subgroup H and a cyclic l-group L. If G is a finite inertia group for some prime p, then G is of this form for every prime l / p Žsee Lemmas 3.2 and 3.3 of w7x.. We write Sp2 d Ž R . for the group of 2 d = 2 d symplectic matrices over a ring R. If E is a field that is a finite extension of Q l , let OE denote the ring of integers. Let n denote a primitive nth root of unity in K. The following theorem is the main result of this paper. THEOREM 1.1 ŽEmbedding Theorem.. If there is an embedding G ¨ Sp2 d Ž K ., and l G 5, then there is an embedding G ¨ Sp2 d Ž OK .. Let m denote the least common multiple of the orders of the elements of H. For l s 3, we can show: THEOREM 1.2 ŽEmbedding Theorem Bis.. embedding G ¨ Sp2 d Ž K .. If
Suppose l s 3, and there is an
Ža. d F 5, or Žb. 9 ¦ 噛G, or Žc. 3 ¦ d and the natural representation of G on K 2 d is irreducible, or Žd. 3 ¦ w K Ž m . : K x Ž e. g., K s K Ž 噛H .., then there is an embedding G ¨ Sp2 d Ž OK .. We will make use of the following result, which we prove in Section 2. Our proof was inspired by Sect. 17.6 in w5x and the proof of Lemma 1.1 in Sect. 1 of Chap. X in w2x. THEOREM 1.3 ŽExtension Theorem.. Suppose that W is a finite dimensional K-¨ ector space, f : W = W ª K is a non-degenerate alternating Ž resp., symmetric. K-bilinear form, and : H ª AutŽW, f . is a group homomorphism that makes W into a simple K w H x-module. Assume that, for e¨ ery g g G, the representation of H
g : H ª Aut Ž W . ,
h ¬ Ž ghgy1 .
is isomorphic to . Suppose T is a H-stable OK -lattice in W. Then
: H ª Aut Ž T , f . ; Aut Ž W , f . can be extended to a homomorphism
G : G ª Aut Ž T , f . ; Aut Ž W , f . .
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We will say that two bilinear forms have the same parity if they are either both symmetric or both alternating. Let denote a uniformizer of OK . 2. PROOF OF THE EXTENSION THEOREM Since the H-module W is simple and H is an l⬘-group, the H-module Tr T is also simple. Replacing f by i f if necessary, we may assume that f ŽT, T . s OK and f : T = T ª OK is perfect Žsince the simplicity of Tr T implies that the induced non-zero pairing on Tr T is non-degenerate.. Let E denote the centralizer of H in End K ŽW .. By Cor. 1 on p. 179 of w3x, K w H x is decomposable Ži.e., splits into a direct sum of matrix algebras over fields.. Therefore, since W is a simple K w H x-module, E is a field that is a finite extension of K. It follows from Theorem 74.5 Žespecially a description of the center K in Žii.. of w1x that E is the field of definition of a certain character of H; in particular, E : K Ž m . : K Ž 噛H . .
Ž 1.
Thus ErK is unramified, since 噛H is prime to l. Therefore ErQ l is also unramified, since KrQ l is unramified. The simplicity of the H-module Tr T implies that the H-stable OK sublattice OE T of W is of the form aT with a g K * Žsee Exercise 15.2 of w5x.. Thus T s ay1 Ž OE T . is OE -stable; i.e., T is an OE -lattice in the E-vector space V. It follows from the Jacobson density theorem that the image of K w H x in End K ŽW . is End E ŽW .. In other words, End E ŽW . is the K-vector subspace of End K ŽW . generated by the Ž h. for h g H. By the non-degeneracy of f, there is an involution u ¬ u⬘ of End K W . characterized by f Ž ux, y . s f Ž x, u⬘ y .
᭙ x, y g W . y1 .
By the H-invariance of f, Ž h.⬘ s Ž h ᭙h g H. Thus the involution u ¬ u⬘ sends End E ŽW . into itself, and therefore sends E, the center of End E ŽW ., into itself. Let E0 s u g E N u⬘ s u4 . Then K : E0 : E. Either E s E0 or ErE0 is a quadratic extension. Let c be a generator of L. The homomorphisms
: H ª Aut O K Ž T . ,
c : H ª Aut O K Ž T .
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define OK w H x-module structures on T in such a way that the corresponding K w H x-modules are isomorphic. Since H is an l⬘-group, the corresponding OK w H x-modules are isomorphic Žsee Sects. 14.4 and 15.5 of w5x.; i.e., there exists A g Aut O KŽT . such that
Ž chcy1 . s A Ž h . Ay1
᭙h g H.
Then
Ž c i hcyi . s Ai Ž h . Ayi
᭙h g H , i g Z.
Since c 噛L s 1, we have A噛L g OEU . Further, A End E ŽW . Ay1 s End E ŽW .. Since E is the center of End E ŽW ., AEAy1 s E. In other words, defining Ž c .Ž u. s AuAy1 for u g E induces a homomorphism
: L ª Gal Ž ErK . . Since w E : E0 x divides 2 and l is odd, the composition L ª GalŽ ErK . ¸ GalŽ E0rK . has the same kernel as . Thus 噛 Ž L . divides w E0 : K x .
Ž 2.
Let f AŽ x, y . s f Ž Ax, Ay . for x, y g T. For all h g H, f A Ž Ž h . x, Ž h . y . s f Ž A Ž h . x, A Ž h . y . s f Ž A Ž h . Ay1Ax, A Ž h . Ay1Ay . s f Ž Ž chcy1 . Ax, Ž chcy1 . Ay . s f A Ž x, y . , since chcy1 g H and f is H-invariant. Thus f A is H-invariant and of the same parity as f. Therefore there exists a g E0U such that f Ž Ax, Ay . s f Ž ax, y . s f Ž x, ay . ᭙ x, y g W . Since A g AutŽT . and f : T = T ª OK is perfect, we have a g AutŽT .. This implies easily that a g OEU0 . Let [ Ž c . g GalŽ ErK .. Then aA s A y1 Ž a.. There exists a1 g OEU0 such that a y1 Ž a . s a12 . ŽIndeed, let be a uniformizer for OE . Then y1 Ž a. ' a l Žmod . for 0 some non-negative integer j. Since l is odd, l j q 1 is even, so a y1 Ž a. is a square modulo . Thus a y1 Ž a. is a square, since all elements of OE 0 congruent to 1 modulo are squares.. j
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For all x, y g T, f Ž Ž A2 a1y1 . x, Ž A2 a1y1 . y . s f Ž aAa1y1 x, Aa1y1 y . s f Ž A y1 Ž a . a1y1 x, Aa1y1 y . s f Ž a y1 Ž a . a1y1 x, a1y1 y . s f Ž x, y . . Let A1 s A2 a1y1 g Aut O KŽT .. Then f is A1-invariant, detŽ A1 . s 1, and conjugation by A1 coincides with conjugation by c 2 . Thus A噛L 1 s bI is a scalar operator of determinant 1 on the E0-vector space W. Therefore b g E0U is a root of unity, i.e., b s 1, where is the number of roots of unity in E0 . Since E0 : E, the extension E0rQ l is unramified, so l does not divide . Letting B [ A1, then conjugation by B coincides with conjugation by b1 [ c 2 , and B 噛L s I. Since b1 is a generator of L, sending b1 to B defines the desired extension G of . 3. LEMMAS FOR THE EMBEDDING THEOREMS Assume from now on that we are in the setting of the embedding theorems. Therefore there exist a 2 d-dimensional K-vector space V, a non-degenerate alternating K-bilinear form e: V = V ª K , and a faithful symplectic representation
: G ¨ Aut Ž V , e . . Clearly, to prove the embedding theorems it suffices to show the existence of an embedding G ¨ Sp2 d⬘Ž OK . for some d⬘ F d. PROPOSITION 3.1. Suppose that V is simple as a G-module but not as an H-module. Then either Ži. the H-module V is isomorphic to W r for some simple H-module W and some r ) 1, or Žii. there exist a normal subgroup G 1 of G, and a simple symplectic G 1-module V1 which is a K-¨ ector space of e¨ en dimension 2 d1 s 2 drw G : G 1 x, such that H : G 1 / G and such that if g 1 is a non-identity element of G 1 , then there exists g g G such that gg 1 gy1 is not in the kernel of G 1 ª AutŽ V1 .. Further, d is di¨ isible by l and d1 F drl. Moreo¨ er, if l 2 does not di¨ ide 噛G then G 1 s H. n Proof. We follow the proof of Prop. 24 of w5x. Let V s [is1 Vi be the canonical decomposition of the restriction of to H into a direct sum of
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isotypic representations. Since the G-module V is simple, G permutes the Vi transitively. If V is some Vi , then the H-module V is isotypic and Ži. holds. Assume from now on that Ži. does not hold. Let G 1 s s g G < s Ž V1 . s V1 4 ; G. Then H : G 1 / G. Since G 1 contains H, it is normal in G. Thus for every Vi , G 1 s s g G < s Ž Vi . s Vi 4 ; G. Every Vi is a simple G 1-module, because V is a simple G-module. The kernels of the natural maps G 1 ª AutŽ Vi . have trivial intersection and are conjugate in G. It follows that if g 1 is a non-identity element of G 1 , then it has a conjugate which does not lie in the kernel of G 1 ª AutŽ V1 .. Since w G : G 1 x divides w G : H x, it is an l-power, and therefore odd. Thus n is odd, so at least one of the simple G 1-modules Vi is self-dual. This implies easily that all the Vi are self-dual. Suppose V1 is not symplectic. Then none of the Vi are symplectic. Since Vi is a simple G 1-module, every G 1-invariant alternating bilinear form on Vi is zero. Since the Vi are mutually non-isomorphic simple G 1-modules, every G 1-invariant bilinear pairing between Vi and Vj for i / j induces the zero map Vi ª VjU Žs Vj ., and therefore is zero. Therefore, every G 1-invariant alternating bilinear n form on V s [is1 Vi is zero, contradicting that V is symplectic. Thus V1 is symplectic. Since 2 d is divisible by w G : G 1 x, therefore d is divisible by the odd prime l. If l 2 does not divide 噛G then w G : H x s l so G 1 s H. We leave the next lemma as an exercise. LEMMA 3.2. If G 0 is a finite group, V0 is a finite-dimensional K-¨ ector space which is also a faithful K w G 0 x-module, and T0 is a G 0-stable OK -lattice in V0 , then e0 Ž Ž x, f . , Ž y, g . . s g Ž x . y f Ž y . for x, y g T0 and f , g g T0U [ Hom O K Ž T0 , OK . defines a perfect alternating G 0-in¨ ariant form on T0 [ T0U , and induces a natural embedding G 0 ¨ AutŽT0 [ T0U , e0 .. LEMMA 3.3. If Theorem 1.1 Ž resp., Theorem 1.2. holds for all irreducible faithful symplectic representations of dimension F 2 d o¨ er K of all quotients of G, then it holds for all faithful symplectic representations of G o¨ er K of dimension F 2 d.
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Proof. The G-module V splits into a direct sum of G-modules V ⬘ such that the restriction of e to V⬘ is non-degenerate, and either V ⬘ is simple or V⬘ s V0 [ V0U , where V0 is simple. In the latter case choose a G-stable OK -lattice T0 in V0 and apply Lemma 3.2. LEMMA 3.4. Suppose V is simple as both a G-module and an H-module. Suppose T is a G-stable OK -lattice in V, and choose i g Z so that i eŽT, T . s OK . Then i e: T = T ª OK is a perfect G-in¨ ariant alternating bilinear form and induces an embedding G ¨ Aut Ž T , i e . ( Sp2 d Ž OK . . Proof. Let e: Tr T = Tr T ª OK r OK be the non-zero pairing induced by i e. Its kernel is an H-submodule of Tr T, so is zero by the H-simplicity of Tr T ; i.e., e is non-degenerate. By Nakayama’s lemma, i e is perfect. LEMMA 3.5. Suppose G 0 is a finite group and G 1 is a normal subgroup of G 0 . Suppose there exist a free OK -module T1 of rank 2 d1 , an alternating perfect form e1: T1 = T1 ª OK , and a homomorphism f : G 1 ª AutŽT1 , e1 . such that whene¨ er g 1 is a non-identity element of G 1 there exists g g G such that gg 1 gy1 f kerŽ f .. Then there exist a free OK -module T of rank 2 d1w G 0 : G 1 x, an alternating perfect form e: T = T ª OK , and an injecti¨ e homomorphism : G 0 ¨ AutŽT, e .. Proof. Let T s u: G 0 ª T1 N u Ž xs . s sy1 u Ž x .
᭙s g G 1 , ᭙ x g G 0 4 ,
choose a section p: G 0rG 1 ª G 0 , and let e Ž u, ¨ . s
Ý
␥gG 0rG 1
e1 Ž u Ž p Ž ␥ . . , ¨ Ž p Ž ␥ . . .
for u, ¨ g T .
Note that e is independent of the choice of section p. Define a homomorphism : G 0 ª AutŽT, e . by Ž g .Ž u.Ž x . s uŽ gy1 x . for g g G 0 , u g T, x g G 0 . Then the desired conditions are all satisfied. COROLLARY 3.6. Suppose G 0 is a finite group and G 1 is a normal subgroup of G 0 . Suppose there exists an injecti¨ e homomorphism G 1 ¨ Sp 2 d 1Ž OK .. Then there exists an injecti¨ e homomorphism G 0 ¨ Sp2 d 1wG 0 : G 1 xŽ OK .. LEMMA 3.7. If ⌳ is a finite cyclic group of order l m , then there exists an injecti¨ e homomorphism ⌳ = "14 ¨ Sp Ž l m .ŽZ l ..
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Proof. Let C denote the subgroup of ⌳ of order l. By Lemma 3.7 of w7x there is an injective homomorphism C = "14 ¨ Sp ly1ŽZ l .. Now apply Corollary 3.6 to G 1 s C = "14 : ⌳ = "14 s G 0 . LEMMA 3.8. Suppose S is a normal l-subgroup of G. Then S : L, and S is central in G. If 噛S ) 1, then e¨ ery element of order l in G is contained in S. Proof. By the conjugacy of Sylow-l-subgroups, S : L. The natural map  : L ¨ G ¸ GrH ( L is the identity map. Suppose x g S : L and g g G. Then gxgy1 g S : L. Since L is commutative,  Ž x . s  Ž gxgy1 .. Thus x s gxgy1 , so S is central in G. Suppose y g G has order l. Then y is conjugate to some w g L of order l. Suppose 噛S ) 1. Since L is a cyclic l-group, w g S. Since S is normal, y g S.
4. PROOF OF THE EMBEDDING THEOREMS Note that the embedding theorems are trivial for d s 1. By Lemmas 3.3 and 3.4, we may assume from now on that the G-module V is simple and the H-module V is not simple. If Žii. of Proposition 3.1 holds, then we may induct on d, applying Lemma 3.5. Note that if l s 3, then Ža. d1 F dr3 and therefore d1 s 1 if d F 5; Žb. if 噛G is not divisible by 9 then G 1 s H; Žc. 3 divides d. By Proposition 3.1, we may now assume that V ( W r for some simple H-module W, where r ) 1. It follows that W is self-dual; i.e., there is an H-invariant non-degenerate alternating or symmetric K-bilinear form f : W = W ª K. We may choose an H-stable lattice T in W and Žreplacing f by i f for suitable i, if necessary . we may assume that f : T = T ª OK is perfect. Let w s dim K Ž W . . Assume first that w s 1. Then H ; Aut K ŽW . s K *. Since V s W r and H : G : SpŽ V ., we have H : "14 , and Lemma 3.7 gives the desired embedding. Assume from now on that w G 2. Let
: H ¨ Aut Ž W , f . ; Aut Ž W . be the injective homomorphism defining the H-module structure on W. Since H is normal in G, the subspace gW ; V is an H-submodule of V
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for every g g G. The natural representation H ª AutŽ gW . is isomorphic to the representation
g :
H ª Aut Ž W . ,
h ¨ Ž ghgy1 . .
On the other hand, since V ( W r as H-modules, therefore gW ( W as H-modules. ŽIndeed, every H-submodule in V is isomorphic to a direct sum of copies of W, and w s dimŽ gW ... Now apply the extension theorem and extend to a homomorphism
G : G ª Aut Ž T , f . : Aut Ž W , f . . Since f is perfect, we have T * s T. Lemma 3.2 gives a perfect alternating G-invariant form e0 on T [ T, and we let
0 : G ª Aut Ž T [ T , e0 . ( Sp2 w Ž OK . be the direct sum of two copies of G . Suppose G is injective. If f is alternating Žresp., symmetric., then G Žresp., 0 . gives the desired embedding. Now assume G is not injective. Then kerŽ G . l H is the identity, so kerŽ G . is a normal l-subgroup of G. By Lemma 3.8, kerŽ G . : L. Now retain the notation from the proof of the extension theorem. Suppose dim E 0ŽW . s 1. Then H ; E0U , and for h g H, f Ž x, y . s f Ž hx, hy . s f Ž h 2 x, y . ᭙ x, y g W. Thus h 2 s 1, i.e., H : 1, y14 , and we are done by Lemma 3.7. Now assume that dim E 0ŽW . G 2. Then
w E0 : K x F wr2.
Ž 3.
Let L0 s ker Ž . s x g L: G Ž x . y s y G Ž x . ᭙ y g E 4 . Then kerŽ G . : L0 : L. Let G 0 s HL0 , a normal subgroup of G. Since V s W r , the restriction of g to G 0 can be viewed as a homomorphism
G : G 0 ª Aut E Ž W . ; Aut E Ž V . . Then s G on H ; G 0 , and
Ž z . Ž h. Ž z .
y1
s G Ž z . G Ž h . G Ž z .
y1
s G Ž z . Ž h . G Ž z .
for every z g L0 , h g H. Thus
: L0 ª End H Ž V . * s GL r Ž E . ,
z ¬ G Ž z . Ž z .
y1
y1
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is a homomorphism. If ␥ is an element of order l in the cyclic l-group L0 , then a homomorphism from L0 is injective if and only if it does not kill ␥ . It follows that is injective, since G is not injective on L0 but is. Write 噛L0 s l t. Then r G Ž l t . s Ž l y 1 . l ty1 G l y 1.
Ž 4.
Since 噛L s 噛kerŽ .噛 Ž L. s l t噛 Ž L., we have, by Ž2., Ž3., and Ž4.,
Ž 噛L . s Ž l t . 噛 Ž L . F rwr2.
Ž 5.
Let
: G ¸ GrH ( L ¨ Sp Ž噛L. Ž OK . be the map from Lemma 3.7. If f is alternating, we are done by taking G [ . Suppose from now on that f is symmetric. In particular, G is an orthogonal representation. The direct sum of 0 and will give the desired embedding if 2 w q Ž 噛L . F 2 d
Ž s dim K Ž V . s rw . .
If l G 5, then by Ž4. and Ž5. we have Ž噛L. F Ž r y 2. w, and we are done. Now suppose l s 3. We will prove that either the desired embedding exists, or d G 6, 9 <噛G, 3 < d, and 3
G m : G ª Aut Ž T , f . = Sp Ž噛L. Ž OK . ª Aut Ž T Ž噛L. , h . . ŽTo see it is injective, suppose x g G has prime order q. If q / l, then x g H; so G Ž x . / I and Ž x . s I, where I denotes the identity. If q s l, then x g kerŽ G . : L0 s L by Lemma 3.8, and x f H s kerŽ .; so G Ž x . s I but Ž x . / I. Thus in both cases, x f kerŽ G m .; so kerŽ G m . is trivial.. Thus we may assume that L / L0 s kerŽ .. So 3 divides 噛 Ž L., which by Ž2. divides w E0 : K x, which divides w s 2 drr. Thus 3 divides d, and w G w E0 : K x G 3. Therefore w s w E0 : K xdim E 0ŽW . G 3 ⭈ 2 s 6 and 2 d s rw G 2 w G 12, so d G 6. Since 1 / kerŽ G . : L0 « L, therefore 噛L cannot be 1 or 3. Thus 9 <噛G. If w K Ž m . : K x is not divisible by 3, then by Ž1., neither is w E0 : K x. By Ž2. we have L s L0 , and we are done as before.
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REFERENCES 1. C. Curtis and I. Reiner, ‘‘Methods of Representation Theory,’’ Vol. II, Wiley, New York, 1987. 2. W. Feit, ‘‘The Representation Theory of Finite Groups,’’ North-Holland, AmsterdamrNew York, 1982. ´ 3. J.-M. Fontaine, Sur la decomposition des algebres de groupes, Ann. Sci. Ecole Norm. Sup. ´ ` 4 Ž1971., 121᎐180. 4. J.-P. Serre, Sur la rationalite d’Artin, Ann. of Math. 72 Ž1960., ´ des representations ´ 405᎐420. 5. J.-P. Serre, ‘‘Linear Representations of Finite Groups,’’ Springer-Verlag, New YorkrHeidelberg, 1977. ŽTranslation of second French edition.. 6. A. Silverberg and Yu. G. Zarhin, Subgroups of inertia groups arising from abelian varieties, J. Algebra 209 Ž1998., no. 1, 94᎐107. 7. A. Silverberg and Yu. G. Zarhin, Polarizations on abelian varieties and self-dual l-adic representations of inertia groups, available at http:rrwww.math.uiuc.edurAlgebraicNumber-Theoryr0171. ŽTo appear in Comp. Math..