Some Cyclo-quaternionic Lattices

199, 472]498 Ž1998. JA977163

JOURNAL OF ALGEBRA ARTICLE NO.

Some Cyclo-quaternionic Lattices Gabriele Nebe* Lehrstuhl B fur ¨ Mathematik, RWTH Aachen, Templergraben 64, D-52062, Aachen, Germany Communicated by E¨ a Bayer-Fluckiger Received August 20, 1996

The faithful lattices of rank 2Ž p y 1. of the groups SL 2 Ž p . are described. For small primes p these and related lattices are investigated by computer. In particuQ 1998 lar a new extremal even unimodular lattice of rank 48 is constructed. Academic Press

INTRODUCTION The study of finite integral matrix groups is one source for producing nice lattices. In particular representations of the group PSL2 Ž p . for p ' 3 Žmod 4. have been studied in w10x in connection with globally irreducible representations. Gross gives an interpretation of some of the invariant lattices as Mordell]Weil lattices. Since the real Schur index of the faithful rational representations of SL 2 Ž p . of degree 2Ž p y 1. is two, they can be viewed as representations over totally definite quaternion algebras. The present article grew out of the investigation of finite quaternionic matrix groups w13x. The paper is organized as follows: After introducing notation and general arguments the faithful lattices of degree 2Ž p y 1. of SL 2 Ž p . are described. For p ' 1 Žmod 4. these lattices can be constructed as cyclotomic lattices over quaternion algebras, as described in Section 3. In the next section it is shown that some of these lattices coincide with the very dense Mordell]Weil lattices discovered by Elkies and Shioda w23x. The * E-mail address: [email protected]. 472 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

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concluding section deals with those SL 2 Ž p .-lattices for which the endomorphism ring is a maximal order in the quaternion algebra with center Qw p x ramified only at the two infinite places. This article was written during a fellowship at the university of Bordeaux financed by the Deutsche Forschungsgesellschaft. I am grateful to both organizations. Finally I want to thank K. Belabas for his help in solving problems with PARI.

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1. GENERAL NOTATION AND PROPERTIES Let G F GLnŽQ. }following P. Hall the symbol F is used to denote ‘‘is a subgroup of’’}be a finite subgroup of the group of rational invertible n = n matrices. There are two important sets that describe the G-invariant Euclidean lattices in Q n : the set L Ž G . [  L F Q n N L is a full Z-lattice in Q n and Lg F L for all g g G 4 of G-invariant lattices and the vector space of G-invariant quadratic forms F Ž G . [  F g Mn Ž Q . N F s F t r and gFg t r s F for all g g G 4 . F Ž G . is a subspace of the vector space of the rational symmetric n = n matrices and contains the non-empty subset F) 0 Ž G . consisting of the positive definite invariant quadratic forms. The space F Ž G . can be viewed as the space of symmetric homomorphisms from the natural representation g ¬ g to its contragredient representation g ¬ Ž gy1 . t r. Its dimension can be calculated by decomposing the representation over the reals. The representation is irreducible over R if and only if dim Q Ž F Ž G .. s 1. In this case the matrix group G is called uniform, because then there is up to scalar multiples a unique G-invariant quadratic form. If L g L Ž G . is a G-invariant lattice and F g F) o Ž G . then the dual lattice La [ La , F [  ¨ g Q n N lF¨ t r g Z for all l g L4 is again G-invariant, i.e., La g L Ž G .. Two lattices Ž L, F . and Ž L9, F9. are called isometric if there is an isometry T g GLnŽQ. with LT s L9 and TF9T t r s F. The automorphism group AutŽ L, F . is the group of all isometries of Ž L, F . with itself. Isometries of definite lattices in medium-sized dimensions may be calculated using the algorithm described in w18x. Ž L, F . is called integral if L : La and unimodular if L s La . If p g N, the rescaled lattice Ž L, pF . of the Euclidean lattice Ž L, F . is denoted by

474

GABRIELE NEBE

Ž p.

L. Generalizing the notion of unimodularity, a lattice Ž L, F . is called p-modular if the rescaled dual lattice Ž p. La s Ž La , pF . is isometric to Ž L, F .. For primes p Žor p s 1. with k 1 [ 24rŽ p q 1. g N it follows from w19x that the minimum minŽ L, F . [ min l t r Fl N 0 / l g L4 of an even p-modular lattice Ž L, F . of dimension n is at most 2w nrŽ2 k 1 .x q2. An even p-modular lattice is called extremal if its minimum equals 2w nrŽ2 k 1 .x q 2. One further measure of the quality of a Euclidean lattice Ž L, F . is its Hermite parameter g Ž L, F . [ minŽ L, F .rŽdetŽ L, F ..1r n where detŽ L, F . is the determinant of L, i.e., the determinant of a Gram matrix of L Žwith respect to F . and n [ dimŽ L. is the dimension of L. This is related to the density of the lattice. The lattice Ž L, F . is said to be primiti¨ e if the ideal generated by the values of the bilinear form F on L is Z. In abuse of notation a subgroup G F GLnŽQ. is called irreducible if the natural representation is irreducible. By Schur’s Lemma G is irreducible if and only if the endomorphism algebra EndŽ G . [  x g MnŽQ. N xg s gx for all g g G4 is a skewfield. LEMMA 1.1. Let G be an irreducible subgroup of GLnŽQ. and assume that dimŽ F Ž G .. F 2. Let F g F) 0 Ž G ., L g L Ž G .. Ži. If G is uniform then C [ EndŽ G . is isomorphic to either Q, an imaginary quadratic number field, or a positi¨ e definite quaternion algebra with center Q. If dimŽ F Ž G .. s 2 then C is either an abelian number field with maximal real subfield of degree 2 o¨ er Q or a positi¨ e definite quaternion algebra o¨ er a real quadratic number field. The mapping x ¬ xF is an isomorphism from the maximal real subfield of the center of C to F Ž G .. Žii. The anti-automorphism i : C ª C; x ¬ Fx t r Fy1 is independent of the choice of F g F) 0 Ž G .. If C is commutati¨ e then i induces the complex conjugation on C. If C is a quaternion algebra then i is the canonical in¨ olution of C. Žiii. If the endomorphism ring M [ End LŽ G . [  x g C N Lx : L4 is stable under the map i of Žii. then M is also the endomorphism ring of the dual lattice La, F . The condition holds if M contains the maximal order of the fixed field of the restriction of i to the center of C and in particular if G is uniform. Proof. Ži. This is well known Žcf., e.g., w14, Remark ŽII.1.x.. Žii. Let x g C, g g G. Then g i Ž x . gy 1 s gFx t r F y 1 gy 1 s yt r t r t r y1 Fg x g F s i Ž x ., and hence i Ž x . g C. Moreover i 2 is the identity on C and i Ž xy . s i Ž y . i Ž x . for all x, y g C. Since for any F9 g F Ž G . the matrix F9Fy1 lies in the center of C, the map i is independent of F.

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Moreover, x g C is a fixed point of i if and only if xF is symmetric. Therefore i Ž x . s x m x lies in the maximal real subfield of the center of C. Galois theory resp. w20, Theorem 8.11.2x now implies Žii.. Žiii. Let x g M, l g L, and ¨ g La . Then lF Ž ¨ x . t r s lFx t r Fy1 F¨ t r g Z since Fx t r Fy1 s i Ž x . g M. So ¨ x g La and therefore M : End La Ž G .. Equality follows since Laa s L. Now assume that M contains the maximal order of the fixed field of the restriction of i to the center of C. If C is a quaternion algebra and tr denotes the reduced trace tr : C ª ZŽ C . then i Ž x . s trŽ x . y x for all x g C. Since trŽ M . lies in the maximal order of the fixed field of the restriction of i to the center of C, M is stable under i . If C is commutative then similarly i Ž M . s M. If G is uniform, the fixed field of i is Q and i Ž M . s M, because M contains Z. If one has a subalgebra Q : H of a quaternion algebra H , one might ask: What happens with the maximal orders? Clearly, if N is a maximal order of Q then there is a maximal order M of H containing N. The case whre Q itself is a quaternion algebra is much easier than the one where Q is abelian, because then ZŽ M . N is of finite index in M. Restricting oneself to the situation occurring in this paper, one finds: PROPOSITION 1.2.

Let 1 - p g N be square-free, H ( Q 'p , `, ` the totally

definite quaternion algebra o¨ er Qw p x only ramified at the two infinite places, Q : H a quaternion algebra with center Q, and N a maximal order of Q. There are exactly 2 s different maximal orders of H containing N where s is the number of ramified finite primes of Q that do not ramify in Qw p x.

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Proof. Let M be a maximal order of H containing N. Consider M as a M y M-bi-lattice. Then the maximal orders of H that contain N are the endomorphism rings End LŽ M . of the M y N-lattices L in H . These lattices L are of the form M Ap , where A is a non-zero ideal in the center Qw p x of H and p a non-zero two-sided ideal of N, because all two-sided M-ideals come from central ideals. Let p be a two-sided ideal of N and M9 [ End M p Ž M .. The M y M9-lattices in H are the lattices M Ap , where A runs through the non-zero ideals of Qw p x. Let r be a rational prime that ramifies in Q and pr the maximal two-sided ideal in N containing r. If r ramifies in Qw p x then rR s A 2r where R is the maximal order of Qw p x and A r 1 ] R. Completing M and N at A r , one sees that Mpr s M A r . Therefore M Appr s M A A r p is again an M y M9-lattice for every non-zero ideal A in Qw p x.

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GABRIELE NEBE

If r does not ramify in Qw p x then r is inert Žsince Q : H . and therefore Mppr is not an M y M9-lattice. The proposition follows since Mpr2 s r M.

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Con¨ ention. If C is a subalgebra of MnŽQ. isomorphic to a number field K, C and K are identified and the matrices in C are referred to as algebraic numbers. In particular an element x g C with x 2 s p is denoted as p and z k also means a primitive kth root of unity in C. The notation of w17, Proposition II.4x is used to describe finite matrix groups whose natural representation is close to a tensor product. In particular if G F GLnŽQ. and H is a subgroup of the unit group of EndŽ G ., then G( H denotes the group generated by G and H. A 2 on top Žsometimes followed by a natural number in brackets. indicates a certain extension of this group by a group of order 2.

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2. THE SL 2 Ž p .-LATTICES OF RANK 2Ž p y 1. The aim of this section is to describe the faithful rational lattices of rank 2Ž p y 1. of the group SL 2 Ž p ., the group of the 2 = 2 matrices over the field with p elements of determinant 1, where p is a prime number. The lattices of SL 2 Ž p . of rank p y 1 Žwhich belong to non-faithful rational representations . have been described in w17, Chap. V Žb.x. They are cyclotomic lattices strongly related to Craig’s lattices Žcf. w5, p. 222x.. The lattices of degree 2Ž p y 1. turn out to have an interpretation as cyclotomic lattices over quaternion algebras. To fix notation let p ) 3 be a prime and z pq 1 g C denote a primitive Ž p q 1.st root of unity. Let A g SL 2 Ž p . be an element of order p q 1. For 1 F i F Ž p q 1.r2 there is a unique character Qi of degree p y 1 of i . Žcf. w21x., which is the restriction of SL 2 Ž p . with Qi Ž A. s yŽ z pq1 q zyi pq1 an irreducible character of SL 2 Ž p ..2. It is irreducible if i / Ž p q 1.r2 and faithful if i is odd. Immediately from the character table of SL 2 Ž p ..2 given in w21x Žsee also w7x. one gets the following LEMMA 2.1. The faithful rational ¨ alued characters of degree p y 1 of the group SL 2 Ž p . are the characters QŽ pq1.r i for i g  2, 4, 64 di¨ iding p q 1 but not Ž p q 1.r2. By w9, Theorem 6.1x, all these faithful characters QŽ pq1.r i have real Schur index 2. They lead to rational representations D i of degree n [

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2Ž p y 1. of G [ SL 2 Ž p .. A closer analysis of this theorem yields the following If p ' 1 Žmod 4. then EndŽ D 2 Ž G .. ( Q 'p , `, ` is isomorphic to the quaternion algebra with center Qw p x only ramified at the two infinite places. Žii. If p ' 5 Žmod 12. then EndŽ D 6 Ž G .. ( Q`, 3 is isomorphic to the quaternion algebra with center Q ramified at 3 and infinity. Žiii. If p ' 3 Žmod 8. then EndŽ D 4 Ž G .. ( Q`, 2 is isomorphic to the quaternion algebra with center Q ramified at 2 and infinity. LEMMA 2.2. Ži.

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The modular constituents of the Qi are already described in w17x and may be obtained from w4x. The only reference I found for the invariant bilinear form of Ži. is w11, Theorem VI.1.1x where the inverse is described:

ˆ i denote the restriction of the character LEMMA 2.3. Let r be a prime and Q Qi to the r-regular classes, i.e., the conjugacy classes of elements of order not di¨ isible by r, of G s SL 2 Ž p .. Ži. The irreducible Fp G-modules are Fp w x, y x s , the spaces of homogeneous polynomials of degree s Ž0 F s F p y 1. with G-action defined by x syj y j Ž ac db . [ Ž ax q by . sy j Ž cx q dy . j for Ž ac db . g G, 0 F j F s. Fp w x, y x s is an absolutely irreducible Fp G-module of dimension s q 1. Its character will be denoted by b sq 1. Up to scalar multiples it has a unique G-in¨ ariant bilinear form Ž,.. The latter is defined by

Žx

syj

j

y ,x

syi

y

i

. [ Ž y1.

sy j

s j

ž/

y1

d i , syj

Ž1 F i, j F s. .

It is alternating if s is odd and symmetric if s is e¨ en. ˆ i s biy1 q b pyi . Žii. If r s p then Q ˆ Ž pq1.r i is Žiii. If r / p let i g  2, 4, 64 di¨ ide p q 1. The character Q ˆ Ž pq1.r i s Q ˆ Ž pq1.r2 s reducible if and only if i s 2 r or i s 2. In these cases Q ˆ q Q9 ˆ Ž where QŽ pq1.r2 \ Q q Q9. decomposes as the sum of 2 characters Q of degree Ž p y 1.r2. Using this information one finds the following three theorems: THEOREM 2.4. Let p ' 1 Žmod 4., G [ SL 2 Ž p ., and L g L Ž D 2 Ž G ... Then the endomorphism ring End LŽ D 2 Ž G .. \ M is a maximal order of the endomorphism algebra C [ EndŽ D 2 Ž G .. ( Q 'p , `, ` . A system of representati¨ es of isomorphism classes of M D 2 Ž G .-lattices is gi¨ en by LA, where 0 / A runs through a set of representati¨ es of the ideal classes of Qw p x s

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GABRIELE NEBE

ZŽ C .. There is a unique ideal class, say represented by A 0 , for which there exists a F0 g F) 0 Ž G . such that Ž LA 0 , F0 . is unimodular. Denote this unimodular Euclidean lattice Ž LA 0 , F0 . by L2Ž py1., 2 Ž M .. Proof. By Lemma 2.3 the restrictions of the C-constituents of the natural character of D 2 Ž G . to the r-regular classes are irreducible Brauer characters of G for all primes r. Since the Sylow-p-subgroups of G are of order p this impies that there is only one genus of D 2 Ž G .-lattices. Therefore M is a maximal order in C. The M D 2 Ž G .-lattices correspond to two-sided ideals of M hence are of the form LA, where 0 / A is an ideal in Qw p x. Choose F g F) 0 Ž D 2 Ž G ... By Lemma 1.1Žiii. the dual lattice La, F is again a M D 2 Ž G .-lattice. Hence La, F s LB for some ideal B in Qw p x. Since by Lemma 1.1Ži. the form aF is symmetric for all a g Qw p x, the dual lattice Ž LA .a, F s LAy1 B for all ideals A of Qw p x. By w26, Theorem Ž10.4.Žb.x the class number of Qw p x is odd. So there is an ideal A 0 of Qw p x and b g Qw p x such that B s A 20 Ž b .. Since the fundamental unit of Qw p x has norm y1 Žcf. w26, Ex. Ž8.3.x. there is a totally positive generator b 0 of Ž b .. Then F0 [ b 0 F g F) 0 Ž D 2 Ž G .. and the lattice Ž LA 0 , F0 . is unimodular. Let Ž LA, xF . Ž x g Qw p x totally positive. be any other unimodular M D 2 Ž G .-lattice. Then LA s Ž LA .a, x F s LB Ay1 xy1. Hence Ay2 B and therefore Ž Ay1 A 0 . 2 is principal. Again since the class number of Qw p x is odd, this implies that the ideal class of A is the one of A 0 .

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Remark 2.5 Žcf. w24x.. With the notation of Theorem 2.4 assume that M contains a maximal order N of the subalgebra Q`, p of C. Then L2Ž py1., 2 Ž M . is invariant under an extension SL 2 Ž p ..2. Proof. By Proposition 1.2, M is the unique maximal order of C containing N. The representation D 2 of G extends to a rational representation D 2 of both extensions G.2, with endomorphism algebra EndŽ D 2 Ž G.2.. ( Q`, p Žwhich may be calculated as the endomorphism algebra of the normalizer of a Sylow p-subgroup in D 2 Ž G.2... Therefore the group D 2 Ž G.2. fixes one of the N D 2 Ž G .-lattices, say L9. Since each D 2 Ž G .-lattice has a maximal order as its endomorphism ring, Proposition 1.2 implies that End L9Ž D 2 Ž G .. s M. Hence L9 is isomorphic to some M D 2 Ž G .-lattice LA. The N D 2 Ž G.2.-sublattices of L9 are of the form L9m or L9mp , where m g Z and p denotes the maximal two-sided ideal of N containing p. Since L9p s L9 p , all these lattices are isomorphic as M D 2 Ž G .-lattices. Now let F g F) 0 Ž D 2 Ž G.2.. be primitive on L9. Then the y1 dual lattice L9a, F is either L9 or L9 p . In the first case Ž L9, F . s y1 L2Ž py1., 2 Ž M . is unimodular, in the second case Ž L9, e p F . s

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L2Ž py1., 2 Ž M . is unimodular for a suitable fundamental unit e g Qw p x. If D 2 Ž G.2. s ² D 2 Ž G ., x : then conjugation with x induces the Galois automorphism on Qw p x. Therefore the group ² D 2 Ž G ., e x : is a subgroup of y1 AutŽ L9, e p F ..

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In the other two cases the group D k Ž SL 2 Ž p .. Ž k s 6 resp. 4. is uniform and the endomorphism algebra has class number 1. THEOREM 2.6. Let p ' 5 Žmod 12. and G [ SL 2 Ž p .. Then the group D 6 Ž G . is uniform. There is a D 6 Ž G .-in¨ ariant lattice L and F g F) 0 Ž G ., such that Ž L, F . \ L2Ž py1., 6 is primiti¨ e of determinant p Ž py5.r3. The endomorphism ring End LŽ D 6 Ž G .. of L is a maximal order M in the endomorphism algebra C [ EndŽ D 6 Ž G .. ( Q`, 3 . If p ) 5 then the two lattices L and La, F represent the isomorphism classes of M D 6 Ž G .-lattices in Q n. The primiti¨ e D 6 Ž G .-lattices ha¨ ing not a maximal order as endomorphism ring are unimodular. If p s 5 then L8, 6 s La8, 6 is unimodular, and hence isometric to E8 . Proof. The irreducible rational representation D 6 remains irreducible when considered over the reals Žcf. Lemma 2.2.. So the matrix group D 6 Ž G . is uniform. Let L g L Ž D 6 Ž G .. be a G-invariant lattice and F the positive definite D 6 Ž G .-invariant form, which is primitive on L. Since Q p C ( M2 ŽQ p . the Q p D 6 Ž G .-module Q p L s V1 [ V2 is a sum of two isomorphic Q p D 6 Ž G .-modules each affording the character QŽ pq1.r6 . Let i s 1, 2 and L i be a Z p D 6 Ž G .-lattice in Vi . The Sylow p-subgroup of G is of order p, so the Z p D 6 Ž G .-sublattices of L i in Vi are linearly ordered. By Lemma 2.3 the irreducible Fp D 6 Ž G .-constituents of L irpL i are absolutely irreducible of degree Ž p y 5.r6 and Ž5 p y 1.r6. Replacing L by a maximal sublattice of p-power index if necessary, one may suppose that Z p L is a direct sum of two isomorphic Z p D 6 Ž G .-lattices. Then Z p M ( M2 ŽZ p .. The Z p M D 6 Ž G .-lattices on Q p L are linearly ordered by inclusion and the Fp M D 6 Ž G . composition factors of LrpL are of dimension 2ŽŽ p y 5.r6. and 2ŽŽ5 p y 1.r6.. By Lemma 1.1Žiii. the dual lattice La, F is also an M-lattice. Therefore, the p-part of the determinant of L is one of p 2Ž py5.r6 or p 2Ž5 py1.r6 . Replacing Ž L, F . by Ž La, F , pF . if necessary, one achieves that the p-part of detŽ L, F . is p Ž py5.r3. ˆ Ž pq1.r6 So it remains to consider the prime 3. As a 3-Brauer character Q ˆ q Q9, ˆ but the character values of Q ˆ Žand Q9 ˆ . generate F 3w p x ( F 9 . sQ So there is up to isomorphism only one Z 3 D 6 Ž G .-lattice in Q 3 L. Therefore M is a maximal order in C and L is invariant under the group D 6 Ž G .( S˜3 F GLnŽQ. generated by D 6 Ž G . and the unit group S˜3 of M. The 3-modular constituents of the natural character of D 6 Ž G .( S˜3 are of degree Ž p y 1.r2, where the lifts of the corresponding character values

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GABRIELE NEBE

generate the biquadratic extension Qw p , z4 x of Q. Since yp is a square in F 3 , the F 3 D 6 Ž G .( S˜3-module Lr3L has two non-isomorphic composition factors LrL9 and L9r3L. The corresponding 3-Brauer characters are complex, so LrL9 is the dual module of L9r3L and therefore 3 does not divide detŽ L, F .. Now let p ) 5. The Fp G-module La, F rL is isomorphic to W1 [ W2 , where Wi ( Fp w x, y x s Ž i s 1, 2. is the Fp G-module of homogenous polynomials of degree s [ Ž p y 11.r6 of Lemma 2.3. Since s is odd the G-invariant bilinear form Ž,. on Wi Ž i s 1, 2. is alternating. Hence the symmetric bilinear form induced by F on La, F rL is ² w 1 q w 2 , wX1 q wX2 : s Ž w Ž w 1 ., wX2 . q Ž w Ž wX1 ., w 2 . Ž wi g Wi , i s 1, 2. where w : W1 ª W2 is a G-isomorphism. The G-invariant subspaces of La, F rL are Mi [  w 1 q w 2 g W1 [ W2 N w Ž w 1 . s iw 2 4 Ž i s 0, . . . , p y 1. and M p [ W1 [  04 . Hence MiH s Mi Ž0 F i F p . and the G-invariant overlattices containing L of index p Ž py5.r6 are unimodular.

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THEOREM 2.7. Let p ' 3 Žmod 8. and G [ SL 2 Ž p .. Then the group D 4 Ž G . is uniform. There is a lattice L g L Ž D 4Ž G .. and F g F) 0 Ž G ., such that Ž L, F . \ L2Ž py1., 4 is primiti¨ e of determinant 2 py 1 p Ž py3.r2 . The endomorphism ring End L 2Ž py 1.,4Ž D 4Ž G .. is a maximal order M in Q`, 2 . The two lattices L and La, F represent the isomorphism classes of MD 4Ž G . lattices in Q n. The primiti¨ e D 4 Ž G .-lattices ha¨ ing not the maximal order as the endomorphism ring ha¨ e determinant 2 py 1. Proof. The proof is analogous to the one of Theorem 2.6. But here one has to consider the prime 2 Žinstead of 3.. Now the unit group of M is SL 2 Ž3.. Adjoining the lifts of the character values of the 2-modular constituents of the natural character of D 4 Ž G .( SL 2 Ž3. one obtains the character field Qw y p , z 3 x. Since Žyp .Žy3. ' 1 Žmod 8. the F 2 D 4Ž G .( SL 2 Ž3.-module Lr2 L has 2 Ždifferent. constituents of degree p y 1. But now, the corresponding 2-Brauer characters are real, which implies that the 2-part of detŽ L, F . is 2 py 1.

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Remark 2.8. If p s 11 then AutŽ L20, 4 . s SL 2 Ž11. ( SL 2 Ž3. acts transitively on the 12 D 4Ž SL 2 Ž11..-sublattices of index 118 in L20, 4 . These sublattices are extremal 2-modular lattices with automorphism group 2. M12 .2 Žcf. w17, Lemma ŽIX.3.x.. 2 Ž2 .

If p s 19 then AutŽ L36, 4 . s SL 2 Ž19. ( SL 2 Ž3. has 2 orbits on the 20 D 4 Ž SL 2 Ž19..-sublattices of index 19 14 in L36, 4 of length 12 resp. 8. Both orbits consist of extremal 2-modular lattices Žof minimum 6.. The automorphism groups are SL 2 Ž19..2 resp. SL 2 Ž19.(C3 .

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481

3. CYCLO-QUATERNIONIC LATTICES In this section an explicit construction for the lattices L2Ž py1., 2 Ž M . and L2Ž py1., 6 for primes p ' 1 Žmod 4. is obtained. To find these lattices one has to construct a representation of a metacyclic group, which is certainly easier than constructing D 2 or D 6 , but involves the problem of solving norm equations in abelian number fields if p ' 1 Žmod 8.. The fundamental observation is that for k s 2 and 6 the restriction of D k to the Borel subgroup B [ "C p .CŽ py1.r2 F SL 2 Ž p . Žthe non-split extension of a cyclic group "C p of order 2 p by the subgroup of index 2 in AutŽ C p .. remains rationally irreducible. Note that Ž p y 1.r2 is even because p ' 1 Žmod 4. in these two cases. Since B has only one rational irreducible faithful representation D the restrictions of the two representations D 2 and D 6 to B coincide for p ' 5 Žmod 12.. The endomorphism algebra C [ EndŽ DŽ B .. is Q 'p , `, ` . The center of C will be identified with Qw p x.

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LEMMA 3.1. Let DŽ B . s ² z, a: F GL2Ž py1.ŽQ., where ² z : 1 ] DŽ B . is the normal p-subgroup of DŽ B ., aŽ py1.r2 s y1, and conjugation with a induces a Galois automorphism of order Ž p y 1.r2 of Qw z x ( Qw z p x. Then ` [ ŽŽ1 y z . a.Ž py1.r4 g NG L 2Ž py 1.ŽQ .Ž DŽ B .. normalizes DŽ B .. Proof. This is straightforward. ` s Ž1 y z .Ž aŽ1 y z . ay1 .Ž a2 Ž1 y . z ay2 . ??? Ž aŽ py1.r4y1 Ž1 y z . a1y Ž py1.r4 . aŽ py1.r4 , where the first Ž p y 1.r4 factors pairwise commute. Since these factors also commute with z, one has z ` s zy1. Moreover a ` s Ž1 y zy1 .Ž1 y z .y1a s yzy1a. Note that y` 2 is the norm of Qw z xrQw p x of Ž1 y z . and hence a totally positive generator of the ideal Ž p . of Qw p x.

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PROPOSITION 3.2. Let M be a maximal order of C, F g F) 0 Ž DŽ B .., and L g L Ž DŽ B .. with End LŽ DŽ B .. s M. Let S be a system of non-zero representati¨ es of the ideal classes of ZŽ C . s Qw p x.

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Ži. The M DŽ B . sublattices of p-power index in L are L \ LŽ0. > L > LŽ2. > ??? > LŽŽ py1.r2. s p L with < LŽ i.rLŽ iq1. < s p 2 for all 0 F i Ž p y 1.r2. The lattices LŽ0. A, . . . , LŽŽ py3.r2. A with A g S form a system of representati¨ es of isomorphism classes of M DŽ B .-lattices in Q 2Ž py1.. Ž1.

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Žii. Let ` be as in Lemma 3.1. If i g  0, . . . , Ž p y 5.r44 then LŽ iqŽ py1.r4. A s LŽ i. ` A and Ž LŽ i. ` A, F . is isometric to Ž LŽ i. A, y` 2 F .. Žiii. There is a unique i 0 g  0, . . . , Ž p y 5.r44 for which the lattices LŽ i 0 . A, A g S, are in¨ ariant under D 2 Ž SL 2 Ž p .. F AutŽ LŽ i 0 . A, F ..

482

GABRIELE NEBE

Replacing L by LŽ i 0 . A 0 for a suitable ideal A 0 of Qw p x and choosing an appropriate F g F) 0 Ž DŽ B .. Ž as in Theorem 2.4. one achie¨ es that Ž L, F . s L2Ž py1., 2 Ž M . is unimodular. Živ. Assume that M contains a maximal order M `, 3 of Q`, 3 . If Ž L, F . s L2Ž py1., 2 Ž M . is the unimodular lattice of Žiii., then Ž LŽŽ py5.r12., F . s L2Ž py1., 6 is in¨ ariant under D 6 Ž SL 2 Ž p .. F AutŽ LŽŽ py5.r12., F ..

'

Proof. Ži. Let ² z : 1 ] DŽ B . be the normal p-subgroup of DŽ B .. The ² : order M z is isomorphic to M2 ŽZw z p x.. So the M DŽ B . sublattices of L correspond to the ideals of Zw z p x which are stable under the Galois group GalŽQw z p xrQw p x.. Hence a system of representatives of isomorphism classes of M DŽ B .-lattices is given by LŽ i. A s LŽ1 y z . i A where A g S and 0 F i F Ž p y 3.r2. Žii. By Lemma 3.1, ` g NG L Ž DŽ B .. is an element of determi2Ž py 1. ŽQ . nant p Ž py1.r2 normalizing DŽ B .. Since ` lies in the enveloping algebra of DŽ B ., it commutes with C. Therefore the lattice LŽ i. ` A is a MDŽ B .-sublattice of index p Ž py1.r2 of LŽ i. A and hence equals LŽ iqŽ py1.r4. A by Ži.. Clearly Ž LŽ i. ` A, F . is isometric to Ž LŽ i. A, ` F ` t r .. Since ` normalizes DŽ B ., the form ` F ` t r lies in F) 0 Ž DŽ B ... With Lemma 1.1Ži. one gets that ` F ` t r s yx ` 2 F for some totally positive unit x g Qw p x. Since the fundamental unit of Qw p x has norm y1 the unit x s y 2 is a square and y yields an isometry between Ž LŽ i. A, yx ` 2 F . and Ž LŽ i. A, y` 2 F .. Žiii. The group D 2 Ž SL 2 Ž p .. contains a group DŽ B .. Since C is also the endomorphism algebra of D 2 Ž SL 2 Ž p .. one has F Ž D 2 Ž SL 2 Ž p ... s F Ž DŽ B .. and there is a 0 F i 0 F Ž p y 3.r2 such that M D 2 Ž SL 2 Ž p .. fixes the lattices LŽ i 0 . A Ž A g S .. By Theorem 2.4 there is an ideal A 0 of Qw p x such that Ž LŽ i 0 . A 0 , bF . is unimodular for some totally positive b g Qw p x. If i 0 ) Ž p y 5.r4 one applies the isometry `y1 of Žii. and replaces F by y` 2 F to achieve 0 F i 0 F Ž p y 5.r4. The uniqueness of such an i 0 follows because the determinant detŽ LŽ i 0qi . A 0 , bF . s p 4 i Ž1 F i F Ž p y 5.r4. is ) 1 and not divisible by p py 1 s detŽ` 2 .. Živ. Since Q`, 3 F Q 'p , `, ` , one has p ' 5 Žmod 12. and 3 is inert in

'

'

'

' '

Qw p x. The lattices LŽ i. A together with LŽ i.p A Ž0 F i F Ž p y 3.r2, A g S ., where p is a generator of the maximal two-sided ideal of M `, 3 containing 3, form a system of representatives of isomorphism classes of M `, 3 DŽ B .-lattices. Among these there is a lattice L9, on which D 6 Ž SL 2 Ž p .. acts. But then D 6 Ž SL 2 Ž p .. also fixes the lattice L9p . With Žii. the lattice Ž LŽŽ py5.r12., F . is up to isometry the unique M DŽ B .-lattice of determinant p Ž py5.r6 and Živ. follows from Theorem 2.6.

'

Table I displays some invariants of the cyclo-quaternionic lattices for p s 5, 13, and 17. Here the class number of Qw p x is one. The first line

'

CYCLO-QUATERNIONIC LATTICES

483

TABLE I ps5

detŽ L. minŽ L. < Lmin < g Ž L.

p s 13

L8, 2

LŽ1. 8, 2

L24, 2

LŽ1. 24, 2

LŽ2. 24, 2

LŽ3. 24, 2

LŽ4. 24, 2

LŽ5. 24, 2

1 2 240 2

54 4 120 1.79

1 4 196560 4

34 4 936 2.61

13 8 6 1248 2.55

1312 12 13104 3.33

1316 14 312 2.53

13 20 24 1248 2.83

L32, 2

LŽ1. 32, 2

LŽ2. 32, 2

LŽ3. 32, 2

LŽ4. 32, 2

LŽ5. 32, 2

LŽ6. 32, 2

LŽ7. 32, 2

17 8 8 63648 3.94

17 12 10 4896 3.46

17 16 12 1632 2.91

17 20 20 3264 3.40

17 24 30 4896 3.58

17 28 44 4896 3.69

p s 17

detŽ L. 1 17 4 minŽ L. 4 6 < Lmin < 146880 233376 g Ž L. 4 4.21

contains the name of the lattice, where it is assumed that Ž LŽ0., F . s L2Ž py1., 2 Ž M . is unimodular. The quadratic form F is omitted and L2Ž py1., 2 Ž M . is shortly denoted by L2Ž py1., 2 , if M is unique up to conjugacy. The second row contains the determinant. The last three rows contain the minimum, the number of minimal vectors, resp. the rounded value of the Hermite parameter of the lattice, if these data could be computed. ˜ in the quaterThere are two non-conjugate maximal orders M and M nion algebra Q '29 , `, ` . They correspond to the two different constructions Q '29 , `, ` s Qw'29 x m Q`, 2 s Qw'29 x m Q`, 3 and can be distinguished by ˜ a maximal the property that M contains a maximal order of Q`, 2 and M order of Q`, 3 . One obtains the numerical results for p s 29 in Table II. With programs, developed by H. Napias w12x, which allow us to perform an LLL-reduction over Euclidean domains, one obtains LLL-reduced i. Ž i. Ž ˜ .. . Žresp. LŽ56, bases for the lattices LŽ56, viewed as lattices over the 2 M 2 M ˜ .. maximal order in Q`, 2 Žresp. Q`, 3 . which is contained in M Žresp. M Whereas for the first lattices it is possible to calculate the minimal vectors, i. Ž ˜ . Ž PARI does not produce results for the lattices LŽ56, 1 F i / 7, 8. 2 M after 1 week of calculations. The upper bounds for the minima of these lattices are obtained from an LLL-reduced basis and the minimum of Ž˜. LŽ2. 56, 2 M will be proved in the next section. Remark 3.3. The lattices LŽ2Ži. py1., 2 Ž M . have a purely algebraic interx m ZwŽ1q 'p .r2x M pretation as ideals of the maximal order O [ Zw z p q zy1 p x of the quaternion algebra QO with center Qw z p q zy1 ramified only at p the Ž p y 1.r2 infinite places.

484

GABRIELE NEBE

TABLE II p s 29

detŽ L. minŽ L. < Lmin < g Ž L.

L56, 2 Ž M .

Ž . LŽ1. 56, 2 M

Ž . LŽ2. 56, 2 M

Ž . LŽ3. 56, 2 M

Ž . LŽ4. 56, 2 M

Ž . LŽ5. 56, 2 M

Ž . LŽ6. 56, 2 M

1 6 15590400 6

29 4 6 9744 4.72

29 8 8 58464 4.95

2912 10 146160 4.86

2916 12 204624 4.59

29 20 16 53592 4.81

29 24 20 9744 4.72

p s 29

detŽ L. minŽ L. g Ž L.

˜. L56, 2 Ž M

Ž˜. LŽ1. 56, 2 M

Ž˜. LŽ2. 56, 2 M

Ž˜. LŽ3. 56, 2 M

Ž˜. LŽ4. 56, 2 M

Ž˜. LŽ5. 56, 2 M

Ž˜. LŽ6. 56, 2 M

1 6 6

29 4 6 4.72

29 8 10 6.18

2912 F 12 F 5.83

2916 F 14 F 5.35

29 20 F 16 F 4.81

29 24 F 20 F 4.72

p s 29

detŽ L. minŽ L. < Lmin < g Ž L.

Ž . LŽ7. 56, 2 M

Ž . LŽ8. 56, 2 M

Ž . LŽ9. 56, 2 M

Ž . LŽ10. 56, 2 M

Ž . LŽ11. 56, 2 M

Ž . LŽ12. 56, 2 M

Ž . LŽ13. 56, 2 M

29 28 28 107880 5.20

29 32 28 696 4.09

29 36 28 696 3.21

29 40 28 696 2.53

29 44 58 9744 4.12

29 48 88 19488 4.91

29 52 116 24360 5.09

p s 29

detŽ L. minŽ L. < Lmin < g Ž L.

Ž˜. LŽ7. 56, 2 M

Ž˜. LŽ8. 56, 2 M

Ž˜. LŽ9. 56, 2 M

Ž˜. LŽ10. 56, 2 M

Ž˜. LŽ11. 56, 2 M

Ž˜. LŽ12. 56, 2 M

Ž˜. LŽ13. 56, 2 M

29 28 20 4872 3.71

29 32 32 9744 4.67

29 36 F 42

29 40 F 56

29 44 F 68

29 48 F 98

29 52 F 108

F 4.82

F 5.05

F 4.82

F 5.47

F 4.74

There are two remarkable lattices in these tables, LŽ1. 32, 2 s L 32, 6 and ˜ s L56, 6 , which are denser than the corresponding extremal unimodular lattices. These lattices also have an interpretation as Mordell]Weil lattices as shown in the next section. Ž . LŽ2. 56, 2 M

4. CONNECTION WITH MORDELL]WEIL LATTICES In w23x, Shioda investigates the Mordell]Weil lattice of the elliptic curve E : y 2 s x 3 q 1 q t pq 1

CYCLO-QUATERNIONIC LATTICES

485

over the rational function field k Ž t ., where k is any field of characteristic p containing Fp 2 and p is a prime ' y1 Žmod 6.. He shows that with respect to the canonical height h the group of k Ž t .-rational points EŽ k Ž t .. is a positive definite even lattice of rank 2 p y 2, determinant p Ž py5.r3 , and minimal norm Ž p q 1.r3. ŽThese lattices have been independently discovered by N. D. Elkies.. Let K [ Fp 2 Ž t .. Since the Frobenius automorphism a ¬ a p generates the Galois group GalŽFp 2rFp . the unitary group U2 ŽFp 2 ., isomorphic to an extension of the central product of a cyclic group of order p q 1 and SL 2 Ž p . by a group of order 2, acts on the K-rational points of the Fermat curve X pq 1 : x 1pq 1 q x 2pq 1 s t pq 1 q 1. The mapping Ž x 1 , x 2 . ¬ Žyx 1Ž pq1.r3, x 2Ž pq1.r2 . defines a dominant rational mapping from X pq1 to E. So one is tempted to induce the action of U2 ŽFp 2 . on EŽ K .. But this induced action on EŽ K . is not well defined. However, one gets: THEOREM 4.1. If p ' y1 Žmod 6. then the group G [ U2 ŽFp 2 . acts on the lattice Ž EŽ K ., h. ¨ ia isometries. The kernel of this action is the subgroup of order Ž p q 1.r6 of the center ZŽ G . ( C pq 1. Proof. The strategy of the proof is to define an action of G on a subset S of vectors of norm Ž p q 1.r3 in EŽ K .. The kernel of this action is the subgroup of index 6 in the center ZŽ G .. The image H acts as isometries Žwith respect to h. on this subset S of the free abelian group EŽ K ., hence linear on the Q-vector space spanned by S. Since H has no faithful rational representation of degree - 2Ž p y 1., the subset S generates a full H-invariant sublattice L of EŽ K .. The most difficult part of the proof is the identification of L with EŽ K .. For this purpose a sublattice of rank 4 of L on which a certain element of order p q 1 in G acts as a 6th root of unity is compared with the corresponding sublattices of the other G-lattices of dimension 2Ž p y 1. using an explicit description of the group ring. Elkies pointed out a much easier proof of Theorem 4.1 which is sketched in Remark 4.8. The Subset S : EŽ K .. With w23, Proposition 5.3x one finds that the elements Ž x 1 , x 2 . g X pq1ŽFp 2 w t x. such that x 1 and x 2 are of degree 1 are of the form x 1 s at q b, x 2 s ct q d, where Ž ac db . g G is an element of the unitary group G s U2 ŽFp 2 .. Define S to be the set of images of these points: S [ Ž x, y . s Ž y Ž at q b .

½

Ž pq 1 .r3

, Ž ct q d .

Ž pq1 .r2

.

ž

a c

b g U2 Ž Fp 2 . d

/

5

.

The Action of G on S. Let z denote a primitive Ž p q 1.st root of unity in Fp 2 . The preimages of Ž x, y . s ŽyŽ at q b .Ž pq1.r3 , Ž ct q d .Ž pq1.r2 . g S

486

GABRIELE NEBE

are of the form Ž z 3 a Ž at q b ., z 2 b Ž ct q d ... They form a full right coset of the subgroup U [ ²Ž z03 01 ., Ž 10 z02 .: ( CŽ pq1.r3 = CŽ pq1.r2 F G. Identifying S with the set of right cosets U _ G, one gets an action of G on S induced by right multiplication in G. The kernel is the largest normal subgroup of G contained in U, coreŽU . s U l ZŽ G . ( CŽ pq1.r6 . That this action of G on S induces isometries of the lattice Ž L, h., may be seen by a direct calculation of the scalar products: Let P, Q g S. Viewing the points on EŽ K . as elements Ž P ., Ž Q . of the Neron]Severi ´ group of the corresponding elliptic surface and taking into account that this surface has no reducible fibers and that P and Q do not intersect with the zero divisor, one calculates ² P, Q : s Ž p q 1.r6 y ŽŽ P .Ž Q .. Žcf. w22x., where ŽŽ P .Ž Q .. is the intersection number of the two divisors Ž P . and Ž Q . and ² ? , ? : is the bilinear form whose associated quadratic form ² P, P : s hŽ P . is the canonical height. Let P [ ŽyŽ at q b .Ž pq1.r3 , Ž ct q d .Ž pq1.r2 . and Q [ ŽyŽ a t q b .Ž pq1.r3, Žg t q d .Ž pq1.r2 . be in S. The value t g P 1 ŽFp 2 . s Fp 2 j  `4 gives an intersection point of Ž P . and Ž Q ., if Ž at q b . s z 3 r Ž a t q b . and Ž ct q d . s z 2 s Žg t q d . for some r, s. If

ž

a g

zi b / d 0

/ ž

0

z

j

/ž

a c

b d

/

for all 1 F i , j F p q 1 Ž gen .

then ŽŽ P .Ž Q .. is the number of solutions 0 F r - Ž p q 1.r3, 0 F s Ž p q 1.r2 of

Ž ad y bg . z 3 rq2 s q Ž b c y a d . z 3 r q Ž bg y ad . z 2 s q ad y bc s 0. Ž w. Namely then each solution of Žw. gives an intersection at t s t Ž r, s . s Ž bz 3 r y b .rŽ a y az 3 r . s Ž dz 2 s y d .rŽ c y gz 2 s . g Fp 2 j  `4 . If equality holds in the condition Žgen. for some Ž i, j . and P / Q, then P [ wŽy1. j xQ g S, hence ² P, Q : s Žy1. jq1 Ž p q 1.r6. Clearly the condition Žgen. is invariant under right multiplication with elements of G. If one replaces P by Pg and Q by Qg, Eq. Žw. for ŽŽ Pg .Ž Qg .. is multiplied by the determinant of g g G. Therefore G acts as isometries, hence linear, on L s ² S :. We now want to show that L equals EŽ K .. To this aim let A g SL 2 Ž p . 0 . s SU2 ŽFp 2 . an element of order p q 1, A [ Ž 0z zy1 . Let X [ ² Ž yt Ž pq1.r3 , 1 . , Ž y Ž z t .

Ž pq1 .r3

Ž yz Ž pq1.r3 , t Ž pq1.r2 . : F L.

, 1 . , Ž y1, t Ž pq1.r2 . ,

CYCLO-QUATERNIONIC LATTICES

487

Then Ž X, h. is isometric to ŽŽ pq1.r6.A22 a rescaling of an orthogonal sum of two copies of the hexagonal lattice. In particular minŽ X, h. s Ž p q 1.r3 and detŽ X, h. s ŽŽ p q 1.r6. 4 ? 3 2 . The eigenvalues of the matrix describing the action of A on X are primitive 6th roots of unity. The normalizer 0 1 .: in SL 2 Ž p . of ² A: is N [ ² A, Ž y1 , a quaternion group of order 0 2Ž p q 1.. It also fixes the lattice X. To identify L, the sublattices T of the SL 2 Ž p .-invariant lattices corresponding to the suitable homogenous component of N are investigated. Let c Ž b . denote the complex irreducible character of ² A: defined by b Ž . Ž . c Ž b . Ž A. [ z pq 1 0 F b F p where z pq1 g C is a primitive p q 1 st root of unity and let cb be the irreducible character of N whose restriction to ² A: is c Ž b . q c Žy b . Ž0 - b - Ž p q 1.r2. and c , c 9 be the two different extensions of c ŽŽ pq1.r2. to N. LEMMA 4.2. With the notation abo¨ e let p ' y1 Žmod 6., N s ² A:.C2 ( C pq 1.C2 be the quaternion group of order 2Ž p q 1., q a prime, and q a the maximal q-power di¨ iding p q 1. If q s 3 and p ' 1 Žmod 4. let K [ Q 3w p x be the unramified extension of degree 2 of Q 3 , K [ Q q in all the other cases, and R the maximal order in K. Let a [ Ž p q 1.r6, ea the centrally primiti¨ e idempotent in KN belonging to ca , and L the block of RN containing a nonzero multiple of ea . Assume a G 1.

'

Ži. If q G 5, then ea LrŽ ea L l L . ( Ž Rrq a R . 4 as R-modules. Žii. Let q s 3 and e resp. e9 be the centrally primiti¨ e idempotents of KN belonging to c resp. c 9. Let L a [ Ž1 y e y e9. L. Then ea L arŽ ea L a l L a . ( Ž Rr3 ay 1 R . 4 and ea LrŽ ea L l L . ( Ž Rr3 a R . 2 [ Ž Rr3 ay 1 R . 2 as R-modules. Žiii. Let q s 2 and p ' 3 Žmod 4.. View ca as a character of N [ NrZŽ N . ( Dpq 1. Let ea be the corresponding idempotent and L be the corresponding block of Z 2 N. Then ea LrŽ ea L l L . ( ŽZr2 ay 1 Z. 4 . Proof. In all cases the defect group of L is C q a F ² A: and RrqR is a minimal splitting field for LrqL. Ži. From w16, Theorem ŽVIII.5.x one finds that L is of the form L s  Ž x, y . g Ž 1 y ea . L [ ea L N n Ž x . s m Ž y . 4 , where m and n are epimorphisms of the resp. orders onto M2 Ž Rrq a R .. Hence the amalgamation module of ea L and Ž1 y ea . L is ea LrŽ ea L l L . ( Ž ea L q Ž1 y ea . L .rL ( Ž Rrq a R . 4 . Žii. Now q s 3 and the character ca belongs to the exceptional vertex of the Brauer tree of L.

488

GABRIELE NEBE

For s s 1, . . . , a let ea r3 sy 1 be the centrally primitive idempotent of KN belonging to the character 3s

Ca r3 sy 1 [

Ý

c x a r3 sy 1

xs1, 3¦x

s and L s [ Ý is1 ea r3 iy 1 L. According to w16, Theorem ŽVIII.5. Ži.x

L s s  Ž x, y . g L sy1 [ ea r3 sy 1 L N ns Ž x . s m s Ž y . 4 , where m s : ea r3 sy 1 L ª ea r3 sy 1 LrJacŽ ea r3 sy 1 L . 3 y1 and ns : L sy1 ª sy 1 L sy 1r3L sy1 ( ea r3 sy 1 LrJacŽ ea r3 sy 1 L . 3 y1 are the canonical epimorphisms. We claim that L 1 l L s s 3 sy1L 1. This is trivial for s s 1. Assume that it is true for some a ) s G 1. Let x s x 1 q ??? qx sq1 g L 1 l L sq1 , where x i g ea r3 iy 1 L Ž1 F i F s q 1.. Since x g L 1 and L sq1 is a subdirect sum of the ea r3 iy 1 L Ž1 F i F s q 1., this implies x 2 s ??? s x sq1 s 0. Hence x 1 g L 1 l L s with nsq1Ž x 1 . s x 1 q 3L s s 0. Therefore x s x 1 s 3 y with y g L s . Now L s l ŽQ 3 L 1 . : L 1 , again because L s is a subdirect sum. Hence y g L s l L 1 s 3 sy 1L 1 and x g 3 sL 1. Since L 1 s ea L a , this implies ea L arŽ ea L a l L a . ( L 1r3 ay 1L 1 ( Ž Rr3 ay 1 R . 4 . As in Ži. the block L of RN is of the form L s Ž x, y . g L a [ Ž e q e9. L N n Ž x . s m Ž y .4 . Here m and n are homomorphisms of the resp. orders onto Ž Rr3 a R . [ Ž Rr3 a R .. Hence the total amalgamation module ea LrŽ ea L l L . is isomorphic to Ž Rr3 a R . 2 [ Ž Rr3 ay 1 R . 2 . sy 1

Žiii. Since the degree of the irreducible Brauer character belonging to L is even, a defect group of L is isomorphic to C2 ay 1 F N. The idempotent ea belongs to the exceptional vertex of the Brauer tree of L. Let e2 a be the centrally primitive idempotent of KN belonging to the character c 2 a Žof N . and L ay 1 [ Ž1 y e 2 a . L. As in Žii., ea L ay1rŽ ea L ay1 l L ay1 . ( ŽZr2 ay 2 Z. 4 . As in Ži., L s Ž x, y . g Ž1 y e2 a . L [ e2 a L N n Ž x . s m Ž y .4 where m and n are epimorphisms of the resp. orders onto ŽZr2 ay 1 Z. 4 from which one gets Žiii.. LEMMA 4.3. Let p ' 5 Žmod 12. and Ž M, F . be a primiti¨ e lattice of dimension 2Ž p y 1. with Ž SL 2 Ž p .( S˜3 ..2 F AutŽ M, F ... With the notation abo¨ e let T [ Ž ea q ey a . M l M F M where a [ Ž p q 1.r6 and ea Ž resp. eya . is the centrally primiti¨ e idempotent in C² A: corresponding to c Ž a . Ž resp. c Žy a . ..

CYCLO-QUATERNIONIC LATTICES

489

Ži. Let M be a maximal order of Q 'p , `, ` containing a maximal order of the subalgebra Q`, 3 . If Ž M, F . s L2Ž py1., 2 Ž M . is unimodular, then ŽT, Ž1ra . F . is isometric to the e¨ en unimodular lattice of dimension 8. If ` is a totally positi¨ e prime element o¨ er p in the maximal order of the endomorphism algebra EndŽ SL 2 Ž p .( S˜3 . ( Qw p x then the rescaled lattice ŽT, Ž`ra . F . is p-modular of minimum G 4. Žii. If Ž M, F . s L2Ž py1., 6 , then ŽT, Ž1ra . F . is isometric to A22 . If Ž M, Ž1rp . F . s La2Ž py1., 6 , then ŽT, Ž1rpa . F . is isometric to A22 .

'

Proof. Using the character tables given in w21x one calculates p

Ž Qi . N ² A : s 2

Ý

c Ž b . y c Ž i. y c Ž pq1yi. ,

b s0, b 'i Žmod 2 .

where Qi is the character of SL 2 Ž p . described in Section 2. In case Ži., the restriction of the natural character to SL 2 Ž p . is 2QŽ pq1.r2 s 2Ž Q q Q9. and in case Žii. it is 2Qa , hence dimŽT . s 8 resp. 4. To derive the index of T [ T H in M, let q be a prime and q a the largest q-power dividing p q 1. If a s 0, then Z q T is a direct summand of Z q M. Therefore assume that a G 1. Let R and N [ NS L 2 Ž p.Ž² A:. be as in Lemma 4.2. In case Ži. the lattice M and its sublattice T can be viewed as ZwŽ1 q p .r2x-modules of rank p y 1 resp. 4. In case Žii. the lattices M and T have a structure over a maximal order of Q`, 3 and hence over Zw'y 1 x. In particular for q s 3, Z 3 M is regarded as an RSL2 Ž p .-module of R-rank p y 1 via an embedding R ¨ End Z 3 M Ž SL 2 Ž p ... First let q s 2. Since a and Ž p q 1.r2 s 3 a are odd, the character c 2 a , which is the only other Frobenius character in the block of Z 2 N containing ca , does not occur in the restriction of Qa or QŽ pq1.r2 to N. Therefore Z 2 T is a direct summand of Z 2 M and 2 does not divide the index w M : ŽT [ T H.x . Now let q ) 2. A defect group of the block of Qa as well as the one of the block containing Q and Q9 in RSL2 Ž p . is the Sylow q-subgroup C q a ( D F ² A: and has normalizer N. Since N is also the normalizer of the simple subgroup C q F D w8, Lemma ŽVII.1.5.x states that the RN-module Z q M decomposes as

'

˜ [ P, Z q MN R N s M where P is a projective RN-module and the number of indecomposable ˜ equals the number of indecomposdirect summands of the RN-module M able direct summands of the RSL2 Ž p .-module Z q M.

490

GABRIELE NEBE

If q ) 3 then in both cases the character afforded by the Q q N-module ˜ does not contain ca . Therefore Z q T is a Z q N-sublattice of P. Qq M Lemma 4.2Ži. now implies that in both cases Ži. and Žii. the Sylow q-subgroup of MrŽT [ T H. is ŽZrq a Z. dimŽT ., which is also the Sylow q-subgroup of T a, F rT, because q does not divide the determinant of Ž M, F .. Now let q s 3. The character afforded by the Q 3w p x SL 2 Ž p .-module Q 3 M is 2Q Žor 2Q9. in case Ži. and it is Qa in case Žii.. Hence the one of ˜ is 2Ýass1 Ca r3 sy 1 in case Ži. and it is c q c 9 q the Q 3 w p x N-module Q 3 M a Ý ss2 Ca r3 sy 1 in case Žii. where Ca r3 sy 1 is as defined in the proof of Lemma 4.2 Žii.. The RSL2 Ž p .-module Z 3 M is decomposable in case Ži.. The Green correspondent Žcf., e.g., w6, Theorem Ž20.6.x. of an indecomposable sum˜ mand of Z 3 M is one of the two isomorphic indecomposable summands M9 ˜ ˜ Ž . of M. With the notation introduced in Lemma 4.2 ii the RN-module M9 is isomorphic to Ž1 y e y e9. P1 , where P1 is a projective indecomposable ˜ Lemma 4.2Žii. shows that the L-module. Since Z 3T is a sublattice of M, H. Ž Sylow 3-subgroup of Mr T [ T is ŽZ 3 w p xr3 ay 1 Z 3 w p x. 4 ( ŽZr3 ay 1 Z. 8. In case Žii. the character ca only occurs in the character afforded by the projective RN-lattice P. Lemma 4.2Žii. yields that the Sylow 3-subgroup of MrŽT [ T H . is ŽZ 3w'y 1 xr3 a Z 3w'y 1 x. [ ŽZ 3 w'y 1 xr3 ay1 Z 3w'y 1 x. ( ŽZr3 a Z. 2 [ ŽZr3 ay 1 Z. 2 . The image of N( S˜3 acting on T is the central product N˜ [ S˜3 ( S˜3 . In case Žii., this group N fixes only one lattice of determinant 3 2 and dimension 4. To determine the p-part of the determinant the explicit description of the irreducible p-modular representations of SL 2 Ž p . as the space of homogenous polynomials of degree s in two variables Fp w x, y x s , with character b sq 1 Žcf. Lemma 2.3. is used. Extending scalars one may choose a basis Ž x i y j N i q j s s . of Fp 2 w x, y x s such that the action of A is x i y j A s z Ž iyj. x i y j. Hence the character c Ž a . Žand hence c Žy a . . occurs in b sq1 if and only if there are integers 0 F i, j F s with i q j s s and i y j s a , i.e., a F s and s ' a Žmod 2.. In particular c Ž a . and c Žy a . do not occur in bay1. Since G acts on La2Ž py1., 6rL2Ž py1., 6 with character 2 bay1 , one has ŽT, Ž1ra . F . ( A22 if Ž M, F . s L2Ž py1., 6 and ŽT, Ž1rpa . F . ( A22 if Ž M, Ž1rp . F . s La2Ž py1., 6 . Since the trivial 2-modular constituent does not occur in the representation of N on QT, the rescaled primitive lattice ŽT, Ž1ra . F . is even. Hence in the case Ži. ŽŽ M, F . s L2Ž py1., 2 Ž M .., it is isometric to E8 , the unique even unimodular lattice of dimension 8. The rescaled lattice Ž M, ` F . can be viewed as a unimodular lattice over M. In particular the rescaled lattice

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'

'

CYCLO-QUATERNIONIC LATTICES

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ŽT, Ž`ra . F . comes from an even unimodular ZwŽ1 q therefore of minimum G 4.

'p .r2x-lattice and is

To state the corresponding result for p ' y1 Žmod 12. the notation for finite rational matrix groups and their lattices as defined in w17, Proposition ŽII.4.x are used. 2 Ž3.

2 Ž 3. The groups PSL2 Ž p . D = D 12 and PSL 2 Ž p . m D 12 F GL2Ž py1. ŽQ. are extensions of PSL2 Ž p . = D 12 by an automorphism of order 2 acting on both direct factors, where the restriction of the natural character to PSL2 Ž p . is 2QŽ pq1.r2 in the first case and 2QŽ pq1.r6 in the second Ž3. 2 Ž 3. ˆŽŽ pq1.r12. 2m case. The corresponding lattices AŽŽpypq1.r4. A2 = A 2 resp. A py1 1 D 2 Ž3 . ŽŽ pq1.r4. Žor A10 m A 2 if p s 11. contain the tensor product A py1 m A2 ŽŽ pq1.r12. Ž py1.r2 ˆ Ž . resp. A py 1 m A 2 or A10 m A 2 if p s 11 of index 3 cf. w17, x Chap. V .

LEMMA 4.4. Let p ' y1 Žmod 12. and Ž M, F . be a primiti¨ e lattice of dimension 2Ž p y 1. with Ž PSL2 Ž p . m D 12 ..2 F AutŽ M, F .. Using the notation of Lemmas 4.2 and 4.3 let T [ Ž ea q ey a . M l M F M. 2 Ž3 .

Ži. If Ž M, F . s AŽŽpypq1.r4. m A 2 is of determinant p py1 , then dimŽT . 1 is 8. The rescaled lattice ŽT, Ž1ra . F . is p-modular of minimum G 4. 2 Ž3 .

2 Ž3 .

Žii. If Ž M, F . s AˆŽŽpypq1.r12. m A 2 Ž or A10 m A 2 if p s 11. is of 1 determinant p 2Ž ay1., then ŽT, Ž1ra . F . is isometric to A22 . If Ž M, Ž1rp . F . s 2 Ž3 . a 2 Ž3 . AˆŽŽpypq1.r12. m A 2 Ž or A10 m Aa2 if p s 11. is the dual lattice, the sublattice 1 ŽT, Ž1rpa . F . is isometric to A22 . Proof. Most of the proof is completely analogous to the one of Lemma 4.3 and need not be repeated. The main difference is the argumentation in the case q s 2. As in Lemma 4.2 let 2 a be the maximal 2-power dividing p q 1. Ži. The restriction of the natural character of AutŽ M, F . to PSL2 Ž p . is 2Q 3 a s 2Ž Q q Q9.. Let R [ Z 2 wŽ1 q y p .r2x if p ' 3 Žmod 8. and R [ Z 2 if p ' y1 Žmod 8.. Let M9 be an indecomposable summand of the RPSL2 Ž p .-module Z 2 M. Since the R-rank of M9 is Ž p y 1.r2 ' 1 Žmod 2., a vertex of M9 is the Sylow 2-subgroup D of N, where N is as in ˜ [ P, where P is a Lemma 4.2Žiii.. By w6, Theorem Ž20.6.x, Z 2 M N N s M sum of indecomposable RN-lattices with vertex of the form x D l N where x g PSL2 Ž p . y N. Since ca belongs to a block with defect group C2 ay 1 , Z 2 T is an R-sublattice of P. Ža. Assume that p ' 3 Žmod 8., i.e., a s 2. If Z 2 T is a direct summand of P then w6, Corollary Ž20.8.x says that M9 and an indecomposable summand of Z 2 T have a common vertex. But ca does not belong to a

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GABRIELE NEBE

block of RN of maximal defect, so this is a contradiction. The only other Frobenius character in the block of ca of RN is c 2 a . So Z 2 T, not being a direct summand of P, is a sublattice of a projective RN sublattice of P. Now Lemma 4.2 Žiii. implies Ži.. Žb. Assume that p ' y1 Žmod 8., i.e., a ) 2. then C2 ay 1 ( C 1 ] D is a characteristic subgroup of D. Since N s NP S L 2 Ž p.Ž C ., C is not of the form x D l N for some x g PSL2 Ž p . y N. Now N is also the normalizer of the simple subgroup C2 F C, hence by w8, Lemma ŽVII.1.4.x the indecomposable Z 2 N-modules belonging to blocks with defect group C have either a vertex C or are projective. Hence Z 2 T is a sublattice of a projective Z 2 N-submodule of P and one gets Ži. with Lemma 4.2 Žiii.. To get the minimum of the lattice ŽT, Ž1ra . F . one has to view it as ˜ If the minimum of ŽT, Ž1ra . F . is unimodular ZwŽ1 q y p .r2x-lattice T. 2, this lattice T˜ represents 1. Since the action of the normalizer NG Ž² A:. on T˜ is irreducible Žwith image containing "S3 m S3 . T˜ contains 4 linearly independent vectors of length 1 and hence is isometric to ZwŽ1 q y p .r2x 4 . But then AutŽT, F . s C2 X S4 does not contain S3 = S3 , which is a contradiction.

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Žii. The block of Z 2 PSL2 Ž p . containing Qa is of defect 2 ay 1 and has defect group C2 ay 1 . As in Lemma 4.3 for q G 5, one gets that Z 2 T is a sublattice of a projective Z 2 N-module. Hence by Lemma 4.2Žiii. the Sylow 2-subgroup of MrŽT [ T H. is ŽZr2 ay1 Z. 4 . The rest is as in the proof of Lemma 4.3. We now finish the proof of Theorem 4.1: COROLLARY 4.5. The restriction of the representation D of G [ U2 ŽFp 2 . on Q L to the subgroup SL 2 Ž p . has character 2QŽ pq1.r6 . The lattice L has determinant p Ž py5.r3 , so L s EŽ K . Proof. As seen in the beginning of the proof of Theorem 4.1 the image DŽ G . is SL 2 Ž p . : 2 = C3 if p ' 5 Žmod 12. and "PSL2 Ž p ..2 = C3 if p ' y1 Žmod 12.. An element g g G y Ž G9 ? ZŽ G .. can be chosen as g [ Ž y0z 3 10 .. The square g 2 g coreŽU . lies in the kernel of the action if Ž p q 1.r2 is odd and yg 2 g coreŽU . if Ž p q 1.r2 is even. In both cases there is a central subgroup ² v : F ZŽ DŽ G .. : EndŽ DŽ G .. with v 2 q v q 1 s 0. Since the values of the characters Qi of degree p y 1 of SL 2 Ž p . are real, the restriction of D to SL 2 Ž p . has character 2Qi where i g Ž p q 1.r2, Ž p q 1.r4, Ž p q 1.r3, Ž p q 1.r64 l Z has the same parity as Ž p q 1.r2 Žcf. Section 2.. The characters QŽ pq1.r4 and QŽ pq1.r3 therefore only have to be considered if p ' y1 Žmod 12.. They extend to the non-split

CYCLO-QUATERNIONIC LATTICES

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extension "PSL2 Ž p ..2 with character field Qw'y 2 x and Qw'y 1 x Žcf. w21x. contradicting that v commutes with DŽ G .. Hence the character of D is one of 2QŽ pq1.r2 or 2QŽ pq1.r6 . To apply Lemma 4.3 resp. Lemma 4.4 note that there is an additional automorphism f g AutŽ L. defined by ŽyŽ at q b .Ž pq1.r3 , Ž ct q d .Ž pq1.r2 . f [ ŽyŽ a p t q b p .Ž pq1.r3, Ž c p t q d p .Ž pq1.r2 .. Clearly f maps the set S into itself and fixes the condition Žgen. and the numbers of solutions of Žw.. 0 1 .. Hence f g AutŽ L.. The element f 1 [ DŽŽ y1 f g AutŽ L. centralizes 0 DŽ SL 2 Ž p .. and induces the outer automorphism on the center ZŽ DŽ G .. ( C6 . Moreover f 12 s DŽyI .. Hence H [ ² DŽ G ., f : is isomorphic to Ž SL 2 Ž p .( S˜3 ..2 if p ' 5 Žmod 12. and H s Ž PSL2 Ž p . m D 12 ..2 if p ' y1 Žmod 12.. In all four cases H is an absolutely irreducible subgroup of GL2Ž py1.ŽQ. and the H-invariant primitive lattices are isometric to M or M a, where M is as in Lemma 4.3 resp. Lemma 4.4. Since the lattice L s ² S : is a sublattice of EŽ K ., L is integral and its determinant is divisible by p. Assume now that the restriction of D to SL 2 Ž p . has character 2QŽ pq1.r2 . pq1.r4. 2 Ž 3 . Then L is isometric to Ž ` m. L2Ž py1., 2 Ž M . resp. Ž m.AˆŽŽpy1 = A 2 for D some totally positive m g ZwŽ1 q p .r2x resp. m g Z ) 0 as described in Lemma 4.3Ži. resp. Lemma 4.4Ži.. Therefore the lattice X ( ŽŽ pq1.r6.A22 Ždefined just before Lemma 4.2. is a sublattice of the lattice ŽT, ` mF . resp. ŽT, mF . Žof Lemma 4.3Ži. resp. Lemma 4.4Ži.. of minimum G 4ŽŽ p q 1.r6.. Since the minimum of X is Ž p q 1.r3 this is a contradiction. Hence D N S L 2 Ž p. has character 2QŽ pq1.r6 . Using parts Žii. of Lemmas 4.3 and 4.4 one easily concludes that Ž L, h. s L2Ž py1., 6 resp. Ž L, h. s 2 Ž3 . 2 Ž3 . AˆŽŽpypq1.r12. m A 2 Žor A10 m A 2 if p s 11. is of determinant p Ž py5.r3 s 1 detŽ EŽ K ., h.. Therefore L s EŽ K ..

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Theorem 4.1 and Lemma 4.3 resp. Lemma 4.4 yield the following two corollaries: COROLLARY 4.6. If p ' 5 Žmod 12. then Ž EŽ K ., h. is isometric to L2Ž py1., 6 . In particular, the minimum of L2Ž py1., 6 is Ž p q 1.r3. The automorphism group AutŽ L2Ž py1., 6 . contains the absolutely irreducible subgroup 2 Ž3 .

SL 2 Ž p . ( S˜3 . In Theorem 2.6 it was also shown that the subgroup SL 2 Ž p . of the automorphism group of the Mordell]Weil lattice acts on p q 1 unimodular overlattices of L2Ž py1., 6 . It would be nice to see these lattices in EŽ K . Q to have an estimation for their minimum. 2 Ž3 .

For p s 17 the automorphism group AutŽ L32, 6 . s SL 2 Ž17. ( S˜3 has two orbits on these 18 lattices of length 12 resp. 6 represented by extremal

494

GABRIELE NEBE

unimodular lattices L resp. L9. The automorphism group of L is SL 2 Ž17., whereas L9 is the Barnes]Wall lattice with automorphism group 1q 10 q Ž . 2q .O10 2 . If p ' y1 Žmod 12., one uses w17, Theorem ŽV.9.x to show COROLLARY 4.7.

If p ' y1 Žmod 12. then Ž EŽ K ., h. is isometric to

2 Ž3 . 2 Ž3 . AˆŽŽpypq1.r12. m A 2 Ž resp. A10 m A 2 for p s 11. with automorphism group 1 2 Ž3 .

containing PSL2 Ž p . m D 12 . For p s 11 the lattice Ž EŽ K ., h. is described in w17, p. 50x, because its 2 Ž3 . automorphism group PSL2 Ž11. m D 12 is a maximal finite subgroup of GL20 ŽQ.. It has determinant 112 and 12540 vectors of minimal length 4. For p s 23 the rank of EŽ K . is 44. The lattice has determinant 23 6 and 2708112 vectors of minimal length 8. Remark 4.8. As pointed out by Elkies, there is a shorter proof of Theorem 4.1 using the elliptic curve E9 : Y 2 s X 3 q T p y T, which is equivalent to E over any extension of Fp that contains elements j , j 2 , j 3 , 2 and h with j p / j s j Ž p ., j 22 s j 33 s j p y j , and h pq 1 s y1 via the transformation T [ Ž j t q j ph .rŽ t q h ., X [ j 3 xrŽ t q h .Ž pq1.r3 , and Y [ j 2 yrŽ t q h .Ž pq1.r2 . For g [ Ž ac db . g SL 2 Ž p . Ž a, b, c, d g Fp , ad y bc s 1., let g ŽT . [ Ž aT q b .rŽ cT q d .. Then the group SL 2 Ž p . acts on E9Ž K . via Ž X ŽT ., Y ŽT .. ? g [ Ž X Ž g ŽT ..rŽ cT q d .Ž pq1.r3, Y Ž g ŽT ..rŽ cT q d .Ž pq1.r2 ., since g ŽT . p y g ŽT . s ŽT p y T .rŽ cT q d . pq 1. 5. SOME LATTICES CONSTRUCTED WITH D 2 Let p be a prime p ' 1 Žmod 4.. Consider the matrix group G [ D 2 Ž SL 2 Ž p .. F GL2Ž py1.ŽQ.. Since the representation D 2 decomposes into two non-equivalent representations over R, F Ž G . is of dimension 2 and may be identified with the center Qw p x of the endomorphism algebra C [ EndŽ G . ( Q 'p , `, ` Žcf. Lemma 1.1Ži... As in Theorem 2.4 let Ž L, F . s L2Ž py1., 2 Ž M . be unimodular. Multiplying F by integral totally positive elements a g ZwŽ1 q p .r2x : C, one obtains infinitely many integral Euclidean lattices Ž L, a F .. If p s 5, 13, or 17, the class number of the quaternion algebra Q 'p , `, ` is one, so G fixes up to isomorphism only the lattice L.

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THEOREM 5.1. Let p s 5, 13, or 17, and b [ Ž5 q 'p .r2 g ZwŽ1 q 'p .r2x \ R : ZŽC . the totally positi¨ e generator of R with minimal trace. Then the norm N of b is N s 5, 3, or 2 in the respecti¨ e cases. Consider the

CYCLO-QUATERNIONIC LATTICES

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lattices L i [ Ž L, Ž b q i . F . for i g Z G 0 . Then L i is an e¨ en Ž i 2 q 5i q N .modular lattice. The minimum of L i is 2 i q 4 if p s 5 resp. 4 i q 6 if p s 13 or p s 17. In particular L0 is an extremal N-modular lattice. For p s 5, the lattices L i are the densest lattices in their genus. Proof. By Remark 2.4 the normalizer of G in the automorphism group AutŽ L, F . of the unimodular lattice contains an element n which induces the Galois automorphism on the center Qw p x of the endomorphism algebra of G. Therefore one gets for any 0 / a g Qw p x:

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Fy1 n Ž a F . n t r Fy1 s Fy1 a nFn t r Fy1 s Fy1 a s Ž aa . Ž a F .

y1

.

So if a is primitive Ži.e., R s Zw a x. and totally positive, the lattice Ž L, a F . is Ž aa .-modular. To obtain the minimum of L i consider the unimodular even R-lattice L [ Ž L, p F ., where p g R is a totally positive prime element dividing p, with quadratic form f : L m L ª R. Then L i is the Z-lattice with quadratic form Ž b q i . F s Tr(ŽŽ b q i .rp . f ., where Tr denotes the trace from Qw p x to Q. Assume first that p s 5. Then b s p s Ž5 q '5 .r2 and Ž b q i . F s Tr(Ž1r10.Ž10 q 5i y i'5 . f . Let x g L with f Ž x, x . s a q b'5 . Then x t r Ž b q i . Fx s TrŽŽ1r10.Ž10 q 5i y i'5 .Ž a q b'5 .. s 2 a q iŽ a y b .. Since f is even, one has a, b g Z and a ' b Žmod 2.. Moreover the minimum of the Z-lattice L0 is 4, so a G 2 and there is x g L with f Ž x, x . s 2, because 2 is the unique even totally positive element of R with trace 4. Finally Ž a q b'5 . is totally positive, therefore a2 ) 5b 2 . Distinguish 3 cases If b F 0, then 2 a q iŽ a y b . G aŽ i q 2. G 2 i q 4, with equality if Ž a, b . s Ž2, 0.. If b s 1, then a is odd and hence a G 3. Therefore a y b G 2 and 2 a q iŽ a y b . G 2 i q 6. Finally if b ) 1 then a y b ) Ž'5 y 1. b ) 1 and therefore a y b G 2. Since a ) '5 b one has 2 a q iŽ a y b . ) 2'5 b q 2 i ) 4 q 2 i. That L i is the densest lattice in its genus follows from the observation that the Hermite parameter of a Ž i 2 q 5i q 5.-modular lattice of minimum G 2 i q 6 is G Ž2 i q 6.r 'i 2 q 5i q 5 ) 2 contradicting the fact that E8 s L8, 2 Ž M . is the densest lattice in dimension 8 w3x. Next consider the case p s 13. As in the case p s 5 one gets that the minimum of L i is the minimum of f i Ž a, b . [ Ž a y b . q Ž a y 3b . i, where a q b'13 are the values of the quadratic form f on L . Hence a ) '13 b ) 3.6b, a ' b Žmod 2. and a G 6. If x g L , then also e x g L for a fundamental unit e of R, one obtains the inequality TrŽ e 2 Ž a q b'13 .. s 11a y 39b G 12. With this, one obtains that minŽ f i Ž a, b .. G 6 q 6 i if b F 0, minŽ f i Ž a, b .. G 6 q 4 i if b s 1, and minŽ f i Ž a, b .. G 8 q 4 i ) 6 q 4 i for b G 2.

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The integral lattice Ž L, p F . represents 14. Since AutŽ L, p F . contains an element inducing the Galois automorphism on the center ZŽ C . the R-lattice L represents both even totally positive elements 7 " '13 of R with trace 14. So again this minimum is attained because f i Ž7, 1. s 6 q 4 i. The case p s 17 may be dealt with similarly. Here f i Ž a, b . s a y 3b q iŽ2 a y 8 b . and it is helpful to compute that the minimum of L0 is 6 and the one of L1 is 10 and to use this restriction on the representation numbers of L . In particular L represents 18 q 4'17 so the minimum f i Ž18, 4. s 6 q 4 i is attained. Remark 5.2. If p s 29, 37, or 41 the algebra Q 'p , `, ` contains two conjugacy classes of maximal orders Žcf. w25x.. For p s 37 both unimodular lattices L 72, 2 Ž M . contain vectors of length 6. For p s 41 only one of the two unimodular lattices L80, 2 Ž M . Žthe one, where M contains a maximal order of Q`, 3 ., remains a candidate to be extremal. For p s 29, 37, and 41 none of the modular lattices Ž b . L2Ž py1., 2 Ž M . Žwhere b s Ž7 q p .r2. is extremal. A further operation one might apply to the lattices L2Ž py1., 2 Ž M . is taking tensor products over maximal anti-identifiable subrings O of the endomorphism rings. The automorphism group of this lattice contains a subgroup D 2 Ž SL 2 Ž p .. m D 2 Ž SL 2 Ž q .., where S [ QO is the Q-subalgebra

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S

spanned by O, isomorphic to the central product of SL 2 Ž p . and SL 2 Ž q .. To obtain a unimodular lattice one usually has to choose an appropriate quadratic form in F) 0 Ž D 2 Ž SL 2 Ž p .. m D 2 Ž SL 2 Ž q .... S

Clearly for p s q s 5, S s Q '5 , `, ` one again obtains the lattice L8, 2 Ž M ., but for p s 5, q s 13 Žwhere one may take the subalgebra S to be Q`, 2 . one obtains a new extremal unimodular lattice L48 of dimension 48. Choosing p s 5, q s 17 Ž S ( Q`, 3 . one otains a unimodular lattice Ž L, F . but also a 3-modular lattice Ž L, p 3 F . of dimension 64, where p 3 is a totally positive prime element in Qw'17 , '5 x dividing 3. Both lattices remain candidates to be extremal.1 To show the 3-modularity of Ž L, p 3 F ., one constructs an element n g AutŽ L, F . which induces the Galois automorphism of Qw'17 , '5 x over Qw'17 x. Since np3 ny1 s Ž4 q '17 . 2 py1 3 , the 3-modularity of Ž L, p 3 F . can be seen as in the proof of Theorem 5.1. One might hope to construct an extremal unimodular lattice of dimension 72 with p s q s 13, S s Q '13 , `, ` , but this lattice contains vectors of length 6. An easy construction for the extremal unimodular lattice L48 in dimension 48 is the following: Let M be a maximal order in Q '13 , `, ` and F3 1

Meanwhile I proved the extremality of the unimodular lattice Ž L, F ..

CYCLO-QUATERNIONIC LATTICES

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denote a Gram matrix of the extremal 3-modular lattice ŽŽ5 q i g M24ŽQ. be an element of order 4 in the endomorphism ring End L 24 , 2 Ž M .Ž D 2 Ž SL 2 Ž13.... Since iF3 is skew symmetric, one has i t r s yF3y1 iF3 Žcf. Lemma 1.1Žii... Using this one easily checks that the matrix F3 Ž1 q i. F[ tr 3F3y1 Ž1 q i.

'13 .r2. L24, 2 Ž M .. Let

ž

/

is a Gram matrix of a positive definite 48-dimensional unimodular lattice M, where 1 denotes the 24 = 24 unit matrix. Moreover the group SL 2 Ž13. is a subgroup of the automorphism group AutŽ M . via the representation r Ž . D 2 q Dyt 2 . One computes e.g., with PARI that the minimum of M is 6. The identification of L48 with M may be obtained computing the automorphism group of M: THEOREM 5.3. The lattice M is an extremal e¨ en unimodular lattice. AutŽ M . contains a subgroup SL 2 Ž13. such that the normalizer N [ NAutŽ M .Ž SL 2 Ž13.. is the absolutely irreducible group Ž SL 2 Ž13. m SL 2 Ž5...2 2 , S

where S ( Q`, 2 . In particular M s L48 is not isometric to one of the two known extremal unimodular lattices P48 p and P48 q Ž cf. w5x. of dimension 48. Proof. It remains to compute N. Using only calculations in dimension 24, one may obtain a g GL24 ŽZ., such that 3aF3y1 a t r s F3 and aŽ1 q i . ay1 s Ž1 q i . t r. Then the matrix Ž ayt0 r 0a . lies in AutŽ M . and induces the outer automorphism on SL 2 Ž13. and the Galois automorphism on the center of the endomorphism algebra A [ EndŽ SL 2 Ž13.. ( M2 Ž Q '13 , `, ` .. The centralizer C [ CAutŽ M .Ž SL 2 Ž13.. consists of all elements in the simple algebra A, fixing M and the quadratic form F. Let 0 / ¨ g M be any vector. Then ¨ A l M is a lattice of rank dividing 32 Žs dim Q Ž A... One computes C as the subgroup of AutŽ ¨ A l M, F . which lies in A as C ( SL 2 Ž5..2. Therefore N s ² SL 2 Ž13., a, C : s Ž SL 2 Ž13. m SL 2 Ž5...2 2 . S

Remark 5.4. Since the automorphism group AutŽ M . contains an irreducible subgroup ( C65 , the lattice M has a structure over the 65th cyclotomic field. The existence of many root free unimodular lattices having such a structure has been predicted in w2x. However, it is shown in w1x that an extremal unimodular lattice may not be obtained from a principal ideal in Zw z65 x. REFERENCES 1. C. Bachoc and C. Batut, Etude Algorithmique de Reseaux Construits avec la Forme ´ Trace, Experimental Math. 1 Ž1992., 183]190.

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