Classifying lattices using modular forms { a prelimimary report Rudolf Scharlau and Boris B. Venkov February 1, Introduction and main results F

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1 Classifying lattices using modular forms { a prelimimary report Rudolf Scharlau and Boris B. Venkov February 1, Introduction and main results For some years now, computer programs are available ([SchH] and MAGMA) which generate representatives for all isometry classes of lattices (integral quadratic forms) in a genus of dimension say up to 16 and not too large determinant. In some cases, when all lattices turn out to have a rich structure, it is desirable to have a better insight and a computer-free alternative to the classication. This is in particular possible if the lattices L have a large root system R(L) (as introduced in [SchBl]). For instance, if L is even and of prime level `, then there are short and long roots of norms 2 and 2`, respectively. The long roots correspond to the short roots of the rescaled dual lattice L \ (see below for unexplained notions). Now one makes use of the classical fact that the theta series of L and L \ are transformed into each other, up to a factor, by the action of the Fricke involution. Analysing the formulae leads one to introduce a modication of the Coxeter number of a root system. Using the action of the Fricke involution on theta series with harmonic coecients, one can show for small ` and appropriate dimensions, that this modied Coxeter number is the same for all irreducible components of the root system R = R(L). In this way, one can determine all possibilities for the root system. By `lattice' we mean as usual a Z-lattice L of full rank in a rational vector space with a positive denite scalar product (x; y). It is called integral if (x; y) 2 Z for all x; y 2 L, and even, if (x; x) 2 2Z for all x 2 L. Any even lattice is integral. A lattice is called p-elementary, for some prime number p, if pl # L, where L # = fy 2 QL j (x; y) 2 Z for all x 2 Lg denotes the dual lattice of L. By a left upper index 2 Q +, we denote scaling: L equals L as a lattice, but the form is multiplied by. If L is p-elementary, then p (L # ) is again (integral and) p-elementary. We write p (L # ) = L \ for short. For an even lattice L, the smallest number ` s.t. `(L # ) is again even is called the level of L. If ` is a prime number 6= 2, an even lattice L has level ` if and only if L is `-elementary and L 6= L #. An even lattice has level 2 if and only it is 2-elementary, L 6= L #, and L is totally even in the sense that both L and 2 L # are even. If L has level ` (prime 1

2 R. Scharlau and B.B. Venkov { Lattices and Modular Forms 2 or not), then its theta series is of level `, i.e. a modular form for the congruence subgroup 0 (`) of SL 2 (Z); see e.g. [Miy]. The lattices of level 1 are the well-studied even unimodular lattices; the lattices of level 3 and dimension at most 12 or of level 2 and dimension at most 16 are the proper object of investigation in this paper. From the point of view of concrete classications of lattices, the following three theorems are the main results of this paper. Before stating the results, we recall from [SchBl] the denition R(L) := fv 2 L j v primitive; 2(v; L)=(v; v) Zg of the root system of an arbitrary lattice L. The quadratic form is part of the structure of such root systems R; therefore, scaling R makes sense. The lattice generated by a root system A; B; C; : : : is denoted by A; B; C; : : :. The root system of a lattice is (when neglecting the scaling) a root system in the usual sense of Lie algebra theory as treated in [Bou]; in particular it decomposes uniquely into indecomposable components with the standard names A r ; B r ; G 2. Specializing again to lattices of prime level `, we observe, that R(L) = fv 2 L j (v; v) = 2g _[ fv 2 `L # j (v; v) = 2`g =: R sh _[ R lo (`short' and `long' roots). Thus, the possible irreducible components of R(L) are A r, `A r, D r, `D r, E 6, `E 6, E 7, `E 7, E 8, `E 8 and the mixed types 2 B r ; C r (` = 2) and G 2 (` = 3). Theorem 1 There are 10 isometry classes of lattices of rank 12, level 3 and determinant 3 6. Representatives are the Coxeter-Todd lattice and 9 lattices having the following root systems: 6A A 1 3A A 2 2A A 3 6G 2 D 4 3 D 4 2G 2 A 5 3 A 5 G 2 A 6 3 A 6 D 6 3 D 6 E 6 3 E 6 Notice that all but exactly one of these lattices are reective in the sense that the root system has maximal rank 12. This is completely analogous to the famous case of 24-dimensional, even unimodular lattices [Ven], or 16-dimensional lattices of level 2 and determinant 2 8 [SchV] (genus of 4D 4 or the Barnes-Wall lattice). Theorem 2 There are isometry classes of lattices of rank 16, level 2 and determinant 2 6 ; 2 4 ; 2 2, respectively. They are all reective and have the root systems shown in Tables 2, 3 and 4 below, and are uniquely determined by their respective root system. Theorem 3 There are isometry classes of lattices of rank 12, level 3 and determinant 3 4 ; 3 2, respectively. They are all reective, have the root systems 6A 2 A 5 D A 1 A A 2 D 7 3 A 3 G 2 E 6 3G 2 E 7 3 A 5 2E 6 D A 1 E 8 2G 2 and are uniquely determined by their respective root system.

3 R. Scharlau and B.B. Venkov { Lattices and Modular Forms 3 2 Lattices of level 3: the root systems Our aim is the classication of lattices of level 3 in dimensions n at most 12. In view of the unique decomposition of an arbitrary (positive denite) lattice as an orthogonal sum of indecomposable ones, which are necessarily also of level 1 or 3, the classication immediately reduces to the case n = 12. This is covered by Theorems 1and 3 above. In the following, we present a detailed proof of Theorem 1, which was announced in [SchV] and deals with the most dicult case. We will not treat Theorem 3 in this paper, which can be obtained along the lines of Theorem 2, but with fewer computations. For a given dimension n and determinant det L = jl # =Lj = 3 r, all 3-elementary even lattices form one genus. They exist if and only if either r 2 f0; ng and n 0(8), or 0 < r < n; 2r n (mod 4): If n = 2m, the mapping L 7! L \ is an involution on (the isometry classes of) this genus. A \-invariant representative (i.e. a `modular' lattice in the sense of [Que]) is ma 2 = A 2? ::::? A 2, where A 2 as usual denotes the A 2 -root lattice. In the case of Theorem 1 (n = 12; r = 6) we shall call this genus H for short. Another, prominent, lattice in H is the Coxeter-Todd lattice C (see [CT]) which has no vector of norm 2 and again satises C = C \. It is in fact a consequence of our classication that this holds for all lattices in H. Following [SchV] we now derive a list of `possible' root systems of lattices in H which are subject to two essential restrictions. In the next section we shall show that for each root system in our list, there exists a unique lattice. As usual, h denotes the Coxeter number of an irreducible root system R. It is characterized by the following formula: (1) X v2r (v; x) 2 (v; v) = h (x; x) for all x 2 Q R: See e.g. [Bou], Chap.V, x6.2, p We now collect the properties of the genus H, and of the Coxeter-Todd lattice, which will be used in the sequel. Lemma 4 The theta-series of an arbitrary lattice K in H coincides with the theta-series of its 3-scaled dual lattice: K = K \. Proof: This is essentially contained in [Que], we briey recall the argument. The theta-series K is a modular form of weight 6 for the congruence subgroup 0 (3) SL 2 (Z) The Fricke involution t 3 2 SL 2 (Z) normalizes 0 (3) and thus acts on the space of those modular forms as well as on the subspace of cusp forms. By denition, t 3 interchanges K and K \. Furthermore, the dierence of any two -series of lattices in H, in particular K K \, is a cusp form. The special fact now is that the space of cusp forms of weight 6 is one-dimensional; see [Miy], Table A, p Thus t 3 xes every cusp form, and consequently K = K \.

4 R. Scharlau and B.B. Venkov { Lattices and Modular Forms 4 Let K be any lattice in H, and T (K) := K # =K its discriminant group. The induced bilinear form T (K) T (K)! Q=Z (discriminant form) is regular and takes its values in Z 1. After multiplication by 3, it can be considered as an 3 ordinary inner product on the F 3 -vector space T (K). Clearly, this inner product is of determinant 1 (look at K = 6A 2 ), and thus isometric to H?H?h1; 1i, where H denotes a hyperbolic plane over F 3 and h1; 1i a two-dimensional space with orthonormal basis. The number of elements of norm 6= 0 2 F 3 in such a space is known to be q 5 + q 2 with q = 3, i.e Combining these general properties of H with the special properties of the Coxeter-Todd lattice C, we arrive at the following lemma. Lemma 5 Let C be the Coxeter-Todd lattice and T = T (C) its discriminant group, equipped with the discriminant bilinear form. a) The orthogonal group O(C) maps surjectively onto the orthogonal group O(T ). b) The minimal norm of C # is 4. Every anisotropic element in T has precisely 3 three representatives of norm 4. 3 For the proof of a), one has to invoke one of the explicit constructions of C, see e.g. [CT]. Part b) is then derived as follows: The minimal norm 4 of 3 C# follows from Lemma 4, using the basic fact that C itself has minimum 4. It is known from either the explicit construction, or from modular forms theory, that there are 756 = minimal vectors. Since the orthogonal group of T acts transitively on the elements of norm 1 + Z, it follows form a) that the vectors of norm 3 4 in 3 C# are equidistributed modulo C. The second claim of part b) follows. We shall use the standard glueing construction of unimodular lattices K e = K 1 ' K 2 = f(x; y) 2 K # 1 K # 2 j 'x = yg where ' : T (K 1 )! T (K 2 ) is a bijection (see Proposition 2 of [SchV]) in the following special case: K 1 is an arbitrary lattice in the genus H, and K 2 = C. In this case, the glueing map ' exists. (By Lemma 5, part a), it is even essentially unique, i.e., two possible ''s can be transformed into each other by an automorphism of K 1 K 2 preserving the components; this will not be needed in the sequel.) Thus, to any K 2 H we have an associated Niemeier lattice K e := K ' C. An essential step to the proof of Theorem 1 is provided by the follwing lemma. Lemma 6 Let K be a lattice in H and R = R(K) its root system. Then the following properties hold: a) R = ; or rank R = 12: b) If R 6= ;; then (i) the Coxeter number h is the same for all irreducible components of R,

5 R. Scharlau and B.B. Venkov { Lattices and Modular Forms 5 (ii) jr sh j = jr lo j, (iii) jr sh j = 6h. Proof: Property (iii) is readily derived from properties (i) and (ii), but will also be obtained in the course of the proof of (i). The equality (ii) jr sh j = jr lo j follows immediately from Lemma 4 and the basic observation that the short roots of K # (or K \ ) are in one-to-one correspondence with the long roots of K, by mapping v 7! 3v. Now assume that R = R(K) 6= ;. Then also R sh 6= ;, and thus e R := R( e K) 6= ;. Let us describe e R = fv 2 e K j (v; v) = 2g explicitly in terms of K. The vectors ev in e K of norm 2 are of two kinds: either ev 2 K, or ev is of the form ev = v 1 + v 2 ; v 1 2 K # ; v 2 2 C # ; (v 1 ; v 1 ) = 2 3 ; (v 2; v 2 ) = 4 3 ; 'v 1 = v 2. Now we use Lemma 5: for every element in C # =C which is anisotropic with respect to the quadratic form 3(x; x) + 3Z; there exist exactly 3 representatives in C of norm (x; x) = 4 3 : It follows that the roots of e K of the second kind are of the shape v 1 + v 2, where 3v 1 runs through the long roots of K, and to every v 1 there are exactly three v 2. We see that (2) je Rj = jr sh j + 3jR lo j = 4jR sh j: Now we use the following formula from [Ven], Proposition 1: (3) X (v; x) 2 = 1 12 (x; x)j Rj e for all x 2 K: e v2 e R The above description of R e gives X X X (4) (v; x) 2 = (v; x) ( v X 3 ; x)2 = 2 v2r sh v2r lo v2r v2 e R (v; x) 2 (v; v) for all x 2 e K. Now specify an irreducible component R i of R and choose x non-zero in R i. Substituting (4) into (3) and comparing with (1) gives h i = 1 24 j e Rj for the Coxeter number h i of R i. This proves (i). (In view of (2), it also proves that jr sh j = 6h, i.e. the additional property (iii).) Recall that the Coxeter numbers of the irreducible root systems are as follows: h(a l ) = l + 1; h(d l ) = 2(l 1); h(g 2 ) = 6; h(e 6 ) = 12 < h(e 7 ) < h(e 8 ) Having this in mind, it is easily veried that the 12-dimensional root systems consisting of roots of norm 2 and 6 only, and having properties (i) and (ii), are precisely the root systems listed in the theorem.

6 R. Scharlau and B.B. Venkov { Lattices and Modular Forms 6 3 Constructing lattices of level 3 In this section, we will complete the proof of Theorem 1 by constructing, for each root system R of that list, a lattice K in H with R(K) = R. Unlike in the case n = 16; ` = 2 [SchV] where most of the constructions were omitted, we give a full treatment here. For each possible root system R, we specify the \glue code" K (= K=hRi), generating the desired lattice K over its sublattice hri, verify its uniqueness (up to isomorphism), and determine \by hand" the automorphism group of K, and thus the orthogonal group of K. The uniqueness of the code and thus of the lattice also follows from the mass formula. We adopt the following notational conventions. A glue vector of a component X of R is a vector in hxi # with minimal norm in its coset modulo hxi. The relevant factor group is denoted by T (X) := hxi # =hxi, the discriminant group of X; similarly T (R). If R(K) contains a component G 2, then the sublattice A 2 = hg 2 i K splits o, by Proposition 2.10 of [SchBl]. Therefore, all root systems to be considered are essentially of the form R = Y 3 Y, where Y is unscaled, i.e. exists of vectors of norm 2. We shall refer to T (Y)f0g and f0gt ( 3 Y) as the rst and second factor of T (R). Glue vectors of Y will be denoted by g 1 ; g 2 ; : : : ; those of 3 Y by g1; 0 g2; 0 : : : : We occasionally refer to the standard constructions for the root systems D l and A l ; see e. g. [Bou], Tables, or [CoSl], Chapter 4, where also the glue vectors are shown. The additional generators of K will be linear combinations (in practice: sums) of the g i and gj. 0 More formally, K will be the inverse image in hri # of a subgroup K T (R) which has to be specied. By Theorem 2.2 of [SchBl], the second projection of K is contained in the subgroup T (Y) T ( 3 Y) (this inclusion comes from the equality h 3 Yi # = 1 3 hyi# ). Concretely, this means that the gj 0 to be used are just copies of the g j, without factors 1. In many cases, 3 a bit more can be said in advance, namely the following. Lemma 7 Assume that det Y = jt (Y)j is not divisible by 3. Then K is of the form K = f(x; 'x) j x 2 T (Y)g T (Y) T ( 3 Y), for an appropriate isomorphism ' : T (Y)! T (Y). The proof is easy and well known from the unimodular case; observe that K is locally unimodular at all primes 6= 3. The following two general arguments will be used without further mentioning in the treatment of the individual cases. In order to ensure that the root system does not enlarge, R(K) = R, it is sucient that no new vectors of norm 2 are introduced. Thus the precise condition on K is as follows: for any x 2 Knf0g, the length, i.e. the minimal norm of a representative vector in hri #, is integral, even, and at least 4. The second general argument deals with the orthogonal group O(K). It is intermediate between the Weyl group W(R) = W(Y) W( 3 Y) and the group Aut(Y)Aut( 3 Y). The relevant factor group A(K) = O(K)=W(R) can be considered as a subgroup of the outer automorphism group (group of diagram automorphisms) A(Y) A( 3 Y) = A(Y) A(Y), acting on the discriminant group

7 R. Scharlau and B.B. Venkov { Lattices and Modular Forms 7 T (Y)T (Y). Observe furthermore that id X is not in the Weyl group for X = A l. In the following, S n denotes the symmetric group on n letters, Z m the cyclic group of order m, and Di 2k the dihedral group of order 2k. R = 6A A 1 The discriminant group T (Y) = T (6A 1 ) is isomorphic to Z2, 6 with generators g i, i = 1; : : : ; 6 of norm 1 (since g 2 i = e i =2 in standard coordinates). According to Lemma 7, the code is given by an isomorphism ' : Z 6 2! Z2; 6 this isomorphism is specied by the images '(g i ), i = 1; : : : ; 6. If c denotes the number of nonzero components of '(g i ), then the length of (g i ; '(g i ), equals (1 + 3c)=2. This number must be even and 6= 2 and thus c = 5. Therefore '(g i ) is the sum of the g j, 0 j 6= (i) for some permutation 2 S 6. Since S 6 acts by automorphisms of 3 Y R, the lattice K given by ' is unique up to isometry. Every element of S 6, acting by outer automorphism on Y, uniquely extends via ' to 3 Y and thus to the whole lattice. I.e. O(K) = W(R)oS 6 = Z 12 2 o S 6, of order ! = R = 3A A 2 We have T (A 2 ) = Z 3, with non-zero elements g i, of length `(g i ) = 2, in the 3 i-th component (i = 1; 2; 3) of Y. The glue code K is three-dimensional over the eld F 3. Let K 1 be the projection onto the rst factor T (Y) = Z3. 3 First observe that K 1 6= 0; otherwise, there would be additional roots (of norm 6 in the rst factor). Since the length function takes integral values on the second factor, the same must be true for K 1. Consequently, K 1 must be one-dimensional over F 3, generated by g 1 g 2 g 3 (of length 2). The sign can be normalized to K 1 = hg 1 + g 2 + g 3 i by automorphisms of the root system Y. The intersection K 2 of K with the second factor T ( 3 Y) = Z 3 3 is of dimension 2 and contains no single g 0 2 (in view of the length function). Clearly, such a subgroup K 2 Z 3 3 can be normalized to K 2 = fx 1 g x 2 g x 3 g 0 3 j x j 2 F 3 ; P x j = 0g = hg 0 1 g 0 2; g 0 2 g 0 3i. The whole glue code K is now uniquely determined: the third generator is of the form g 1 +g 2 +g 3 +g 0, with g 0 6= 0 and therefore unique (mod K 2 ); take e.g. g 0 = g 0 1. Since all sign changes except id on all of T (R) have been used to normalize K, the outer automorphism group is h idi S 3 S 3, where the symmetric group S 3 acts by permutation on the rst and second factor Z 3 3. Thus, O(L) is of order R = 2A A 3 We have T (Y) = Z4, 2 with generators g 1 ; g 2 of length `(g i ) = 3, and `(2g 4 i) = 1. According to Lemma 7, K is generated by g 1 + '(g 1 ), g 2 + '(g 2 ), for appropriate elements '(g i ) in the second factor. Since 3 + `('(g 4 i)) must be integral and even, the only choice, up to permuting the two components of the second Z4, 2 and changing signs, is '(g 1 ) = g g 0 2; '(g 2 ) = g g 0 1

8 R. Scharlau and B.B. Venkov { Lattices and Modular Forms 8 The glue code thus dened has indeed the property that all its elements have even length at most 4. Every element of the outer automorphism group A(Y) = Di 8 of the rst factor extends uniquely to the second factor via '. Thus O(K) is of order 2 3 (4!) 4. R = D 4 3 D 4 2G 2 We have to construct an 8-dimensional lattice M with R(M) = D 4 3 D 4 ; the desired 12-dimensional lattice will then be K = M?A 2?A 2. The discriminant group T (D 4 ) has 3 non-zero elements given by glue vectors g 1 ; g 2 ; g 3 g 1 + g 2, all of norm 1, and permuted transitively by A(D 4 ) = S 3. As glue vectors of K, we may take g 1 + g 0 1, g 2 + g 0 2, (and thus also g 3 + g 0 3), which are of norm 4 and satisfy all other requirements. Moreover, this choice is unique up to isomorphism, in view of the transitivity properties of A(D 4 ) A(D 4 ). For the same reason, each element of A(D 4 ) = S 3 acting on the rst factor can be extended to an automorphism of K, i.e. A(M) = S 3. Thus jo(k)j = 3! (2 3 4!) R = 6G 2 The lattice is the root lattice 6A 2 itself, with zero glue code, and orthogonal group W(G 2 ) 6 o S 6, i.e. the wreath product of S 6 and Di 12, of order 6! R = A 5 3 A 5 G 2 We have to construct a 10-dimensional lattice M with R(M) = A 5 3 A 5. The cyclic discriminant group T (A 5 ) decomposes as T (A 5 ) = hg 1 i hg 2 i where g 1 is of norm 3 2 and order 2 (mod A 5) and g 2 of norm 4 3 and order 3 (mod A 5). Because of the desired integrality of M, the unique choice of (generating) glue vectors is g 1 + g 0 1, g 0 2. The automorphism group of M does not contain (id; id) and thus equals h id M i W(A 5 ) W(A 5 ). The order of O(K) = O(M) O(A 2 ) is 2 (5!) R = A 6 3 A 6 T (A 6 ) = Z 7 has non-zero elements g, 2g, 3g of lenght 6, 10, 12. The glue code K must be generated by g jg 0 for some j 2 f1; 2; 3g. Since l(jg 0 ) = 18, 30, 36, only the case j = 3 is possible. The two choices of the sign can be transformed into each other by the outer automorphism id, acting on the second factor T ( 3 A 6 ). Thus, K is unique up to isomorphism, with orthogonal group h idi W (A 6 ) W (A 6 ), of order 2 (7!) 2. R = D 6 3 D 6 T (D 6 ) = Z 2 2 is generated by elements g 1 ; g 2 of length `(g i ) = 3 2, with `(g 1+g 2 ) = 1. Again, the structure of K is described by Lemma 7. The only solution, up to permuting g 0 1 and g 0 2 by the outer automorphism of 3 D 6, is K = hg 1 + g 0 1; g 2 + g 0 2i :

9 R. Scharlau and B.B. Venkov { Lattices and Modular Forms 9 The automorphism group is Z 2 W (D 6 ) W (D 6 ), of order 2 (2 5 6!) 2. R = E 6 3 E 6 Since T (E 6 ) is of order 3, the glue code is cyclic, generated by g + g 0 say. Since every g 0 in the second factor has integral length, also g must be of integral length. This forces g = 0, and consequently g 0 6= 0, i.e. hg 0 i = T ( 3 E 6 ). Thus K = E 6?E \ 6, and O(K) = O(E 6 ) O(E 6 ), A(K) = Z 2 2, jo(k)j = 2 2 ( ) 2. We summarize our results in the followng table, where we list all our lattices K together with the Coxeter number h, the structure of the outer automorphism group, and the order of O(K). Using the orders of the orthogonal groups, one can readily conrm our enumeration by the mass formula. The mass of H is equal to = lattice h root system outer group order of O(K) K 1 = C ; K 2 2 6A A 1 S K 3 3 3A A 2 Z 2 S 3 S K 4 4 2A A 3 Di K 5 6 D 3 4 D 4 2G 2 S 3 Z K 6 6 6G 2 S K 7 6 A 3 5 A 5 G 2 Z K 8 7 A 3 6 A 6 Z K 9 10 D 3 6 D 6 Z K E 3 6 E 6 Z 2 Z Table 1: The genus II 12 (3 6 ) of lattices of level 3, dimension 12, determinant Lattices of level 2: summary of results The dimension of a lattice of level 2 is known to be divisible by 4, its determinant is of the form 2 2 with n=2, and n is divisible by 8 if = 0 or = n=2. Conversely, for any such pair (n; ) there exists exactly one genus of totally even 2-elementary lattices with these parameters. This section complements the results of [SchVe94] by nishing the classication lattices of level 2 in dimensions n at most 16. In view of the unique

10 R. Scharlau and B.B. Venkov { Lattices and Modular Forms 10 decomposition of an arbitrary (positive denite) lattice as an orthogonal sum of indecomposable ones, which in our case have to be of level 2 as well, the classication problem immediately reduces to the case n = 16. It is shown that, except for the Barnes-Wall lattice (n = 16, = 4), all these lattices are reective, i.e. their root system is of full rank (cf. [SchBl96]). The root system R(L) consists of short roots of norm 2 and long roots of norm 4. The investigation of the possible root systems, in particular the relation between short and long roots, is more complicated than in the case = 4 treated in [SchV] or the case ` = 3; n = 12; det = 3 6 treated above. It is carried out using ordinary theta series and theta series with harmonic coecients (of degree 2). One uses the well known transformation formula relating the theta series of L and L #, which can be rewritten in terms of the so called Fricke involution z 7! 1 acting on 2z mouldar forms, and the explicit knowledge of modular forms for 0 (2) of weight 10. It turnes out that a certain modied Coxeter number is the same for all components of R(L), which gives serious restrictions leading to an explicit list of possible R(L)'s. Recall the root system R(L) of a totally even 2-elementary lattice can be described as R(L) = R sh (L) [ R lo (L) (\short" and \long" roots) with R sh (L) = fv 2 L j (v; v) = 2g and R lo (L) = f2w j w 2 L # ; (w; w) = 1g. The possible irreducible components of R(L) are A`, D` (` 4), E` (` = 6; 7; 8), C`, 2 B`, F 4, where the upper index denotes scaling and in our present case equals 1 or 2. Theorem 8 All 16-dimensional totally even 2-elementary lattices of determinant 6= 2 8 are reective. Possible root system of such lattices of determinant 2, = 3; 2; 1 have the following properties: a) The numbers of short roots jr sh j and of long roots jr lo j are related by jr sh j = 8 >< >: 2jR lo j + 32 if = 3 4jR lo j + 96 if = 2 8jR lo j if = 1 b) There is a natural number h (\modied Coxeter number of R") with the following properties, where we set t = 2 4 : (1) all irreducible components contained in R sh have the Coxeter number h (2) all irreducible components contained in R lo have the Coxeter number h =t (3) all components of type 2 B` have the same ` and h = 2 + 2(` 1)t (4) all components of type C m have the same m and h = 2(m 1) + 2t

11 R. Scharlau and B.B. Venkov { Lattices and Modular Forms 11 (5) components of type F 4 could be in R only if h = 6(t + 1) Elementary, but lengthy considerations lead to the following Proposition 9 There are exactly root systems of rank 16 satisfying the conditions of Theorem 8. These are the root systems shown in Tables 2, 3 and 4 below. Proceeding similarly as in the previous section one can show with some patience the following theorem which, together with the previous Proposition 9 completes the proof of theorem 2. Theorem 10 Each root system R of Theorem 8 and Propositon 9 is realizable as R(L), for an appropriate lattice L. The lattice L is uniquely determined (up to isometry) by its root system. The following three tables summarize the main properties of the lattices of Theorem 10 resp. Theorem 2: The \Coxeter number" h according to Theorem 8, the root system, the index [L : hr(l)i], the order of the orthogonal group O(L), and the order of the \outer automorphism group" A(L) := O(L)=W(L), where W(L) denotes the Weyl group (of R(L)).

12 R. Scharlau and B.B. Venkov { Lattices and Modular Forms 12 no. h root system ja(l)j jo(l)j A A A D 4 4C A A 2 C D A A 2 7 A 3 2C D 6 C B A 2 9 A 2 4 B C E 2 6 A 5 C C 2 6 B D B D 2 9 A E 2 7 B 5 F C 8 2F C 2 10 B E 2 8 B Table 2: The genus II 16 (2 6 II) of lattices of level 2, dimension 16, determinant 2 6

13 R. Scharlau and B.B. Venkov { Lattices and Modular Forms 13 no. h root system ja(l)j jo(l)j 1 6 4D A A D 6 2C E A A 11 2 A 2 C D 8 2C E 7 C 6 2 B D B C E 8 2F C 12 F Table 3: The genus II 16 (2 4 II)of lattices of level 2, dimension 16, determinant 2 4 no. h root system ja(l)j jo(l)j D A 15 2 A E 7 C D 12 C C 8 E C Table 4: The genus II 16 (2 2 II)of lattices of level 2, dimension 16, determinant 2 2

14 R. Scharlau and B.B. Venkov { Lattices and Modular Forms 14 References [Bou] [CoSl] [CT] N. Bourbaki: Groupes et Algebre de Lie, Chap. IV, V et VI.: Hermann, Paris (1968). J.H. Conway, N.J.A. Sloane: Sphere packings, lattices and groups. Springer, New York 1988, Second Edition, H.S.M. Coxeter, J.A. Todd: An extreme duodenary form. Canadian Journal of Math. 5 (1953), 384{392. [Miy] T. Miyake: Modular forms. Springer-Verlag, Berlin (1989). [Que] [SchH] H.-G. Quebbemann: Modular Lattices in Euclidean Spaces. J. of Number Theory 54 (1995), 190{202. R. Scharlau, B. Hemkemeier: Classication of integral lattices with large class number. Math. of Computation Vol. 67, No. 222 (1998) 737{749. [SchBl] R. Scharlau, B. Blaschke: Reective integral lattices. J. of Algebra 181 (1996), 934{961 [SchV] [Ven] R. Scharlau, B.B. Venkov: The genus of the Barnes-Wall lattice. Comment. Math. Helvetici 69 (1994), 322{333. B.B. Venkov: On the classication of integral even unimodular 24-dimensional quadratic forms. Proc. Steklov Inst. Math. 4 (1980), 63{74, also reprinted as chapter 18 in [CoSl]. Rudolf Scharlau Fachbereich Mathematik Universitat Dortmund D Dortmund Germany Boris B. Venkov St. Petersbourg's departement of the Steklov mathematical institute Fontanka 27 St. Petersbourg Russia