Arch. Math. 83 (2004) 528 535 0003 889X/04/060528 08 DOI 10.1007/s00013-004-1076-1 Birkhäuser Verlag, Basel, 2004 Archiv der Mathematik On the unimodularity of minimal vectors of Humbert forms By R. Baeza 1) and M. I. Icaza 2) 1. ummary. If = ( 1,..., m ) is an m-tuple of n n positive definite symmetric real matrices (= Humbert form) and K is a real number field of degree m with ring of integers O K, a vector u OK n is called minimal if [u] = Min{[v] v On K {0}}, where [v] = i [v i ] and v i = i-th conjugate of v. In this note we show that any i Humbert form has a unimodular minimal vector over K if and only if the class number of K is 1. In [2] we introduced a constant M K,n which measures the non-unimodularity of minimal vectors. We estimate here the constant M K,2 in terms of known constants of K. As a by-product we obtain a lower bound for the classical Hermite constant γ 2m. 2. Introduction. Let K/Q be a totally real number field of degree m, with ring of integers O K and discriminant d K. Let h(k) be the class number of K and σ 1,...,σ m : K R the embeddings of K into the real numbers. A Humbert form of rank n over K is an m-tuple = ( 1,..., m ) of n n positive real symmetric matrices 1,..., m ([2]). For any matrix A with entries in K we denote by A (i), the i-th conjugate σ i (A) of A. If u = (u 1,...,u n ) OK n we define the value of at u by [u] = m i [u (i) ], where B[C] means CBC t whenever this product is defined for matrices B and C. The minimum of is defined by m() = min{[u] 0 = u OK n }, and its determinant by det = m det i. We define the generalized Hermite constant of K by γ K,n = sup m()/(det()) 1/n Mathematics ubject Classification (2000): 11H31, 11H55. 1) Partially supported by Proyecto FONDECYT 1030218, FONDECYT 1970214 and Programa de Formas Cuadraticas. Universidad de Talca, Chile. 2) Partially supported by Proyecto FONDECYT 1020959, FONDECYT 1990897 and Programa de Formas Cuadraticas. Universidad de Talca, Chile.
Vol. 83, 2004 On the unimodularity of minimal vectors of Humbert forms 529 where runs over all positive definite Humbert forms of rank n (s. [5], [2]). In particular γ Q,n = γ n is the classical Hermite constant. A vector u OK n is called a minimal vector of if [u] = m(). In this paper we are concerned with the unimodularity of minimal vectors. Recall that a vector u = (u 1,...,u n ) OK n is unimodular if u =O K u 1 + +O K u n = O K. If u is a minimal vector of a form, then u is an ideal of minimal norm in its class. Conversely any ideal of minimal norm in its class is of this form for a suitable binary Humbert form (s. 3). In [2] we introduced a constant M K,n which measures the non-unimodulartity of minimal vectors (s. 3 for the definition). We show that M K,n = 1 for all n 2 if and only if h(k) = 1. As a by-product of the proof of this result one obtains a lower bound for γ K,2 which is used in ection 4 to estimate from below γ 2m. In ection 4 we estimate M K,2 in terms of known invariants of the field K and in ection 6 we introduce another obstruction for the unimodularity of minimal vectors of Humbert forms. We would like to thank the referee for simplifying the original proof of Theorem 3.1 and Corollary 3.2. 3. Unimodularity of minimal vectors. Let us recall the definition of the constant M K,n introduced in [2], which in some sense measures the non-unimodularity of minimal vectors of Humbert forms. For any Humbert form over K of rank n, let M() be the set of minimal vectors. For any u = (u 1,...,u n ) OK n we define the norm of u by N(u) = N K/Q (u i ) u i =0 where N K/Q : K Q is the usual norm. et and N() = inf{n(u) u M()} N[] = inf{n([u]) U GL(n, O K )}, where [U] = ( 1 [U (1) ],..., m [U (m) ]) runs over all equivalent forms to. In [2] it is shown that N[] is bounded above by a constant depending only on K and n, so that we can define the constant M K,n = sup N[] where runs over all Humbert forms over K of rank n. Ifh(K) = 1 it is clear that M K,n = 1 for all n 2 and every form has a unimodular minimal vector. Conversely we have Theorem 3.1. For any totally real number field K the following assertions are equivalent 1. h(k) = 1 2. M K,n = 1 for all n 2
530 R. Baeza and M. I. Icaza arch. math. Proof. (2)follows from (1) by the definition of M K,n. To prove the other implication it suffices to show that M K,2 = 1 implies h(k) = 1. Let us assume that h(k) > 1. Let I O K be a non-principal ideal and let J O K be an ideal in its inverse class, i.e., IJ is principal. We consider the O K -lattice = Ie 1 Je 2 K 2 where e 1 = (1, 0) t,e 2 ( = (0, 1) ) t is the standard basis and where K 2 1 0 is equipped with the standard quadratic form. 0 1 One associates to a Humbert form as follows. The lattice is free since IJ is principal, so we have = O K e O K f with e = (x 1,x 2 ) t,f = (y 1,y 2 ) t. Then I = x 1,y 1, J = x 2,y 2. Let be the Gram matrix of the quadratic form with respect to this basis and I = ( (1),..., (m) ) be the associated Humbert form. Then it follows that the minimum of I is m( I ) = N(α) 2 where α I or α J is of minimal norm among all β I J,β = 0 and the only minimal vectors of I are of the form αe 1 or αe 2. Assume α I and set αe 1 = a 1 e + a 2 f with a 1,a 2 O K. Then (a 1,a 2 ) OK 2 is a typical minimal vector of I. From α = a 1 x 1 + a 2 y 1, 0 = a 1 x 2 + a 2 y 2 it follows that (a 1,a 2 ) is not unimodular. This implies that I has no unimodular minimal vectors and hence M K,2 > 1. This proves the theorem. As a by-product of the above proof one obtains a lower bound for the generalized Hermite constant γ K,2 of K in the case h(k) > 1. Take I O K to be any non-principal ideal of minimal norm in its class and let J O K be an ideal of minimal norm in the inverse class of I. Then in the construction of the proof of the above theorem, the element α is now in I as well as in IJ. Therefore m( I ) = N(α) 2 N(I) 2 N(J) 2 and since det I = N(det ) = N(I) 2 N(J) 2, we obtain γ K,2 m( I ) N(I)N(J) 2N(I). (det I ) 1/2 If N(K)denotes the maximum of all N(I)where I runs over the ideals of minimal norm in its class, we get Corollary 3.2. If K is a (real) number field with h(k) > 1, then γ K,2 2N(K). For example, for K = Q( 10) we know that N(K) = 2. Therefore γ Q( 10),2 4. The constant N(K) seems to be very difficult to compute. We refer to [1] for some estimates. 4. A lower bound for γ 2m. In this section we will use (3.2) and an idea of H. Cohn (s. [3]) to get a lower bound for the classical Hermite constant γ 2m. Take any (real) number field K of degree m. We assume K to be real just to simplify the notations. Let = ( 1,..., m ) be a Humbert form of rank n over K. For any u = (u 1,...,u n ) OK n,u = 0, we get using the arithmetic geometric inequality [ 1 m m m i [u ]] (i) m i [u (i) ]
Vol. 83, 2004 On the unimodularity of minimal vectors of Humbert forms 531 and hence m m [m()] m 1 i [u (i) ]. Let us fix a Z-basis {w 1,...,w m } of O K. Then each component u i O K can be written u i = m x ik w k with x ik Z. et X = (x ik ) M n,m (Z). With w = (w 1,...,w m ) we k=1 have u t = Xw t and hence u = wx t. For each 1 j m, we obtain u (j) = w (j) X t, where w (j) = (w (j) 1,...,w(j) m ). Then the sum m i [u (i) ] seen as a quadratic form in the variables {x ik } is of rank mn and is given by Q(X) = m w (i) X t i Xw (i)t. Let m(q) be the minimum of Q(X) taken over all X M n,m (Z), X = 0. We have m[m()] 1 m m(q) and therefore m[m()] 1 m γmn (detq) 1 mn by the classical Hermite inequality (s. [11]). Thus we are led to compute det Q. To this end we now describe the lattice in R mn determined by the quadratic form Q(X). We start with the isomorphism φ : K Q R R m given by φ(a λ) = (λσ 1 (a),...,λσ m (a)) where σ i : K R are the embeddings of K in R. We denote K Q R by R K. WehaveinR K the sublattice O K R K given by the embedding a a 1. This lattice in R m has volume d K 1/2.φgives then an isomorphism R m K Rmn = (R n ) m. Write the Humbert form = ( 1,..., m ) as = AA t = (A 1 A t 1,...,A ma t m ) with A = (A 1,...,A m ) (GL n (R)) m and define = Aφ(OK n ) Rmn. Then is the lattice associated to Q, and (and hence Q) has discriminant det(a) 2 d K n = det d K n. Inserting this in Hermite s inequality it follows m[m()] 1 m γmn (det ) 1 mn ( dk ) 1 m and this implies γ K () = m() (det ) n 1 m m γ m mn d K. We have thus shown
532 R. Baeza and M. I. Icaza arch. math. Theorem 4.1. For any (real) number field K of degree m, and for any n 1 it holds γ K,n m m γ m mn d K. Remark 4.2. This result is essentially due to H. Cohn. In the case h(k) = 1ithas also been obtained using other methods by T. Watanabe (s. [9]). Combining (4.1) with (3.2) we conclude Corollary 4.3. For any (real) number field K with h(k) > 1 and m = [K : Q], [ ] 1 2N(K) m γ 2m m. d K Remark 4.4. The special feature of the above lower bound for γ 2m is that it is obtained by purely arithmetical arguments. By the contrary, the well known Hlawka-Minkowski bound for γ n, i.e. γ n [ ] 2 2ζ(n) n,n 1, w n is obtained by analytical considerations ([11]). Let us compare both bounds for the special case m = 2, i.e., for γ 4. The bound in (4.3) allows us to use any quadratic real number field with h(k) > 1. Take for example K = Q( 10). Then d K = 40 N(K) = 2. Thus we obtain γ 4 0.63. On the other hand the H-M bound gives γ 4 0.66. In fact we can prove that (4.3) holds true also for non real number fields. 5. An estimate of M K,2. In this section we will estimate M K,2 in terms of known constants of the the field K. To this end we need the following explicit description of M K,2. Proposition 5.1. Let K be any (real) number field. Then M K,2 = sup[ inf N((α, β))] I I= α,β where I runs over all ideals of minimal norm in its class. Proof. Wemayassume h(k) > 1. For any binary Humbert form over K we have N[] = inf ( inf N(u)). Hence there is some T [] and u M(T) with N[] = T [] u M(T) N(u). et I = u. In particular I has minimal norm among the ideals in its class. If I = v,v OK 2, there is some U GL(2, O K) with U[u] = v and hence v M(T[U]). ince T [U] [], we get by the choice of u, that N(u) N(v). In particular N(u) = inf v =I N(v)
Vol. 83, 2004 On the unimodularity of minimal vectors of Humbert forms 533 therefore N[] sup[ inf N(v)] I v =I where I runs over all ideals of minimal norm in its class. ince the right hand side of this inequality does not depend on, we obtain M K,2 = sup N[] sup( inf N(v)). I v =I Conversely let [I] be any ideal class and let I 1,...,I m be all ideals in [I] of minimal norm. et n(i i ) = inf N(u) and choose I 1 = I such that n(i 1 ) n(i i ),1 i m. u =I i Let I = u, u = (α, β) OK 2 such that n(i 1) = N(u). We can construct a Humbert form with ( α, β) M()and any other vector (x, y) M()satisfies βx+αy = 0 (s. proof of (3.1)). It is easy to check that (x, y) has the same norm as u. For any T [], say T = [U], the minimal vectors of T and correspond to each other through the transformation U, thus they generate the same ideals, and in particular they are equivalent to I. Thus by the choice of I and u we have N[] = N() = n(i). Hence N[] = sup [ inf N(v)]. Let us now 1 i m v =I i consider all the ideal classes [I] 1,...,[I] h of K and let i, be the corresponding Humbert forms associated to each class as before. Then we have sup N[ i ] = sup( inf N(v)), 1 i h I v =I where I runs over all ideals of minimal norm in its class. Thus we get M K,2 = sup N[] sup 1 i n This concludes the proof of the proposition. N[ i ] = sup( inf N(v)). I v =I We use now this formula to estimate M K,2 from above. To this end let us recall the following result due to Rieger([10]): any ideal I O K admits generators α, β (of any given signature) with N(α) < c m d K N(I), N(β) < c 2 m d K 1/2 N(I), where c m is a (computable) constant depending only on m = [K : Q]. Inserting in (5.1) the above inequalities and using Minkowski s bound for N(I)(s. [8]) we obtain Corollary 5.2. For any number field K M K,2 cm 3 m! m m d K 2 5. ince we are only interested in the case h(k) > 1, one can use also the inequality (4.3) instead of Minkowki s bound, and we get Corollary 5.3. If h(k) > 1 then 4N(K) 2 M K,2 cm 3 γ2m 2m 4m 2m d K 2 7. The inequality on the left hand side follows from (5.1) and the fact that N(I) divides the norm of any element γ I, and since we may take I non principal in (5.1), we have 2N(I) N(γ) for any γ I, γ = 0.
534 R. Baeza and M. I. Icaza arch. math. 6. Another obstruction for unimodularity of minimal vectors. Let us define the proper minimum of a Humbert form of rank n over the number field K as m () = min{[u] u O n K,uunimodular}. Of course m() m () and there is a unimodular u O n K with m () = [u]. We will show in this section that the number U K,n := sup m () m() exists, where runs over all Humbert forms of K of rank n. Hence we can define the proper Hermite constant of K, γk,n = sup m () and we obtain γ (det ) 1/n K,n γk,n U K,nγ K,n. All this follows from Theorem 6.1. Let K/Q be a (real) number field. Then for any Humbert form over K of rank n there exists a unimodular vector u O n K with [u] u K,n m(), where u K,n is a constant depending only on K and n. Proof. For any n n matrix A M n (K) with det A = 0, set A =max{ A (i), 1 i m} and l(a) = min{l(a (i) ), 1 i m}, where for any real B GL(n, R), B denotes the usual norm and l(b) = B 1 1 (s.[l-t]). ince is a Humbert form there is some B = (B 1,...,B m ) GL(n, R) m such that = BB t = (B 1 B1 t,...,b mbm t ). Then for any x K n [x] = m x (i) B i 2 and [A][x] = m x (i) A (i) B i 2. Using standard norm estimates we obtain l(a) 2m [x] [A][x] A 2m [x]. Therefore l(a) 2m m() m([a]) A 2m m(). We now use Humbert s reduction theory (s. [4]). It states that for any Humbert form there is a U GL(n, U) with [U] R K,n, where R K,n is the cone of Humbert reduced forms. Moreover there exists a finite set {A 1,...,A t } of non-singular n n-matrices over K, depending only on K and n, such that for any R R K,n there is 1 i t such that e = (1,...0) is a minimal vector of R[A i ]. Thus, let us take U GL(n, O K ) with 0 = [U] R K,n and let 1 i n be such that e is a minimal vector of 0 [A i ]. Let A = A i. Then m( 0 [A]) = 0 [A][e] and hence A 2m m() 0 [A][e]. Let a K,n = max{ A i 2m, 1 i t} and b K,n = inf{l(a i ) 2m, 1 i t} constants depending only on K and n. We get m m a K,n m() o [A][e] = oi [A (i) ][e] l(a (i) ) 2 oi [e] b K,n o [e]. From 0 = [U] we obtain 0 [e] = [u] with u = Ue OK n unimodular. Thus we have a K,n m() b K,n [u] and this proves the claim with u K,n = a K,n bk,n 1. It would be interesting to compare both constants M K,n and U K,n. We do not know of any estimate of U K,n.
Vol. 83, 2004 On the unimodularity of minimal vectors of Humbert forms 535 References [1] E. A. Anfer eva and N. G. Chudakov, Effective estimates from below of the norm of ideals of an imaginary quadratic field. Mat. UR b. 11, no 1, 47 58 (1970). [2] R. Baeza and M. I. Icaza, On Humbert-Minkowski s constant for a number field. Proc. Amer. Math. oc. 125, 3195 3202 (1997). [3] H. Cohn, On the shape of the fundamental domain of the Hilbert modular group. Theory of numbers. A. L. Whiteman, ed., Proc. ymp. Pure Math. 8, 190 202 (1965). [4] P. Humbert, Théorie de la reduction des formes quadratiques définies positives dans un corps algébrique K fini. Comm. Math. Helv. 12, 263 306 (1940). [5] M. I. Icaza, Hermite constant and extreme forms for algebraic number fields. J. London Math. oc. 55, 11 22 (1997). [6] P. Lancaster and M. Tismenetsky, The theory of matrices, 2nd. Ed. 1985. [7] J. Martinet, Perfect lattices in Euclidean spaces. Grundlehren Math. Wiss. 327, 2003. [8] W. Narkievicz, Elementary and analytic theory of algebraic numbers, 2nd. Ed. 1990. [9]. Ohno and T. Watanabe, Estimates of Hermite constants for algebraic number fields. Comm. Math. Univ. t. Paul. 50 (1), 53 63 (2001). [10] G. J. Rieger, Einige ätze über Ideale in algebraischen Zahlkörpern. Math. Ann. 136, 339 341 (1958). [11] C. L. iegel, Lectures on the geometry of numbers. 1989. R. Baeza and M. I. Icaza Instituto de Matemáticas Universidad de Talca Casilla 721 Talca Chile rbaeza@inst-mat.utalca.cl icazap@inst-mat.utalca.cl Received: 28 November 2003; revised manuscript accepted: 13 April 2004