On the computation of Hermite-Humbert constants for real quadratic number fields

Transcription

1 Journal de Théorie des Nombres de Bordeaux 00 XXXX On the computation of Hermite-Humbert constants for real quadratic number fields par Marcus WAGNER et Michael E POHST Abstract We present algorithms for the computation of extreme binary Humbert forms in real quadratic number fields With these algorithms we are able to compute extreme Humbert forms for the number fields Q 3 and Q 7 Finally we compute the Hermite-Humbert constant for the number field Q 3 Introduction A new invariant of number fields called Hermite-Humbert constant was introduced by P Humbert in 940 This constant is an analogue to the Hermite constant for Q In [H] Humbert describes a generalization of the reduction theory for quadratic forms over Z where he considers totally positive forms called Humbert forms with integral entries in a given number field K He deduces an analogous reduction theory and the existence of the Hermite-Humbert constant In this work we compute extreme binary Humbert forms for the number fields Q 3 and Q 7 and the Hermite-Humbert constant γ K for K = Q 3 In [BCIO] it is shown that finding Hermite-Humbert constants of real quadratic number fields is tantamount to looking for extreme Humbert forms Their existence is proved in [Ica] Following the precedence of Voronoï a characterization of extreme Humbert forms is given in [C] by introducing two properties of extreme Humbert forms namely perfection and eutacticity With these properties we are able to make a unique characterization of extreme Humbert forms: a Humbert form is extreme if and only if it is both perfect and eutactic The theoretical background Let P denote the set of positive real binary Humbert forms ie S = S S P with positive definite real matrices S and S If K denotes a real quadratic number field and O K its maximal order we denote with S[x] the product x t S x x t S x for any x O K where x denotes the conjugate vector of x and we let det S be the product of the determinants

2 Marcus WAGNER Michael E POHST det S and det S In the same way U denotes a matrix with conjugated entries of a matrix U K Let ms := min {S[x] : 0 x O K} denote the minimum of a given Humbert form S then denotes a set of equivalence classes MS := {[x] O K S[x] = ms} [x] := {y O K y = ɛx ɛ O K } where the elements of MS respectively their representatives are called minimal vectors of S To avoid superfluous notation we denote the elements [x] of MS only with x If a tuple U := U U GL O K is given we can make unimodular transformations from a Humbert form S to another denoted by S[U] := S[U ] S[U ] := U t S U U t S U t By scaling we mean multiplication of a given Humbert form S = S S with an element λ = λ λ R >0 to obtain another Humbert form λs := λ S λ S We note that for a given Humbert form S the set MS is finite which is shown in [Ica] The following theorem is due to Humbert He proved the existence of Hermite-Humbert constants: Theorem For any S P and any real quadratic number field K of discriminant d K there is a constant C R >0 with C < 0 d K such that S[x] C det S x MS There are better upper bounds for C see [Co] and [Ica] We use the estimate C d K from [Co] With theorem we are able to define a map from P to R >0 in the following way Definition & Proposition Let S P and γ K : P R >0 γ K S = ms det S then γ K is invariant under unimodular transformations and scaling Now we are able to define the Hermite-Humbert constant as γ K = sup γ K S S P Any Humbert form S for which equality holds in is called critical The existence of such forms is shown in [Ica] The value γ K S of a critical Humbert form S is a global maximum of γ K Forms for which γ K achieves

3 On the computation of Hermite-Humbert constants for real quadratic number fields 3 a local maximum are called extreme For characterizing extreme forms we introduce two properties: perfection and eutacticity If S = S S P and x MS then xx t S [x] x x t S [x ] is a semi-positive definite Humbert form The set of such forms of a given S P will be denoted with X S Now perfection means dim RX = 5 X X S and eutacticity means that there is a representation S = S S = ρ X X X X S with ρ X R >0 for all X X S Definition 3 If S S R with ai b S i = i i = b i c i then we define the map Φ : R R 6 S S a b c a b c t Now perfection and eutacticity become dim X X S RΦX = 5 and ΦS = X X S ρ X ΦX This means a Humbert form S is eutactic if ΦS is in the interior of the linear convex hull of the points ΦX where X X S The following theorem of [C] characterizes extreme forms: Theorem 4 A Humbert form is extreme if and only if it is both eutactic and perfect An extreme Humbert form of a real quadratic number field has at least 5 minimal vectors Now we need the definition of being equivalent for two Humbert forms S und T : Definition 5 Two Humbert forms S and T are called equivalent if there is a tupel U := U U GL O K with U = U and T = S[U] or there is λ = λ λ R >0 such that S = λt

4 4 Marcus WAGNER Michael E POHST In [C] is shown that there is only a finite number of extreme forms up to equivalence This suggests to look for a suitable set of representatives of extreme Humbert forms With the next lemmata we are able to determine such a set and a finite set M which contains all minimal vectors for each element of it Before we start we need one more definition: Definition 6 Two elements x y OK are called a unimodular pair if they are a O K -basis of OK ie O K = O Kx O K y With ɛ 0 > we denote the fundamental unit of the real quadratic number field K The following lemma is proved in [BCIO]: Lemma 7 Let h K = If any extreme Humbert form has a unimodular pair of minimal vectors then it is equivalent to a form b b 3 S = c b c b with ɛ c < ɛ 0 0 The standard basis vectors e and e are contained in the set MS and for any other minimal vector x = x x t Ok we have and N K/Q x γ K N K/Q x γ K x < ɛ0 γ K x x < ɛ0 γ K x We denote the obtained set of such representatives with S For a restriction of the finite set M of minimal vectors for each element of S we can make use of the following lemma see [BCIO]: Lemma 8 Let S = S S P with ai b S i = i b i c i for i = and u = u u t O K Then c c S[u] N K/Q u ms γ K N K/Q u a a S[u] γ ms K u u a c S[u] γ ms K u u a c S[u] ms γ K Let u = u u t v = v v t OK be minimal vectors of S with v / O K u and u u U = v v

5 On the computation of Hermite-Humbert constants for real quadratic number fields 5 then N K/Q det U γ K By Theorem 4 we know every extreme Humbert form has at least 5 minimal vectors Now we are able to compute all possible extreme forms as follows if we assume every Humbert form has a unimodular pair of minimal vectors: determine the finite set M of all possible minimal vectors for each element of S for any 3-set T = {u u u 3 } M\{e e } we have to solve polynomial equations S[u i ] = in the unknowns of S S where b b S = c b c b and i = 3 3 finally we have to test whether the obtained Humbert forms are eutactic and perfect A last step should be done: if we have two perfect Humbert forms S and T we need an algorithm which decides whether T and S are equivalent We consider two 3-sets W = {w w w 3 } MT and V = {v v v 3 } MS of minimal vectors If S and T are unimodular equivalent then there is a matrix U GL O K with 4 Uw i = ɛ i v i for i = 3 and suitable units ɛ i O K Now two perfect Humbert forms S and T are equivalent if and only if there exists a matrix U GL O K with U MT = MS 3 The algorithms In this section K always denotes a real quadratic number field with h K = and ɛ 0 > the fundamental unit of the maximal order O K Next we want to develop an algorithm for computing extreme Humbert forms The main algorithm splits in several subalgorithms The first subalgorithm for computing the set M and all 3-sets is algorithm 0 After computing all 3-sets of M we have to compute real solutions of the polynomial equations obtained by all triples of M to construct Humbert forms Finally we compute minimal vectors with algorithm 4 and if necessary eutactic coefficients

6 6 Marcus WAGNER Michael E POHST Algorithm 9 Main Algorithm Input: The maximal order O K of a real quadratic number field K Output: A set of all eutactic and perfect Humbert forms of K up to equivalence having a unimodular pair of minimal vectors T All 3-sets of M H set of Humbert forms obtained by the real solutions of the polynomial equations H {S = S S H ms = } H 3 { h h h H h = minimal vectors of h and h > 4 } H 4 {h = h h H 3 h is perfect and eutactic } H 5 set of non-equivalent Humbert forms Return H 4 3 Computing of all 3-sets of M In this section we describe the algorithm to compute all suitable 3-sets of M We make use of the results of lemma 7 and lemma 8 Note that we assume every Humbert form has got a unimodular pair of minimal vectors Algorithm 0 3-sets of M Input: An integral basis ω of the maximal order O K of a real quadratic number field K the fundamental unit ɛ 0 Output: All suitable 3-sets of M M X Y {[α] α O K with 0 < N K/Q α d K } For all x X do For all y Y do if k Z with yɛ k ɛ0 d 0 < K x M M {x yɛ k 0 t } T set of all 3-sets of M S and yɛ k 0 < ɛ0 d K x then For all A = {α i α i OK i = 3} T do Ψ {α i α j GL K i j { 3}} If Ψ = or for any X Ψ there holds N K/Q det X > d K S S {A} Else then

7 On the computation of Hermite-Humbert constants for real quadratic number fields 7 For X = α i α j Ψ i j { 3} i j do α A\{α i α j } If λ = λ λ t K exists with λ λ = 0 where α = Xλ then S S {A} Return T \S 3 Unimodular equivalent Humbert forms Now we describe an algorithm which decides whether two perfect Humbert forms S and T are unimodular equivalent Let A = {a a a 3 } MS and let us assume there holds 5 λ a = a with λ K If λ = λ λ λ λ O K λ 0 with gcdλ λ = then a = λ µ for some µ OK Now we get S[a ] = N K/Q λ S[µ] and we obtain N K/Q λ = = N K/Q λ Together this means 6 λ a = a [a ] = [a ] The same holds if we change the role of the a i i = 3 If A = {a a a 3 } MS and B = {b b b 3 } MT then let us assume withous loss of generality the first two element of A and B are K-linear independent Now we have a representation of the third element of each set with the both first elements It is easy to see that if a matrix U GL O K exists with respect to 4 then for the matrices X := a a and Y := b b there holds Y X GL O K If we assume a U GL O K with respect to 4 exists we have Ua a a 3 = ɛ b ɛ b b 3 ɛ ɛ O K and with a 3 = Xλ and b 3 = Y µ we get λ = λ λ t µ = µ µ t K with λ λ µ µ 0 because of 6 Now we obtain ɛ = µ λ and ɛ = µ λ Finally we have to test whether ɛ and ɛ are units in O K and ɛ b ɛ b X MS = MT Algorithm Equivalence of Humbert Tupels Input: The sets MS and MT of two perfect Humbert forms S and T of a real quadratic number field K Output: U GL O K with U MS = MT if S and T are unimodular equivalent false otherwise

8 8 Marcus WAGNER Michael E POHST S For all A = {a i a i OK i = 3} MS For all B = {b i b i OK i = 3} MT do Ψ {a i a j GL K i j { 3}} Ψ {b i b j GL K i j { 3}} For X = a i a j Ψ and Y = b k b l Ψ do If N K/Q det Y X = then a A\{a i a j } b B\{b k b l } λ λ λ t K with λ λ 0 where a = Xλ µ µ µ t K with µ µ 0 where b = Y µ ɛ µ λ ɛ µ λ If ɛ ɛ O K If ɛ b k ɛ b l X MS = MT then Returnɛ b k ɛ b l X Return false 33 Computing minimal vectors Next we want to compute minimal vectors of a given S P For doing this we need a constructive proof for the finiteness of minimal vectors of a given Humbert form The reason for this is that we need the quantities which are involved in this proof for the following algorithm which computes minimal vectors Lemma Let S = S S P C R >0 and K be a real quadratic number field Then the set 7 {x O K S [x] + S [x ] C} is finite Proof For all T R we denote with T the value min x 0 If T is a regular matrix then we have T T x = min x 0 x T x x x R = min y 0 y T y = T

9 On the computation of Hermite-Humbert constants for real quadratic number fields 9 with y = T x If we consider the Cholesky decompositions for S i = Ri tr i with R i GL R i = we get 8 S [x] = R x R x = R x and the same holds for S [x ] For the computation of R i we need the well known estimate R i R i R i i = and now we get with 8 S [x] R x R R x R where the same holds again for S [x ] If m := min i= R we get with x = a b t OK 9 m a + a + b + b = m x + x S [x] + S [x ] C and further 0 a + a = a a a a t ω = ω ω ω [a a t ] } {{ } :=A with a = a +a ω for an integral basis { ω} of O K and a a Z We know the matrix A is a positive definite form in Z because the trace of integral elements of a given maximal order are rational integral elements With 9 and 0 we obtain for any element z = x +x ω y +y ω t x x y y Z of the set in 7 the condition i i A[x x t ] C m and A[y y t ] C m and the set of solutions of these inequalities is finite and so the set in 7 is finite too The next lemma motivates lemma Lemma 3 Let S = S S P Then there exists a constant α R >0 such that suitable representants of the elements of the set MS are contained in the set { x O K S [x] + S [x ] α } With the last two lemmata we are able to compute minimal vectors of a given S P

10 0 Marcus WAGNER Michael E POHST Algorithm 4 Minimal Vectors Input: A Humbert form S = S S P the maximal order O K with integral basis { ω} α and m as in lemma and lemma 3 Output: The set J O K of all minimal vectors of S I J ai b S i CS i i = where S i = i and C = max{a b i c a } i ɛ 0 the fundamental unit ɛ of O K with ɛ > R m min i= R i i where S i = Ri tr i means the Cholesky decomposition of S i i = TrK/Q Tr A K/Q ω Tr K/Q ω Tr K/Q ω ln ɛ B 0 ln ɛ 0 L { x x t O K x i = x i + x i ω x i x i Z and A[x i x i t ] α m i = } µ min x L S[x] For x L do If S[x] = µ then I I {x} For x I do k n Z with 0 λ n < where ln λ = λ λ t R S [x] with Bλ = ln S [x ] J J {xɛ k } Return J 34 Computing eutactic coefficients For computing eutactic coefficients we use well known algorithms of combinatorics There exist classical algorithms to compute barycentric coordinates based on linear programming

11 On the computation of Hermite-Humbert constants for real quadratic number fields 4 Examples Now we have the theoretical background and the algorithms for computing extreme Humbert forms The algorithms are implemented in KASH/KANT To continue we need the following lemma: Lemma 5 Let S P and v i = α β t MS for i s := MS and let v ij for i j s be the determinants of the correspnding pairs: αi α v ij = det j β i β j Then for a fixed prime ideal p with corresponding valuation ν p we have: If {i j k} { s} is ordered so that ν p v ij maxν p v ik ν p v jk then ν p v ij ν p v ik = ν p v jk In particular if {i j} is such that ν p is maximal among all pairs {i j} we have ν p v ij ν p v ik = ν p v jk for k i j Proof For a proof see [BCIO] In the case Q 7 we are able to determine two extreme Humbert forms if we assume every extreme Humbert form has got a unimodular pair of minimal vectors Of course these obtained extreme forms must not achieve the Hermite-Humbert constant for Q 7 but they are the first known examples of extreme Humbert forms for this number field With Algorithm 0 we compute triples Now we solve polynomial equations to obtain Humbert forms and compute minimal vectors and if necessary eutactic coefficients We give an example with the triple { 5+ 7 b 3+ 7 If we write a Humbert form S as b b S = c b c } we obtain for every minimal vector x = x x t OK polynomial equations in c b and b S[x] = 0 x + b x x + cx cx + cb x x + x c = 0

12 Marcus WAGNER Michael E POHST We use resultants of polynomials for eliminating b and b to obtain a polynomial in c like fc = c 6 3 c c c Now we obtain by factorizing c fc = c + 7 c c c c and for gc := c + 8 7c we get c = 7 4 ± c Because of = the solutions c L := Q 7 and we obtain by c > 0 are in the field c = Then we fit c into the polynomial c 6c b c c + 8c b c c 3 + 7c c to obtain 3 b b = 0 which leads to b = In a last step we fit these solutions into the polynomial 8cb 5 + 7c c b c b c + 4c and obtain 7 7 9b = 0

13 On the computation of Hermite-Humbert constants for real quadratic number fields3 with b = Now we get the following Humbert form S = S S = with ms = For the eutactic coefficients we get ρ = ρ = ρ 3 = ρ 4 = ρ 5 = and for the minimal vectors we compute MS\{x 5 := e x 3 := e } = { x := Together we obtain 5+ 7 S = x := 3+ 7 x 4 := 5 xi x t i ρ i S [x] x i x t i S [x ] i= and dim X X S RX is equal to 5 Finally we compute as a local maximum of γ K γ K S = } In the same way we obtain an extreme Humbert form S = S S = with ms = For the eutactic coefficients we get ρ = 45 ρ = ρ 3 = + 85 ρ 4 = and for the minimal vectors we compute MS\{x 5 := e x 3 := e }= { } 3 7 x := x := x 4 := 3 7 Together we obtain S = 5 xi x t i ρ i S [x] x i x t i S [x ] i= and dim X X S RX is equal to 5 Finally We compute as a local maximum of γ K γ K S = und ρ 5 =

14 4 Marcus WAGNER Michael E POHST Now we can use our algorithms to compute the Hermite-Humbert constant for the case K = Q 3 We need the following lemma: Lemma 6 If K = Q 3 and γ K d K = 65 then any Humbert form S with more than 4 minimal vectors has got a unimodular pair of minimal vectors Proof Let assume there is no unimodular pair in MS With the notations of lemma 5 we obtain N K/Q det v ij γ K for each v ij and there holds ν pk v ij = for at least k = or k = 3 where p = and p p 3 = 3 Further only one of these prime ideals can divide each v ij by lemma 5 Suppose this is done by the prime ideal p Because of e MS we obtain ν p β = for all α β t MS Now let m := α β t m := α β t MS and M GL K with columns m and m By assumption ν p det M = = ν p α β α β = ν p γ + ν p α ɛ α ɛ with β i = γɛ i where ν p γ = and ɛ ɛ O K Because of O K/p = 4 and s 5 there muste be such m m MS that ν p α α > 0 but then there holds ν p α ɛ α ɛ > 0 a contradiction By the algorithm 0 we obtain 30 triples Because of lemma 6 we know all extreme Humbert forms must have a unimodular pair of minimal vectors We found 55 Humbert forms with more than 4 minimal vectors By algorithm we obtain 3 Humbert forms with more than four minimal vectors up to unimodular equivalence Their minimal vectors are listed in the following table: nr Minimal vectors without e and e

15 On the computation of Hermite-Humbert constants for real quadratic number fields5 We obtain a critical Humbert form S = S S with S = S = with ms = For its eutactic coefficients we compute ρ = ρ = ρ 3 = ρ 6 = ρ 4 = ρ 5 = and for its minimal vectors we get MS\{x 3 := e x 6 := e } = { x = x = x 4 = 3+ 3 x 5 = After verifying S = 6 xi x t i ρ i S [x] x i x t i S [x ] i= and dim X X S RX is equal to 5 we get because of S is critical γ K S = γ K = } Aknowledgement: We thank the referee for various insightful comments References [H] P Humbert Théorie de la réduction des formes quadratique définies positives dans un corps algébrique K fini Comment Math Helv [BCIO] R Baeza R Coulangeon MI Icaza and M O Ryan Hermite s constant for quadratic number fields Experimental Mathematics [C] R Coulangeon Voronoï theory over algebraic number fields Monographies de l Enseignement Mathématique [Co] H Cohn A numerical survey of the floors of various Hilbert fundamental domains Math Comp b [Co] H Cohn On the shape of the fundamental domain of the Hilbert modular group Proc Symp Pure Math [Ica] MI Icaza Hermite constant and extreme forms for algebraic number fields J London Math Soc Marcus Wagner Technische Universität Berlin Fakultät II Institut für Mathematik MA 8- Str d 7 Juni 36

16 6 Marcus WAGNER Michael E POHST D-063 Berlin Germany wagner@mathtu-berlinde Michael E Pohst Technische Universität Berlin Fakultät II Institut für Mathematik MA 8- Str d 7 Juni 36 D-063 Berlin Germany pohst@mathtu-berlinde