A Mordell Inequality for Lattices over Maximal Orders

A MORDELL INEQUALITY F OR LA TTICES O VER MAXIMAL ORDER S STEPHANIE V ANCE Abstract. In this paper we prov e an analogue of Mordell’s inequality f or lat- tices in finite-dimensional complex or quaternionic Hermitian space tha t are modules ov er a m aximal order in an imaginary quadratic num ber field or a to- tally definite rational quaternion algebra. Thi s inequalit y i mplies that the 16- dimensional Bar nes-W all lattice has optimal density among all 16-dimensional lattices with Hurwitz structures. 1. Introduction Lattice sphere packings in n -dimensional Euclidean spa ce a re configuratio ns o f congruent spheres with dis joint in terior s in whic h the sphere cen ters form a la ttice. These geo metric ob jects and mor e sp ecifically their densities, i.e., the prop o rtions of s pace cov ered by the spheres, ha ve been an activ e area o f research ov er t he past several cent uries. One particular problem in this resear ch are a that has r e ceived m uch attention is the n -dimensional lattic e spher e p acking pr oblem . F or this prob- lem one must determine the o ptimal density o f a lattice in n -dimensio nal Euclidea n space, with the density of a la ttice defined to b e the densit y of the la ttice sphere packing obtained by centering at each lattice p oint a sphere with radius equa l to half the length of the shortes t non-zero lattice vectors. Remark ably this problem remains op en for a ll dimensions n > 8 with the exception o f dimension 24 ; s ee [CS1] and [CK] fo r more de ta ils r egarding the known optimal lattices in dimensions n ≤ 8 and dimension 24. In most low dimensions divisible by 2 or 4, a t least one of the densest known lattices has an Eisenstein o r a Hurwitz str uc tur e r esp ectively; i.e., at least one of these lattices is isometr ic to a lattice in a co mplex or q ua ternionic Hermitian space that is a mo dule over the Eisenstein integers E = Z [ 1+ √ − 3 2 ] in the complex field C or the H urwitz integers H = Z [ i, j, 1+ i + j + k 2 ] in the quaternion skew-field H = { a + bi + c j + dk : a, b, c, d ∈ R , i 2 = j 2 = − 1 and ij = − j i = k } , resp ectively . F or example, the densest known lattices in every even dimension up to 2 4 all ha ve a n E isenstein structure, a nd the densest known la ttices in dimensions 4, 8 , 16, a nd 24 all hav e a Hurwitz structure (note that the densest known 12- dimensional lattice does not have a Hu rwitz structure). The ex istence of this extra algebraic s tr ucture among the denses t known la ttices makes it na tur al to consider the 2 m -dimensio nal Eisenstein a nd 4 m -dimensio nal Hurwitz lattice spher e packing problems, i.e., deter mine the optimal density o f 2 m a nd 4 m -dimensional Eisenstein 2000 Mathematics Subje ct Classific ation. Primary 11H06, 11H31. The author was supp orted by an ARCS F ound ation fellowship and a research assistantship funded by Microsoft Research. 1 2 STEPHANIE V ANCE and Hurwitz lattices. Ev en if the densest lattices in certain dimensions do not have an Eisens tein o r a Hurwitz structure , it would still b e interesting to determine the optimal density of the lattices that do satisfy these additional alg ebraic constr aints. In this pap er we address the 2 m and 4 m -dimensio nal Eis enstein and Hur witz lat- tice sphere packing pr oblems simultaneously by cons idering Eisenstein and Hurwitz lattices in the mor e g eneral context of O - lattices, i.e., lattices in finite- dimens ional complex or quaternionic Her mitia n spa ce that are mo dules over a maximal Z -or der O in an imaginary qua dratic num b er field or a totally definite quater nio n Q -algebr a, resp ectively . In the next section we prov e several prop os itio ns for O -lattices and their deter minants, i.e., the squar e d volumes of their fundamental regions. The s e prop ositions are used in Section 3 to prove a Mo rdell inequality fo r O -lattices which relates the optimal v alue of the Hermite inv ariant of O -la ttices in t wo consecutive complex o r q uaternionic dimensions. Note that the Hermite in v ariant of a n n - dimensional lattice Λ is directly pr op ortional to the ( n/ 2) th power of its density and is given by the formula γ (Λ) = N(Λ) det(Λ) 1 /n , where N(Λ) denotes the norm o f Λ (i.e., the norm of the sho rtest non- zero vectors in Λ) and det (Λ) deno tes the determina nt of Λ. Then in Section 4 w e use the Eisenstein and Hurwitz versions of this Mor dell inequality to obta in upp er b ounds for the Hermite inv a riants of Eisenstein and Hurwitz lattices in lo w dimensions, and we show that the 16 -dimensional Bar nes-W all lattice has optimal density as a 16-dimensiona l lattice with a Hurwitz structure . While the focus of this pap er is on Eisenstein and Hurwitz lattices, w e also consider in Sectio n 5 Gaussia n lattices in C m (i.e., O = Z [ i ]) and the lattices in H m that a re mo dules ov er the ring O = Z [1 , i, 1+ j 2 , i + k 2 ]. As w ith the Eisenstein and Hurwitz cases discussed ab ove, some of the densest known lattices in dimensions 2 m and 4 m hav e one of these O -la ttice structures. Note that all four t yp es of O - lattices discussed in this pap er can b e regar ded a s G -lattices in the sense o f [Ma, ch. 13]. 2. O -La ttices in Complex and Qua ternion ic Hermitian sp ace The following notation is us e d thro ughout this pa p e r. • Let K denote either the c o mplex field C or the qua ternionic skew-field H , and let r denote the r ank o f K as an R -algebr a, i.e., r = 2 or r = 4 . Then let x 7→ x denote complex conjugatio n when K = C and qua ternionic conjugation, i.e., a + bi + cj + dk 7→ a − b i − cj − dk for a, b, c, d ∈ R , when K = H . • When K = C let A denote an imaginary q ua dratic num b er field a nd when K = H let A denote a totally definite quater nion Q -a lgebra. In b oth case s we s hall identify K with R ⊗ Q A . • Let O denote the image in K of a maximal Z -or der in the Q -alg ebra A ; i.e., O is a free Z -submo dule and subring of A (wher e A is identified with its image in K ) satisfying rank Z O = rank R K and Q O = A . Then let D O = det   1 2 ( α i α j + α j α i )  1 ≤ i,j ≤ r  , where α 1 , . . . , α r denotes a Z -basis for O . Note that D O is indep endent of the Z -basis for O used to co mpute it, and hence is a n inv ar iant of O . A MORDELL INEQUALITY FOR LA TTICES OVER MAXIMAL ORDERS 3 • Finally let E r m denote a left m -dimensio nal K -vector spa ce with a non- degenerate Her mitia n pro duct h : K × K → K (i.e., h is left linear in the first v a riable and right co njugate linear in the second v ariable) a nd define the inner pro duct of each pair of vectors x, y ∈ E r m to b e h x, y i = 1 r T r K / R ( h ( x, y )) = 1 2  h ( x, y ) + h ( x, y )  so that N( x ) = h ( x, x ) = h x, x i ; i.e., the dista nces defined o n E r m by h ( · , · ) and h· , ·i are the sa me 1 . Using this notation w e define an rm - dimensional O - la ttice to be a left O - inv aria nt lattice in E r m , i.e., a lattice in E r m that is a left O -mo dule. In this section we prov e several pr op ositions concerning an r m -dimensio nal O -lattice Λ and its O - dual, which is defined to b e the O -mo dule Λ # = { x ∈ E r m : h ( x, Λ) ⊆ O } . Prop ositi on 2.1. If Λ is an O -lattic e in E r m , the n Λ # is also an O -latt ic e in E r m , and the determinants of these two lattic es satisfy the pr o duct formula det(Λ) det(Λ # ) = D 2 m O . T o prove Prop o sition 2.1 we us e the following lemma to deal with the ca se when Λ is a no n-free O -lattice (i.e., when Λ is not a free O -mo dule). Lemma 2 .2. F or every O - lattic e Λ in E r m ther e exists two fr e e O -lattic es Λ 1 and Λ 2 in E r m such that Λ 1 ⊆ Λ ⊆ Λ 2 . Pr o of. An O -lattice Λ genera tes E r m as a r eal vector space and so must contain a K -vector spa ce ba sis for E r m . Hence we can choo se Λ 1 to b e the free O -la ttice generated over O by such a K -vector space basis contained in Λ. No w consider the t wo Q -vector s ubspaces Q Λ 1 and Q Λ of E r m . Both subspaces hav e dimensio n r m , and s ince Q Λ 1 ⊆ Q Λ they must b e equal. Thus Λ is a finitely generated s ubmo dule of Q Λ 1 , and so w e can choos e an a ∈ Z such that Λ ⊆ a − 1 Λ 1 , i.e., Λ is c o ntained in the fr ee O - la ttice Λ 2 = a − 1 Λ 1 .  Pr o of of Pr op osition 2.1. W e first verify that the O - dual of every O -lattice in E r m is also an O - lattice in E r m . Every O -bas is for a fr e e O -lattice in E r m is necessa rily a K -vector space basis for E r m , say v 1 , . . . , v m , and the dual K -vector spac e basis v # 1 , . . . , v # m satisfying h ( v # i , v j ) = δ i,j is an O -ba sis for the O -dual. Hence the O - dual o f a free O -lattice in E r m is a lso a free O -lattice in E r m . In particular, if Λ 1 and Λ 2 are the tw o free O -lattices in E r m given by Lemma 2.2, then bo th Λ # 1 and Λ # 2 are free O -lattices in E r m . The inclusion Λ 1 ⊆ Λ ⊆ Λ 2 then implies that Λ # 2 ⊆ Λ # ⊆ Λ # 1 , a nd so Λ # m ust also b e an O -lattice in E r m . T o pro ve the determinant identit y we fir s t consider the cas e when Λ is a free O - lattice with O -ba sis v 1 , . . . , v m . Letting α 1 , . . . , α r denote a Z -ba sis for O , the vectors { α i v j } 1 ≤ i ≤ r, 1 ≤ j ≤ m form a Z -basis for Λ and the vectors { α i v # j } 1 ≤ i ≤ r, 1 ≤ j ≤ m form a Z -bas is for Λ # . W rite the vectors in these tw o Z -ba ses us ing co ordinates 1 The fact or 1 2 in the definition of the inner pro duct h· , · i i s inserted to simplify later calculations and can be omitted without affecting the density of a lattice in E r m . One of the reasons for omitting this factor would b e to ensure that every l attice Λ ha ving the pr operty that h ( x, y ) ∈ O for every x, y ∈ Λ is necessarily inte gral, i.e., h x, y i ∈ Z for every x , y ∈ Λ . 4 STEPHANIE V ANCE with respec t to a fixed orthonormal basis for E r m (the basis being or thonormal with resp ect to the inner product h· , ·i ), and let M be a n r m × rm matrix such that for 1 ≤ i ≤ r a nd 1 ≤ j ≤ m the ( r ( j − 1) + i ) th row o f M is the vector α i v j . Similarly let N b e a n r m × rm matrix such that for 1 ≤ i ≤ r and 1 ≤ j ≤ m the ( r ( j − 1) + i ) th row o f N is the vector α i v # j . Observe that the tw o matrices M and N satisfy det(Λ) = det( M M T ) and det(Λ # ) = det( N N T ). Hence the discr imina nt ideals 2 d (Λ) and d (Λ # ) sa tisfy d (Λ) d  Λ #  = Z det ( M M T ) det( N N T ) = Z det ( M N T ) 2 . (Note that we are computing the pro duct of discr iminant ideals her e ra ther tha n the lattice determinants, so that o ur calculations can b e gener alized for the non-free case.) W e ca n compute the determinant o f the matrix M N T using the fact that fo r 1 ≤ i 1 , i 2 ≤ r and 1 ≤ j 1 , j 2 ≤ m , the ( r ( j 1 − 1 ) + i 1 , r ( j 2 − 1 ) + i 2 ) th ent ry in M N T is equal to h α i 1 v j 1 , α i 2 v # j 2 i = 1 2  h ( α i 1 v j 1 , α i 2 v # j 2 ) + h ( α i 2 v # j 2 , α i 1 v j 1 )  = 1 2 ( α i 1 α i 2 + α i 2 α i 1 ) δ j 1 ,j 2 . In par ticular, M N T is a n r m × rm blo ck-diagonal matrix with m blo cks all equa l to the r × r matrix  1 2 ( α i α j + α j α i )  1 ≤ i,j ≤ r whose determinant is D O . Therefore, d (Λ) d  Λ #  = Z m Y s =1 D O ! 2 = Z D O 2 m , and from these e q ualities the determinant ide ntit y follows. Now supp o se that Λ is a non-free O -lattice. Notice that in the preceding defini- tions a nd ca lculations we may replace Z by any lo calizatio n, in particula r by Z P , where P is a prime idea l in Z . Letting S = Z \ P w e have (Λ P ) # = { x ∈ K : ∀ s ∈ S, h ( x, s − 1 Λ) ⊆ S − 1 O} , and since h ( x, s − 1 Λ) = s − 1 h ( x, Λ), we may replac e h ( x, s − 1 Λ) by h ( x, Λ) in the formula ab ov e , s howing that (Λ P ) # = (Λ # ) P . Mo reov er, since Λ P is a free O P - mo dule ([AG], Prop o sition 3.7) generated by a K -vector space basis of E r m , the pro duct of the discriminant idea ls d (Λ P ) and d ((Λ P ) # ) is equal to Z P D 2 m O P = Z P D 2 m O (here D O P is eq ual to the determinant of the Gram matrix co rresp onding to a Z P -basis fo r O P ). Therefore for every prime ideal P in Z we have Z P d (Λ) d (Λ # ) = d (Λ P ) d ((Λ # ) P ) = d (Λ P ) d ((Λ P ) # ) = Z P D 2 m O , implying that d (Λ) d (Λ # ) = Z D 2 m O . The deter mina nt identit y det(Λ) det(Λ # ) = D 2 m O follows fro m this last equa lity .  Observe from the definition of the O -dua l of an O -lattice Λ that we alwa ys have the co ntainment Λ ⊆ Λ ## . No w since the deter minant identit y in Pro p o sition 2.1 implies that det(Λ) = det(Λ ## ), it then follows that Λ = Λ ## . 2 The discriminant ideal of an n -dimensional l attice Λ is equal to the i deal in Z consisting of the determinan ts of al l n × n Gram matrices ( h w i , w j i ) where w 1 , . . . , w n ∈ Λ. Note that since Λ is a free Z -mo dule of r ank n , it is not hard to sho w that d (Λ) = Z det(Λ). A MORDELL INEQUALITY FOR LA TTICES OVER MAXIMAL ORDERS 5 W e conclude this s ection with another determinant identit y for O -lattices tha t inv olves the in tersection of an rm -dimensional O -lattice Λ with a K -vector sub- space F of E r m and the pro jection of Λ ont o F ⊥ , i.e., the K -vector subspace of E r m per p endicular to F with resp ect to h· , ·i (or equiv alently h ( · , · ) due to the r e- lationship b etw een h· , ·i and h ( · , · )). Note that b elow we r efer to Λ ∩ F as a re lative lattic e in F because it is a lattice in a K -vector subspace o f F that may no t b e full-dimensional. Also, we reg ard both K -vector subspaces F a nd F ⊥ of E r m as K -Hermitian s paces (and hence ca ll them K -Her mitian s ubspaces) b eca us e we ca n restrict the Hermitian pro duct h ( · , · ) to F × F and F ⊥ × F ⊥ , r esp ectively . Lemma 2.3. L et Λ b e an r m -dimensional O -lattic e and let F b e a K -Hermitian subsp ac e of E r m . (1) The r elative O -lattic e Λ ∩ F is an O - lattic e in F if and only if π F ⊥ (Λ) is an O -lattic e in F ⊥ . (2) If Λ ∩ F is an O -lattic e in F , then det(Λ) = det(Λ ∩ F ) de t ( π F ⊥ (Λ)) . Pr o of. This lemma is pro ved a s Prop ositio n 1.2.9 in [Ma] for lattices in finite- dimensional E uclidean spa ce. Because Λ ∩ F a nd π F ⊥ (Λ) ar e b oth O -mo dules, the O - lattice version readily follows.  Prop ositi on 2.4. If Λ is an rm -dimensional O -lattic e and F is a K -Hermitian subsp ac e in E r m , t hen Λ ∩ F is an O -lattic e in F if and only if Λ # ∩ F ⊥ is an O - lattic e in F ⊥ . Mor e over if these c onditions hold and s = dim K F , then det(Λ) = det(Λ ∩ F ) det (Λ # ∩ F ⊥ ) − 1 D O 2( m − s ) . Pr o of. F o r every x ∈ F ⊥ and y ∈ Λ we can wr ite h ( x, y ) = h ( x, π F ( y ) + π F ⊥ ( y )) = h ( x, π F ⊥ ( y )), and from this fact we can co nc lude that ( π F ⊥ (Λ)) # = Λ # ∩ F ⊥ . Prop os itio n 2.1 then implies π F ⊥ (Λ) is an O -lattice in F ⊥ if and only if Λ # ∩ F ⊥ is an O -lattice in F ⊥ . (Note tha t for the backwards direction w e are using the fact that π F ⊥ (Λ) is a n O -submo dule of (Λ # ∩ F ⊥ ) # satisfying R π F ⊥ (Λ) = F ⊥ .) Therefore by Lemma 2.3 (1), Λ ∩ F is an O -la ttice in F if and only if Λ # ∩ F ⊥ is an O -lattice in F ⊥ . Suppo se now that Λ ∩ F and Λ # ∩ F ⊥ are O -la ttices in F a nd F ⊥ , r esp ectively , and reca ll that ( π F ⊥ (Λ)) # = Λ # ∩ F ⊥ . By Lemma 2.3 (2 ) we hav e the ident ity det(Λ) = det(Λ ∩ F ) det(Λ # ∩ F ⊥ ) # , which by P rop osition 2.1 can b e r ewritten as det(Λ) = det(Λ ∩ F ) det (Λ # ∩ F ⊥ ) − 1 D O 2( m − s ) .  3. Mordell ’s inequal ity f or O -la ttices Using the notation in tro duced in the previous section, define the O -Hermite constant γ ( O , r m ) to b e the supr emum of the Her mite inv aria nt γ (Λ) = N(Λ) det(Λ) 1 / ( r m ) for an rm -dimensional O -lattice Λ. Observe t hat this constant is finite sinc e the Hermite inv ariant of an r m -dimensional lattice is dire c tly pr o p ortional to the  r m 2  th power of its density , with the latter quantit y b ounded by 1 . Moreov er, by 6 STEPHANIE V ANCE Mahler’s compactness theore m (s e e [Ma, p. 43 ]) there exists a n rm - dimensional O - lattice Λ with det(Λ) = 1 such that γ ( O , rm ) = γ (Λ), i.e., Λ is an opti- mal r m -dimensional O - la ttice (note that here w e a re using the fact tha t the set of r m -dimensional O -la ttices with determinan t one is closed in the spac e of r m - dimensional lattices, and γ ( O , rm ) is equal to the o ptimal v alue of norms of these lattices which must b e b ounded since γ ( O , r m ) is finite). Below w e use the results of Section 2 to prove an inequalit y r e lating the O - Hermite consta nt s for dimensions r ( m − 1 ) a nd r m , provided m ≥ 3. W e c ho ose to call this inequalit y a Mordell inequality for O -lattices due to its resemblance to the inequa lity γ n − 1 ≤ γ ( n − 1) / ( n − 2) n in Mo rdell’s theorem (see [Ma, p. 4 1]). In Mordell’s theo rem the constant γ n denotes Hermite’s constant for dimension n , i.e., the optimal v alue o f the Hermite inv ar iant of a n n -dimensiona l lattice. Theorem 3.1. F or e ach inte ger m ≥ 3 , (3.1) γ ( O , rm ) ≤ γ ( O , r ( m − 1)) m − 1 m − 2 D 1 r ( m − 2) O . Equality holds if and only if the O -dual lattic e of every optimal r m - dimensional O - lattic e Λ is also optimal, and for al l minimal ve ctors x ∈ Λ # the r elative O -lattic es Λ ∩ ( K x ) ⊥ and Λ # ∩ K x satisfy: (1) N(Λ ∩ ( K x ) ⊥ ) = N(Λ) , (2) Λ # ∩ K x = O x , (3) γ (Λ ∩ ( K x ) ⊥ ) = γ ( O , r ( m − 1)) . Pr o of. Let Λ b e an optimal O -la ttice in E r m with det(Λ) = 1 , let x b e a minimal vector in the O -dual lattice Λ # , and let F denote the ( m − 1)-dimensio nal subspace ( K x ) ⊥ . By Pr op osition 2 .4 the r elative O -lattice Λ ∩ F is a n O -la ttice in F b ecaus e Λ # ∩ F ⊥ is an O -lattice in F ⊥ = K x (note that O x ⊆ Λ # ∩ F ⊥ ). Then by the definition for the Hermite inv a r iant of a lattice γ (Λ) = N(Λ) ≤ N(Λ ∩ F ) = γ (Λ ∩ F ) det(Λ ∩ F ) 1 r ( m − 1) . W e wish to b ound the term det(Λ ∩ F ) in the last equa lit y by an expression inv olving the Hermite inv aria nt o f the O -dua l lattice Λ # . T o do this we first use the determinant ident ity in Pr o p osition 2.4 with the definition of the Her mite in v aria nt to get det(Λ ∩ F ) = det(Λ) det(Λ # ∩ F ⊥ ) D O − 2 ≤ det( O x ) D O − 2 . Then we let α 1 , . . . , α r denote a Z -basis for O so that det( O x ) = det(( h α i x, α j x i ) 1 ≤ i,j ≤ r ) = det  1 2  h ( α i x, α j x ) + h ( α i x, α j x )   1 ≤ i,j ≤ r ! = det  N( x ) 2 ( α i α j + α j α i )  1 ≤ i,j ≤ r ! = N(Λ # ) r D O . A MORDELL INEQUALITY FOR LA TTICES OVER MAXIMAL ORDERS 7 Observe that by Pr op osition 2.1 and the definition of γ (Λ # ), the term N(Λ # ) r in the last expressio n is eq ual to γ (Λ # ) r D O 2 . Hence we hav e shown that det( O x ) = γ (Λ # ) r D O 3 , with which the upp er b o und for det(Λ ∩ F ) computed ab ove implies that det(Λ ∩ F ) ≤ γ (Λ # ) r D O . Substituting the la s t upper bound for det(Λ ∩ F ) in to the initial expr ession inv olving γ (Λ) we obtain γ (Λ) ≤ γ (Λ ∩ F )  γ (Λ # ) r D O  1 r ( m − 1) = γ (Λ ∩ F ) γ (Λ # ) 1 ( m − 1) D O 1 r ( m − 1) ≤ γ ( O , r ( m − 1 )) γ ( O , rm ) 1 ( m − 1) D O 1 r ( m − 1) . Then, since γ (Λ) = γ ( O , r m ), γ ( O , rm ) ≤ γ ( O , r ( m − 1)) γ ( O , r m ) 1 ( m − 1) D O 1 r ( m − 1) , and from this we obtain our desired ineq ua lity , γ ( O , rm ) ≤ γ ( O , r ( m − 1 )) m − 1 m − 2 D O 1 r ( m − 2) . Observe that from the ab ov e pro of of Inequality (3.1), equality holds if and only if all of the inequalities introduced in bo unding γ (Λ) are tight, i.e., if and o nly if γ (Λ # ) = γ ( O , r m ) and (1 ), (2) and (3) hold for every minimal vector x ∈ Λ # (the pro of a b ov e us es an ar bitrary minimal vector x ∈ Λ # ). More generally , this la st statement holds for any o ptimal lattice Λ that do es not necessarily hav e deter minant 1.  W e choose to ca ll the b ound for γ ( O , r m ) g iven by Inequality (3.1) the Mor del l b ound for r m -dimensio nal O -lattices. In Corollary 3.2 b elow we give an iterated version of this bo und which follows from a s imple induction ar gument with Theore m 3.1 as the bas e case. Corollary 3.2. F or e ach p air of inte gers s > m ≥ 2 , γ ( O , rs ) ≤ γ ( O , r m ) s − 1 m − 1 D O s − m r ( m − 1) . 4. Eisenstein and Hur witz La ttices Recall that Hurwitz la ttices a re lattices in quater nionic Hermitian space that ar e mo dules over the maximal Z -or der H = Z [ i, j, 1+ i + j + k 2 ] in  − 1 , − 1 Q  , and we say a 4 m -dimensional lattice (in Euclidean space) has a Hurwitz structure if it is isometric to a Hurw itz lattice. Similarly , E isenstein lattices are la ttices in complex Her mitian space that ar e mo dules ov er the maximal Z -order E = Z [ 1+ √ − 3 2 ] in Q ( √ − 3) (i.e., E is the ring of integers in Q ( √ − 3)), and we say a 2 m -dimensio nal lattice has an Eisenstein str ucture if it is isometric to an Eis enstein lattice. In this s e ction we us e the Mo rdell b ounds for E is enstein and Hurwitz lattices (Theorem 3 .1) to obtain upp er b ounds for the Hermite inv a riants of low-dimensional lattices with an Eisenstein or Hurwitz structure. The upp er b ounds we obtain are compared in T ables 2 and 4 to the b es t upper b ounds previously kno wn and in several instances give an improv e ment . Note tha t for the dimens io ns in which the Eisenstein and Hurwitz-Hermite constants ar e not known, fo r comparis o n purp oses 8 STEPHANIE V ANCE T able 1. The densest known la ttices with Hurwitz structures in dimensions 4 m ≤ 28 4 m Lattice(s) Hermite Inv ariant 4 Λ 4 = D 4 √ 2 ≈ 1 . 414 21 8 Λ 8 = E 8 2 12 Λ min 12 , Λ max 12 2 7 / 6 ≈ 2 . 2449 2 16 Λ 16 2 3 / 2 ≈ 2 . 8284 3 20 Λ 20 2 17 / 10 ≈ 3 . 2490 1 24 Λ 24 4 28 Λ 28 , L L 28 2 27 / 14 ≈ 3 . 8067 8 we use the upp er b ounds for Hermite’s constants in these dimensio ns pr ov ed b y Henry Cohn and Noam Elk ie s in [CE]; the author is not aw a re of any b e tter b ounds computed sp ecifically for lattices with an Eis e nstein or a Hurwitz structure. 4.1. The Mordell Bound for Hurwitz L attices. The densest known lattices with Hurwitz structures in dimensions 4 m ≤ 28 are lis ted in T able 1 with their Hermite inv a r iants 3 . The lattices listed in this table for dimensions 4, 8, 12, and 24 have all b een pr ov en optimal as lattices with a Hurwitz structure, a nd with the exception of dimension 12 this is a corolla ry to their prov en optimality as lattices; the 12- dimensional lattices listed in T able 1 hav e b een prov en o ptimal as lattices with Hurwitz s tructures by F ran¸ cois Sigrist a nd David-Olivier Jaquet-Chiffelle, and a summary of their ca lculations is given in [Si]. 4 In T able 2 we lis t the Mordell b ounds for Hurwitz lattices in dimensions 4 m ≤ 28 alongside the b est upp er b ound for γ ( H , 4 m ) pr e viously k nown. Note that e a ch of the Mor dell bo unds listed hav e b ee n computed itera tively; i.e., if the Morde ll b ound for 4( m − 1)-dimens io nal Hurwitz la ttices is an improvemen t on the best known upper b o und previously known for γ ( H , 4( m − 1)), then it is used to compute the Mordell b ound for 4 m -dimensional Hurwitz lattices. W e hav e also included in T able 2 a conjectured Mor dell b ound for 24-dimensiona l Hur w itz lattices which is obtained by replacing γ ( H , 20) with γ (Λ 20 ) in Inequality (3.1). Notice that the Morde ll b ound for 1 6 -dimensional Hurwitz lattices is no t o nly an impr ovemen t on the best upp er b ound fo r γ ( H , 16) previously known, but it is eq ua l to the Hermite inv aria nt o f the 16-dimensional B arnes-W a ll lattice Λ 16 . Therefore we hav e prov ed the following theorem. Theorem 4.1 . The 1 6 -dimensional Barnes wal l latt ic e Λ 16 has optimal density as a 1 6 -dimensional lattic e with a Hurwitz structur e. Mor e over, every optimal 16 - dimensional lattic e with a Hurwitz structur e c ont ains ei ther Λ min 12 or Λ max 12 as a 12 -dimensional se ction with identic al norm. Note tha t we ha ve to b e ca r eful when tra nslating the conditio ns in Theo r em 3.1 to 4 m -dimens io nal lattices with a Hurwitz struc tur e bec ause it may be p ossible for a 4 m -dimensional lattice to hav e t wo or more inequiv alent Hurwitz structur e s; that is, a lattice may be isometr ic to tw o differ e nt Hurwitz lattices such that there do es 3 See [NS] for further information on the l attices li sted i n T able 1. 4 The det ails of Si gr ist and Jaquet-Chiffelle’s calculations c an also b e found i n Ac hill Sc h ¨ urmann’s pap er [Sch]. A MORDELL INEQUALITY FOR LA TTICES OVER MAXIMAL ORDERS 9 T able 2. The Mo rdell b ound and conjectured Mordell b ound for the Hurwitz Hermite constant for dimensio ns 4 m ≤ 28 4 m Best Known Mordell Bo und Conjectured Upper Bo und Mordell Bo und 4 √ 2 ≈ 1 . 414 21 – – 8 2 – – 12 2 7 / 6 ≈ 2 . 2449 2 2 3 / 2 ≈ 2 . 8284 3 – 16 3 . 0263 9 2 3 / 2 ≈ 2 . 828 43 – 20 3 . 5200 6 2 11 / 6 ≈ 3 . 5635 9 – 24 4 4 . 2139 0 4 28 4 . 4886 3 2 23 / 10 ≈ 4 . 9245 8 – not exist an isometry b etw een them that is an H -mo dule isomor phism. F or example it is currently an op en pr oblem to determine if Λ 16 has tw o inequiv a lent Hurwitz structures (see [Ma, p. 279 ]). With the Hurwitz s tructure presently known for Λ 16 , all of the 1 2-dimensiona l sections p erp endicular to a minimal vector in Λ # 16 are Λ max 12 . Jacques Martinet has shown in [Ma, Ch. 8] that if ano ther inequiv alent Hurwitz structure exis ts fo r Λ 16 , then all of the 12- dimensional sections p erp endicula r to a minimal vector in Λ # 16 (computed with r esp ect to the new structur e ) are Λ min 12 . One final obser v ation we wis h to make concer ning T able 2 is that the co njectured Mordell b ound for 24-dimensio nal Hur witz lattices is tight, and from this we ca n use Theore m 3.1 to make conjectures ab out the conditions sa tisfie d by certain 20-dimensiona l se ctions of the 24-dimens io nal Leech lattice a s was done for the 1 6- dimensional Barnes -W all lattice ab ove. Ho wev er , unlike the la tter case, the Leech lattice has b een proven to have a unique Hurwitz str ucture by H.-G. Q uebb e mann (see [Qu]). 4.2. The M ordell Bound for E isenstei n lattices. W e now list in T able 3 the densest k nown lattices with E isenstein structures in dimensions 2 m ≤ 26 with their Hermite inv ariants 5 , 6 . All of the la ttices in T able 3 are among the densest known lattices in their dimens ion; how e ver, we note that this table is not a complete list of the densest known la ttices for the dimensio ns shown b ecause the density of the Conw ay-Borcherds lattice T 26 is equa l to that of Λ 26 and the for mer la ttice do es not hav e an Eisenstein structure. The la ttices listed for dimensions 2 m ≤ 8 and for dimension 24 hav e a ll b een prov en optimal as or dinary la ttices (a nd hence ar e optimal a s lattices with an Eisenstein structure), and the lattice Λ 10 has recently bee n prov e n optimal a s a lattice with an Eisenstein structure by Achill Sch¨ urmann; see [Sch] fo r details. In T a ble 4 we list the Mordell b ounds fo r Eisens tein lattices in dimensio ns 2 m ≤ 26 next to the b est upp er b o und pr e viously known for γ ( E , 2 m ). Note that the Mo rdell b ounds listed in this table are computed iteratively , a s was done for the Hurwitz case . Included in T able 4 is a column listing the co njectured Mordell 5 See [NS] for further information on the l attices in T able 3. 6 All l attices li sted in T able 3, with the exception of the 26-dimensional laminated Eisenstein lattice Λ 26 , ar e unique as lattices and hav e a unique Eisenstein structure; see [CS2] and [CS3] for more information on the non-uniqueness of Λ 26 as a lattice and for the uniqueness of the Eisenstein structures on the other lattices. 10 STEPHANIE V ANCE T able 3. The dens est known lattices with E isenstein structures in dimensio ns 2 m ≤ 26 2 m Lattice(s) Hermite Inv ariant 2 Λ 2 = A 2 2 / √ 3 ≈ 1 . 154 70 4 Λ 4 = D 4 √ 2 ≈ 1 . 414 21 6 Λ 6 = E 6 2 / 3 1 / 6 ≈ 1 . 66 537 8 Λ 8 = E 8 2 10 Λ 10 2 6 / 5 / 3 1 / 10 ≈ 2 . 05 837 12 K 12 4 / √ 3 ≈ 2 . 309 40 14 Λ 14 2 10 / 7 / 3 1 / 14 ≈ 2 . 48 864 16 Λ 16 2 3 / 2 ≈ 2 . 82 843 18 Λ 18 2 5 / 3 / 3 1 / 18 ≈ 2 . 98 683 20 Λ 20 2 17 / 10 ≈ 3 . 24 901 22 Λ 22 2 21 / 11 / 3 1 / 22 ≈ 3 . 57 278 24 Λ 24 4 26 Λ 26 4 / 3 1 / 26 ≈ 3 . 83 450 T able 4. The Mo rdell b ound and conjectured Mordell b ound for the Eis enstein Her mite constant for dimensions 2 m ≤ 26 Dimension Best Known Mordell Bo und Conjectured 2 m Upper Bo und Mordell Bo und 2 √ 3 / 2 ≈ 1 . 154 70 – – 4 √ 2 ≈ 1 . 414 21 – – 6 2 / 3 1 / 6 ≈ 1 . 66 537 √ 3 ≈ 1 . 732 05 – 8 2 2 – 10 2 . 0583 7 2 · 3 1 / 6 ≈ 2 . 4018 7 – 12 2 . 5217 9 2 5 / 4 ≈ 2 . 378 41 – 14 2 . 775 80 2 13 / 10 · 3 1 / 10 ≈ 2 . 748 22 2 11 / 5 / √ 3 ≈ 2 . 65281 16 3 . 0263 9 3 . 1755 2 2 3 / 2 ≈ 2 . 82843 18 3 . 2743 3 3 . 4730 0 2 11 / 7 · 3 1 / 14 ≈ 3 . 21460 20 3 . 5200 6 3 . 7299 6 2 7 / 4 ≈ 3 . 36359 22 3 . 7640 4 3 . 9841 6 2 16 / 9 · 3 1 / 18 ≈ 3 . 64478 24 4 4 . 23616 4 26 4.2480 4 2 23 / 11 · 3 1 / 22 ≈ 4 . 4783 1 – bo und for Eisenstein lattices in the dimensions for which the optimal densit y of an Eisens tein la ttice with o ne less complex dimensio n is not known. These conjec- tured b ounds are co mputed using Inequality (3.1) in Theorem 3.1 with the Hermite inv ariants o f the densest k nown 2( m − 1 )- dimens ional E is enstein lattices repla cing γ ( E , 2( m − 1)). The b old en tries in T able 4 indicate an improvemen t (actual o r conjectured) on the b est kno wn upper b o und for γ ( E , 2 m ). Sp ecifically , w e hav e improved o n the b es t known upp er b ounds for the Hermite inv ar iants of Eisens tein la ttices in A MORDELL INEQUALITY FOR LA TTICES OVER MAXIMAL ORDERS 11 dimensions 12 and 14 , with the Mordell bo und co mputed for γ ( E , 12) b eing muc h closer to the Hermite inv aria nt of the Coxeter-T o dd lattice K 12 than the uppe r bo und from [CE]. Unfortunately with the improv ed bo und for γ ( E , 12) w e are unable to c onclude that the Coxeter T o dd lattice K 12 is optimal as a 12-dimensio nal lattice with an Eisenstein str ucture, despite the fact that it is widely b elieved to be optimal as a 1 2-dimensional lattice. How ever, one could hav e a nticipated that this bo und would not b e sharp b ecause K 12 do es no t contain the lattice Λ 10 as a 10- dimensional sec tion, but ra ther contains the lattice K ′ 10 . F or if the Mordell bo und for γ ( E , 12 ) were tight, then b y Theo rem 3.1 a ny o ptimal lattice would contain Λ 10 as a 10 -dimensional section b ecause Λ 10 is the unique optimal 10 -dimensional lattice with an Eisenstein s tructure. Moving on to the conjectured Mor dell b ounds for E isenstein la ttices display ed in T able 4 , note that if o ne could pr ov e Λ 14 is o ptimal as a 14-dimensional lattice with an Eisenstein structure, then the Morde ll bo und for 16- dimensional Eisenstein lattices would imply that the 16- dimensional B arnes-W a ll la ttice Λ 16 is optimal as a lattice with an E isenstein structure . Moreover, due to the conjectured tig ht ness of the Mordell b o unds fo r 16 and 2 4-dimensiona l E isenstein lattices, w e ca n make conjectures ab out o ptimal 16 and 24 -dimensional Eisenstein lattices, as was done in the Hur witz ca se ab ove for these sa me dimensions. 5. Remarks an d Open Problems Even thoug h the foc us of this pap er is on lattices with E isenstein and Hurwitz structures, we wish to empha size that Theorem 3.1 applies to o ther type s of O - lattices. O ther O -la ttices one can consider include the 2 m -dimensio nal Gaussian lattices ( O = Z [ i ]) and the 4 m -dimensiona l J -lattices where J is the subring Z [1 , i , 1+ √ 3 j 2 , i + √ 3 k 2 ] in H co r resp onding to the maximal Z -o rder Z [1 , i , 1+ j 2 , i + k 2 ] in the ra tional p ositive definite qua ternion alg ebra  − 1 , − 3 Q  = { a + bi + cj + dk : a, b, c , d ∈ Q and i 2 = − 1 , j 2 = − 3 , ij = − j i = k } . Many of the denses t known even-dimensional la ttices hav e a Gaussian structure, and the 4 m -dimensional J -lattices are o f interest b ecause they include the Coxeter- T o dd lattice K 12 , the densest known 1 2 -dimensional lattice, which cur iously do es not have the structure of a Hurwitz lattice; see [Gr] concerning the existence of a J -lattice structure on K 12 . Note that the J -Hermite constant for dimension 8 is equal to γ ( E 8 ); unfortunately K 12 do es not contain E 8 as an 8 -dimensional section (see Prop os ition 8.7.9 in [Ma]), and so we cannot use Theo rem 3.1 to c o nclude that K 12 is o ptimal as a 12 -dimensional J -lattice. It may b e p ossible to use Theor em 3 . 1 to pr ov e that an r ( m − 1)-dimensional lattice do es not have a particular O -lattice structure. As p ointed out to the author by Jacques Martinet, if we know the v alue of γ ( O , r m ) but do not k now the v a lue of γ ( O , r ( m − 1)), then we can replace γ ( O , r ( m − 1)) in the inequa lity by the Her mite inv ariant of an r ( m − 1)-dimensional lattice Λ a nd chec k if the inequality remains v alid. If the ineq uality is no long e r v a lid, then this implies that Λ do es not hav e an O - lattice structur e . It would b e interesting to find exa mples of lattices that can b e prov en no t to hav e c ertain O -la ttice struc tur es using this metho d. By Theor em 8.7.2 in [Ma], the 8-dimensional lattice E 8 has an O -la ttice structure ov er every maximal order O in an imag inary qua dratic n umber field. Given that 12 STEPHANIE V ANCE the Mordell inequality for Eisens tein a nd Gaussian lattices in dimension 8 is sharp (note that the densest 6- dimensional Gaussian lattice is D 6 ; see [Sch ]), it is na tural to wonder if the Mordell inequality is shar p for O - lattices in dimension 8 whenever O is a maximal order in a n imag inary quadr atic num b er field. Ev en if this is not true in gener a l, it still would b e in teresting if one co uld characterize the maxima l orders O for which this inequality is sha r p. 6. Acknow ledgments First and foremo s t the a uthor thanks her a dvisor, Henry Cohn, for int ro ducing her to this area of mathematics and for his constant supp or t of her research pur- suits. Next the author thanks Achill Sch¨ urmann for sharing his r esults on per fect forms and for ma king himself av aila ble to answer ques tions and to offer s uggestions on how this pap er could b e improv ed. The a uthor also thanks the Haus do rff Re- search Institute for Ma thematics for hosting the workshop titled “ Exp erimentation with, Constructio n of, and Enumeration o f O ptimal Geometric Structures”. This workshop provided the opp or tunit y to pres ent a preliminary version of this work and to get many helpful co mment s a nd sugge s tions on how it could b e generalize d. In par ticular the author thanks workshop par ticipants Jacques Martinet, Gabriele Nebe, and Renaud Coula ngeon for their int erest in her res e arch and fo r suggesting that s he use lo caliz a tion to gener alize her res ults to include O -lattices wher e the maximal Z -order O is not a left principal ideal doma in and hence every O -la ttice is not necessarily a free O -mo dule. Additionally , the a uthor thanks Jacques Mar tinet for his later suggestions and comments and for providing refer ences on maximal orders. References [AG] M. Auslander and O. Goldman, Maximal Or ders , T r ans. Amer. Math. 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Martinet, Perfect L atti c es in Euclide an Sp ac e , Springer-V erlag, 2003. [NS] G. Nebe and N.J.A . Sloane, A Catalo g ue of L attices , published online at http://w ww.research .att.com/ ~ njas/lat tices . [Qu] H. - G. Quebbemann, An applic ation of Sie gel’s formula over quaternion or ders , Mathe- matik a 31 (1984) , 12–16. [Sc h] A . Sch¨ urmann, Enumer ating p erfe ct forms , to appear i n the Pro ceedings of the Second In ternational Conference on the A lgebraic and Arithmetic Theory of Quadratic F orms, Chile 2007, published in the A MS Contemporary M athematics series. [Si] F. Si grist, Quaternionic- p erfe ct forms in dimension 12, pr eprint , 2000. Curr ent A ddr ess: School of Sciences, Adams St ate College, 208 Edgemont Bl vd., Alamosa, CO 81 102 A MORDELL INEQUALITY FOR LA TTICES OVER MAXIMAL ORDERS 13 F ormer A ddr ess: Dep a r tment of Ma thema tics, Un iversity of W ashington, Box 3 54350, Sea ttle, W A 981 95 E-mail addr ess : slvance@ad ams.edu