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ON QUATERNIONIC TORI AND THEIR MODULI SPACE 7 CINZIA BISI∗,GRAZIANOGENTILI∗ 1 0 2 Abstract. Quaternionic tori are defined as quotients of the skew field H of quaternions by rank-4 lattices. Using slice regular functions, these tori are n endowedwithnaturalstructuresofquaternionicmanifolds(infactquaternionic a curves), and a fundamental region in a 12-dimensional real subspace is then J constructedtoclassifythemuptobiregulardiffeomorphisms. Thepointsofthe 8 modulispacecorrespondtosuitablespecial basesofrank-4lattices,whichare 2 studiedwithrespecttotheactionofthegroupGL(4,Z), anduptobiregular diffeomeorphisms. Alltoriwithanontrivialgroupofbiregularautomorphisms ] -andallpossiblegroupsoftheirbiregularautomorphisms-arethenidentified, V and recognized to correspond to five different subsets of boundary points of C themodulispace. . h t a m 1. Introduction [ A new notion of regularity for quaternion-valued functions of a quaternionic 1 variablewasintroducedin2006,in[17,18]. Thenewlydefinedclassof(slice)regular v functions has already proved to be interesting as a quaternionic counterpart of 4 complexholomorphicfunctions. Inthisquaternionicsetting,aCasorati-Weierstrass 0 3 Theorem was proved in [28] and it allowed the study of the group Aut(H) of all 8 biregular transformations of the space of quaternions H. This group turned out to 0 coincide with the group of all affine transformations of H of the form q qa+b, 1. witha,b Handa=0. Aswecansee,notwithstandingthefactthatthe7→algebraic, 0 abstract∈structure o6f the groupAut(C2) of biholomorphic transformations of C2 is 7 still unknown, that of biregular transformations of H = C2 inherits the simplicity ∼ 1 of the group Aut(C). : v The fact that all quaternionic regular affine transformations of H form a group i under composition, the group Aut(H), permits the direct construction of a class X of natural quaternionic manifolds (actually quaternionic curves): the quaternionic r a tori. These tori are studied in the present paper. Together with the quaternionic projective spaces, [26], they are among the few directly constructed quaternionic manifolds, and bear with them the genuine interest that accompanies any analog of elliptic complex Riemann surfaces. In this paper we construct quaternionic tori, realized as quotients of H with respect to rank-4 lattices, and endow them with natural structures of quaternionic Date:January31,2017. 2010 Mathematics Subject Classification. Primary: 30G35Secondary: 32G15, 14K10. Keywordsandphrases. Regularfunctionsoverquaternions,Quaternonictoriandtheirmoduli space. ∗ Both authors are supported by Progetto MIUR di Rilevante Interesse Nazionale “Varietà reali e complesse: geometria, topologia e analisi armonica” and by GNSAGA of INdAM. The firstauthorisalsosupportedbyProgettoFIRBGeometriadifferenzialeeteoriageometricadelle funzioni. 1 2 C.BISI,G.GENTILI 1-dimensional manifolds. We then use the basic features of quaternionic regular maps to characterize biregularly diffeomorphic tori by means of properties of their generating lattices; this approach introduces into the scenery the group GL(4,Z), that plays a fundamental role in this context. In fact the use of classical results on the reduction of Gram matrices, based on the Minkowski-Siegel Reduction Al- gorithm, allows us to express any “normalized” rank-4 lattice of H in terms of a generating special basis. These special bases parameterize the classes of equivalence of biregular diffeomorphism of quaternionic tori, and are used to define a fundamental set (see (7.2)) for this equivalence relation, as the subset of H3 =R12 ∼ = (v ,v ,v ) H3 : 1,v ,v ,v is a special basis . 2 3 4 2 3 4 M { ∈ { } } Wewillnotdefineaspecialbasishereinthe Introduction(seeDefinition6.10),but we want to say that special bases have properties that urge a comparison with the complex case of elliptic curves, like the following: if 1,v ,v ,v is a special basis 2 3 4 { } of a rank-4 lattice, i.e., if (v ,v ,v ) , then, in particular 2 3 4 ∈M (1) 1 v ,v v ,v v ,v ; 2 2 3 3 4 4 ≤h i≤h i≤h i (2) 1 Re(v ) 1, for all k =2,3,4; −2 ≤ k ≤ 2 (3) 1 v ,v v ,v 1 v ,v , for all (k,l) 2,3,4 2,3,4 such that −2h l li≤h k li≤ 2h l li ∈{ }×{ } l =k. 6 The fundamental set H3 of the equivalence relation of biregular diffeomor- M ⊂ phism among quaternionic tori has some boundary points that are equivalent. In facttherearedifferentelementsbelongingtotheboundary∂ ofthefundamental M set, that correspond to the same torus; as an example we can take the distinct points (i,j,k) and (j,i,k): the two special bases 1,i,j,k and 1,j,i,k gener- { } { } ate the same lattice (the ring of Lipschitz quaternions) and hence the same torus. However, in (7.4) we define the proper subset of the fundamental set , which T M coincides with the interior of , and which is a moduli space for the subset of M equivalenceclassesofthe socalledtame tori. The complete quotientofthe boundary∂ ,withrespecttotheequivalencerelationofbiregulardiffeomorphismofthe M correspondingtori,is stillunknown. However,asithappens inthe complex caseof elliptic curves, the classification of all the boundary tori of ∂ having non trivial M groupsofbiregularautomorphismsisanimportantsteptowardstheunderstanding of the subtle features of the geometry of the moduli space. In this perspective, by exploiting the classification of the finite subgroups of unitary quaternions, we identify all the groups that can play the role of groups of biregular automorphisms of tori, i.e., 2T, 2D , 2D , 2C , 2C , 2C , 4 6 1 2 3 called respectively tetrahedral, 8-dihedral, 12-dihedral, trivial-cyclic, cyclic-dihedral and cyclic group. We then find those points of the boundary of the fundamental set which correspond to tori whose group of biregular automorphisms either M contains, or is isomorphic to, one of the groups listed above. We then prove Theorem 1.1. The only quaternionic tori T with a non trivial group of biregular automorphisms Aut(T)= 1 correspond to the following elements of : 6 {± } M (I,α ,α I) with α 1 and I2 = 1 [ 2C Aut(T) ]; 3 3 3 2 • ∈M | |≥ − ⊆ (eπ3I,α3,α3eπ3I) with α3 1 and I2 = 1 [ 2C3 Aut(T) ]. • ∈M | |≥ − ⊆ The most structured groups of biregular automorphisms appear in the tori with therichestsymmetries: examplesarethetorusgeneratedbythelatticeofLipschitz QUATERNIONIC TORI 3 quaternions and the one generated by the lattice of Hurwitz quaternions, whose groups of automorphisms are the 8-dihedral group 2D and the tetrahedral group 4 2T,respectively. Notice thatthe complexcounterpartofthe torilistedinTheorem 1.1 consists of the harmonic and equianharmonic tori, having moduli i and eπ3i, respectively. Appendices present a computational approach to the study of the modulus of a torus: for example, in the last appendix, an algorithm is produced that checks if a given basis of a lattice is tame. To conclude the Introduction, wepoint outthatthe theoryofslice regularfunctions,presentedindetail inthe monograph[16], has beenapplied to the study ofa non-commutative functional calculus, (see for example the monograph [7] and ref- erences therein) and to address the problem of the construction and classification oforthogonalcomplexstructuresinopensubsetsofthespaceHofquaternions(see [15]). Recent results of geometric theory of regular functions appear in [2], [3], [4], [5],[10],[11],[19],[20],[21]. Paper[12]isstrictlyrelatedtothetopicofthepresent article. 2. Preliminary results Inthis sectionwewillbrieflypresentthoseresultsonsliceregularfunctionsthat are essential for what follows. The 4-dimensional real algebra of quaternions is denoted by H. An element q in H can be expressed in terms of the standard basis, denoted by 1,i,j,k , as { } q = x +x i+x j +x k, where i,j,k are imaginary units, i2 = j2 = k2 = 1, 0 1 2 3 related by the multiplication rule ij = k. To every non-real quaternion q H−R ∈ \ we can associate an imaginary unit, with the map Im(q) q I = . q 7→ Im(q) | | If instead q R, we can set I to be any arbitrary imaginary unit. In this way, for q any q H th∈ere exist, and are unique, x,y R, with y 0 (y = 0 if q R), such ∈ ∈ ≥ ∈ that q =x+yI . q The set of all imaginary units is denoted by S, S= q H q2 = 1 { ∈ | − } and, from a topological point of view, it is a 2-dimensional sphere sitting in the 3-dimensional space of purely imaginary quaternions. The symbol B will denote theopenunitball q H: q <1 ofthe spaceHofquaternions,andthe 3-sphere of all the points of{its∈boun|da|ry ∂}B will be denoted by S3. Remark 2.1. It is worthwhile recalling that the two possible multiplications of a quaternion q by any fixed element u ∂B, that is q qu and q uq, both represent Euclidean rigid motions (rota∈tions) of H = R7→4.Moreover, i7→f v H is ∼ ∈ ∗ any fixed non zero quaternion, then both multiplications, q qv and q vq, can be decomposed as the composition of a Euclidean rigid mo7→tion of H =7→R4 and a ∼ multiplication by a strictly positive real number: the reason is immediate, since v = v v with v ∂B. | | v v ∈ | | | | 4 C.BISI,G.GENTILI To each element I of S there corresponds a copy of the complex plane, namely LI = R+IR ∼=C. All these complex planes, also called slices, intersect along the real axis, and their union gives back the space of quaternions, H= (R+IR)= L . I I[S I[S ∈ ∈ Since LI = R+IR ∼= C, with the symbol eαI, we will mean cosα+Isinα. The following definition appears in [17, 18]. Definition 2.2. Let Ω be a domain in H and let f : Ω H be a function. For all I S let us consider Ω = Ω L and f = f . T→he function f is called (slice)∈regular if, for all I SI, the r∩estrIictionfIhas c|oΩIntinuous derivativesandthe I function ∂¯ f :Ω H defi∈ned by I I → 1 ∂ ∂ ∂¯ f(x+Iy)= +I f (x+Iy) I I 2(cid:18)∂x ∂y(cid:19) vanishes identically. The same articles introduce the Cullen (or slice) derivative ∂ f of a slice regular c function f as 1 ∂ ∂ (2.1) ∂ f(x+Iy)= I f(x+Iy) c 2(cid:18)∂x − ∂y(cid:19) for I S, x,y R. ∈ ∈ Remark 2.3. The Cullenderivativeofasliceregularfunctionturnsouttobe still a slice regular function (see, e.g, [16, Definition 1.7, page 2], [18]). Using the Cullen derivative, it is possible to characterize the slice regular functions defined in the entire space H, or on a ball B(0,R) = q H : q < R centered at 0 H, as follows (see, e.g., [16]). { ∈ | | } ∈ Theorem 2.4. A function f is regular in B(0,R) if and only if f has a power series expansion 1 ∂nf f(q)= qna with a = (0) n n n!∂xn nX0 ≥ converging in B(0,R). Moreover its Cullen derivative can be expressed as 1 ∂n+1f ∂ f(q)= qnb with b = (0) c n n n!∂xn+1 nX0 ≥ in B(0,R). The existence of the power series expansionyields a Liouville Theorem, that we will use in the sequel: Theorem 2.5 (Liouville). Let f : H H be slice regular. If f is bounded then f → is constant. QUATERNIONIC TORI 5 3. Lattices in the space of quaternions Let ω ,...,ω (with m 4) be R-linearly independent vectors in H. 1 m ≤ Definition 3.1. Theadditivesubgroupof(H,+)generatedbyω ,...,ω iscalled 1 m a rank-m lattice, generated by ω ,...,ω . 1 m We will focus our attention on (topologically) discrete subgroups of (H,+), for which the following result holds: Lemma 3.2. Let M be a discrete (infinite) subgroup of (H,+). Then M has no accumulation points. Proof Since M is discrete, there are no accumulation points of M belonging to M. By contradiction, assume that there exists an accumulation point q of M, belonging to H M. Then there exists a sequence qn n N M converging to q. Since qn n N \ M is a Cauchy sequence, for all m{ }N,∈th⊆ere exist rm, sm N { } ∈ ⊆ ∈ ∈ such that q =q and that rm 6 sm 1 (3.1) 0< q q < . | rm − sm| m If we define α = q q M, inequality (3.1) would imply that 0 M is not isolated, beingmthe lrimmi−t ofsmα∈m m N when m + . ∈ (cid:3) { } ∈ → ∞ As a straightforwardconsequence we obtain: Corollary 3.3. Let M be a subgroup of (H,+), let R R+ and let B(0,R) be the closure of B(0,R). Then M is discrete if and only∈if, for all R R+, the ∈ intersection B(0,R) M is a finite set. ∩ The following classical characterization of discrete subgroups of (H,+) will be usedasabasicfactinthesequel(foraproofsee,e.g.,[27,Theorem6.1,page136]). Theorem 3.4. A subgroup of (H,+) is a lattice if and only if it is discrete. Thislasttheoremimplies thatthe studyofallpossiblequotientspacesof(H,+) with respect to a discrete additive subgroup M is reduced to the case in which M is a lattice. With the aim of classifying these quotients, let Tm denote the direct product of m copies of the unit circle S of R2, and call it the m-dimensionaltorus. It is well known that, given the rank m of a lattice in H=R4, there exists only ∼ “one”quotient, upto realdiffeomorphisms (see, e.g., [27, Theorem6.4,page140]): Theorem 3.5. Let L be a rank-m lattice in H (with m 4). Then the group H/L is isomorphic to Tm R4 m. ≤ − × Thecaseofarank-4latticeistheoneinwhichthe quotientoriginates“the”real 4-dimensional torus: Corollary 3.6. Let L be a rank-4 lattice in H. Then the group H/L is isomorphic to T4. As we can see, up to real diffeomorphisms the classification is quite simple. Followingthe guidelines ofthe classicaltheoryofcomplex elliptic functions we will workat the classificationof4-(real)-dimensional,quaternionictori, up to biregular diffeomorphisms. Inthe nextsectionwe will define slice quaternionicstructures on tori, that will be the object of our classification. 6 C.BISI,G.GENTILI 4. A regular quaternionic structure on a 4-(real)-dimensional torus Since quaternionic regular affine transformations form a group with respect to composition,andanalogouslytowhathappensforcomplextori,thefieldHinduces on a quaternionic torus T4 = H/L a structure of quaternionic manifold. This structure will be called a regular quaternionic structure or simply a quaternionic structure on T4. From now on, a torus T4 endowed with a quaternionic structure will be called a quaternionic torus. Moreover, the 4-(real)-dimensional torus T4 will always be denoted simply by T. To construct such a structure, we will first of all consider the classical atlas of the real torus T, and then adopt the procedure used in the complex case U(see, e.g., [6, 14, 29]).Let L be a rank-4 lattice of H, generated by ω ,ω ,ω ,ω . 1 2 3 4 Consider the canonical projection π : R4 = H H/L = T and, for any p H, ∼ → ∈ an open neighborhood U of p small enough to make π an homeomorphism of p |Up U onto its image π(U ). The atlas will consist of the local coordinate systems p p U π(U ),(π ) 1 . If we suppose that, for p,q H, the intersection π(U ) n(cid:16) p |Up − (cid:17)op H ∈ p ∩ π(Uq) is (open and) co∈nnected, then the change of coordinates is such that π−1 |Up ◦ π (x) = x+ 4 n ω for fixed n ,n ,n ,n . Hence the change of coordinates |Uq l=1 l l 1 2 3 4 is a regular funPction. Therefore we obtain a quaternionic structure on T. Using the classical approach, regular maps between quaternionic tori can be defined in thenaturalmanner,aswellasbiregular diffeomorphisms betweenquaternionictori, andbiregular automorphisms ofaquaternionictorus. Wecanthenproceedtostudy the quaternionic tori up to biregular diffeomorphisms, and give the following: Definition 4.1. If there is a biregular diffeomorphism of a 4-(real)-dimensional torus T onto a (4-(real)-dimensional) torus T , we will say that the two tori are 1 2 equivalent. To proceed, we recall that the group Aut(H) of biregular transformations (or automorphisms) of H consists of all slice regular affine transformations, that is Aut(H)= f(q)=qa+b:a,b H,a=0 { ∈ 6 } (see [28]). The result stated in the next proposition has a complete analog in the complex setting,[14,Theorem4.1,page10]. Neverthelesswewillproduceaproof,toacquire familiaritywiththequaternionicenvironmentandtoestablishnotationstobeused. Proposition 4.2. Let L and L be two rank-4 lattices in H, let π :H H/L = 1 2 1 1 T and π : H H/L = T be the projections on the quotient tori→. For any 1 2 2 2 F Aut(H) such→that F(L )= L there exists a biregular diffeomorphism f of T 1 2 1 ∈ onto T which allows the equality f π = π F. Conversely, for any biregular 2 1 2 diffeomorphism f of T onto T , ther◦e exists F ◦ Aut(H) such that f π =π F 1 2 1 2 ∈ ◦ ◦ and F(L )=L . 1 2 Proof. Let F(v) = va+b. Since 0 L we have F(0)= b L and hence we can 1 2 ∈ ∈ suppose b=0. By definition of regular map between tori, to show that F(v)=va QUATERNIONIC TORI 7 induces a biregular diffeomorphism f of T onto T , 1 2 (4.1) H F //H π1 π2 (cid:15)(cid:15) (cid:15)(cid:15) // T T 1 2 f itisenoughtoshowthatq pimpliesF(q) F(p). Indeed,ifq pthenq p L 1 ∼ ∼ ∼ − ∈ and hence F(q) F(p)=qa pa=(q p)a=F(q p) L . 2 − − − − ∈ To provethe conversestatement,we startbyrecallingthatthe mapf :T T 1 2 lifts to a continuous map F : H H, in such a way that the diagram→(4.1) → commutes. Moreover the map F is regular since f is a regular map of T onto T . 1 2 For any λ L , considerG (q)=F(q+λ) F(q). Since F lifts a map between 1 λ ∈ − the quotients, F maps L -equivalent points into L -equivalent points. Hence the 1 2 image of G is contained in the (discrete, see Theorem 3.4) lattice L and, being λ 2 continuous, is therefore constant. At this point it is clear that, taking the Cullen derivative, we obtain ∂ F(q+λ) = ∂ F(q), for all q H. Thus the map ∂ F is c c c ∈ regular(seeRemark2.3)andL -periodic,whichmakesitbounded. BytheLiouville 1 Theoremfor regularfunctions (see Theorem 2.5) the Cullen derivative ∂ F of F is c constant. Since F expands as a power series (see Theorem 2.4) 1 ∂nF F(q)= qn (0) n! ∂xn nXN ∈ converging in the entire H, we obtain (again by Theorem 2.4) 1 ∂n+1F ∂F ∂ F(q)= qn (0)= (0) c n! ∂xn+1 ∂x nXN ∈ and hence ∂F F(q)=F(0)+q (0)=b+qa ∂x is a first degree regular polynomial. Again, since F lifts a map between quotients, necessarily L a L . If the inclusion L a L is proper, then f is not injective: 1 2 1 2 ⊆ ⊂ indeed if some q L satisfies qa 1 L then there exists p ,p L and p L 2 − 1 1 1 2 2 ∈ 6∈ ∈ ∈ such that (qa 1+p ) / L and f(qa 1+p ) = π (q+p a+b) = π (p ) = f(p), − 1 1 − 1 2 1 2 2 ∈ with qa 1+p =p. − 1 Now we know6 that L a = L , that is L a 1 = L . The map F 1 : H H 1 2 2 − 1 − → defined by F 1(w)=(w b)a 1 induces the map f 1 :T T . Indeed − − − 2 1 − → f 1(f(q+L ))=f 1(qa+b+L )=(qa+L )a 1 =q+L a 1 =q+L . − 1 − 2 2 − 2 − 1 This concludes the proof. (cid:3) 5. Equivalence of quaternionic tori Toclassifythe4-(real)-dimensional,quaternionictori,uptobiregulardiffeomor- phisms, we start with the following: Theorem 5.1. Two rank-4 lattices L , L of the space H, generated respectively 1 2 bythe bases α ,α ,α ,α and ω ,ω ,ω ,ω , determine equivalent tori T ,T if 1 2 3 4 1 2 3 4 1 2 { } { } 8 C.BISI,G.GENTILI and only if there exist a H =H 0 and a linear transformation A GL(4,Z) ∗ ∈ \{ } ∈ such that ω α 1 1 A ω2 = α2 a ω α  3   3   ω   α  4 4     Proof. By Proposition 4.2, if f is a biregular diffeomorphism of T onto T , then 1 2 there exists a biregular transformation F of H such that the diagram (4.1) commutes. Since F is biregular on H, then F(q)=qa+b, with a H , and b H. As ∗ ∈ ∈ we pointed out in the proof of Proposition 4.2, without loss of generality, we can suppose both that b=0 and that the function F maps the set of generators of L 1 to a set of generators of L . Taking into account that 2 F(α ) = α a 1 1  F(α2) = α2a  F(α3) = α3a F(α ) = α a 4 4 there exists a matrix  n n n n 11 12 13 14 A= n21 n22 n23 n24  n n n n 31 32 33 34    n41 n42 n43 n44    with integer entries, such that α a= n ω + n ω + n ω + n ω 1 11 1 12 2 13 3 14 4 (5.1)  α2a= n21ω1 + n22ω2 + n23ω3 + n24ω4  α3a= n31ω1 + n32ω2 + n33ω3 + n34ω4 α a= n ω + n ω + n ω + n ω 4 41 1 42 2 43 3 44 4 or, more concisely, α ω 1 1 (5.2)  α2 a=A ω2 . α ω 3 3      α   ω  4 4     The same argument applied in the opposite direction, implies the existence of a matrix B with integer entries such that ω α 1 1 (5.3)  ω2 a 1 =B α2  ω − α 3 3      ω4   α4      and hence, substituting equation (5.3) into equation (5.2), we get α α 1 1  α2 a=AB α2 a α α  3   3   α   α  4 4     which implies AB =I and hence that A (and B) is such that det(A)= 1, i.e. A 4 (and B) belongs to GL(4,Z). ± QUATERNIONIC TORI 9 On the other side, suppose there exists a matrix A GL(4,Z), of this form: ∈ n n n n 11 12 13 14 A= n21 n22 n23 n24  n n n n 31 32 33 34    n n n n  41 42 43 44   such that ω α 1 1 A ω2 = α2 a ω α 3 3      ω   α  4 4     then we can compute F(q)=qa in four different ways: F(q)=qα−11(n11ω1+n12ω2+n13ω3+n14ω4) F(q)=qα−21(n21ω1+n22ω2+n23ω3+n24ω4) F(q)=qα−31(n31ω1+n32ω2+n33ω3+n34ω4) F(q)=qα−41(n41ω1+n42ω2+n43ω3+n44ω4) for all q H. Simple computations show that: ∈ F(q+α )=F(q)+n ω +n ω +n ω +n ω , 1 11 1 12 2 13 3 14 4 F(q+α )=F(q)+n ω +n ω +n ω +n ω , 2 21 1 22 2 23 3 24 4 F(q+α )=F(q)+n ω +n ω +n ω +n ω , 3 31 1 32 2 32 3 34 4 F(q+α )=F(q)+n ω +n ω +n ω +n ω . 4 41 1 42 2 43 3 44 4 Hence F defines a biregular diffeomorphism f between T and T . (cid:3) 1 2 It is natural at this point to give the following Definition 5.2. Two rank-4 lattices L , L of the space H are called equiva- 1 2 lent if the generated quaternionic tori H/L and H/L are equivalent. A basis 1 2 ω ,ω ,ω ,ω ofarank-4latticeL andabasis α ,α ,α ,α ofarank-4lattice 1 2 3 4 1 1 2 3 4 { } { } L are called equivalent if L and L are equivalent lattices, i.e. if (according to 2 1 2 Theorem 5.1) there exist a H and a linear transformation A GL(4,Z) such ∗ ∈ ∈ that ω α 1 1 (5.4) A ω2 = α2 a. ω α 3 3      ω4   α4      Notice that two (different) equivalent bases ω ,ω ,ω ,ω and α ,α ,α ,α 1 2 3 4 1 2 3 4 { } { } of rank-4 lattices may generate exactly the same lattice, and hence exactly the same quaternionic torus. This happens when there exists a linear transformation A GL(4,Z) such that ∈ ω α 1 1 A ω2 = α2  ω α  3   3   ω   α  4 4     i.e., when a=1 in (5.4). 10 C.BISI,G.GENTILI 6. Minkowski-Siegel Reduction Algorithm: reduced and special bases InthissectionwewillspecializetothequaternionicsettingthegeneralMinkowski- Siegel Reduction Algorithm presented in [22, Section 4], [24, Section 9], and use it to construct reduced Gram matrices and reduced bases associated to lattices. In turn,reducedbaseswillbeusedtofindspecial basesforlattices,usefulinthesequel to identify and parameterize equivalence classes of quaternionic tori. Weexplicitlypresentheresomebasicfactsofthisalgorithmicconstruction,both to make the paper as much self-contained as possible, and to have a starting point for the proofs of the results that will follow. Let , denote the usual scalar product of R4. Let p = x +x i+x j +x k 0 1 2 3 h· ·i and q =y +y i+y j+y k be two quaternions. We set, and use in what follows, 0 1 2 3 3 (6.1) p,q = x y . ℓ ℓ h i Xℓ=0 Let v ,v ,v ,v beabasisofthelatticeL H=R4. Foranyu=(n ,n ,n ,n ) Z4 t{he1sq2uar3ed4n}orm of the element v = n⊂v +∼n v +n v +n v1 2L c3an4be∈ 1 1 2 2 3 3 4 4 ∈ expressed by v,v =v tv =uS tu where the matrix 0 h i v ,v v ,v v ,v v ,v 1 1 1 2 1 3 1 4 h i h i h i h i (6.2) S0 = hvv2,,vv1i hvv2,,vv2i hvv2,,vv3i hvv2,,vv4i  3 1 3 2 3 3 3 4  h i h i h i h i   v ,v v ,v v ,v v ,v  4 1 4 2 4 3 4 4  h i h i h i h i  issymmetricandpositivedefinite,andisusuallycalledtheGrammatrix associated to the basis v ,v ,v ,v . In this setting and with the notations established, we 1 2 3 4 { } will use the following procedure (see, e.g., [24], page 122): Algorithm 6.1 (Minkowski-Siegel Reduction Algorithm). This algorithm acts on a Gram matrix S and produces a matrix U = U(S ) 0 0 belonging to GL(4,Z) and a Gram matrix R = R(S ) = US tU. The produced 0 0 matrix R has, and is in fact characterizedby, the properties which follow. Here are the steps of the algorithm: The Gram matrix S (of a certain basis v ,v ,v ,v ) is given. 0 1 2 3 4 • Consider the function Q :Z4 R+ defi{ned as } 1 • → Q (u)=uS tu. 1 0 By our assumption, Q is (the restriction to Z4 of) a positive definite qua- 1 dratic form, and hence it attains its strictly positive minimum value at a point u =(n ,n ,n ,n ) Z4. 1 11 12 13 14 ∈ To proceed, we need to recall that there exist infinitely many matrices of • GL(4,Z) having the first row equal to u (see e.g. [22, Section 4, pages 1 191-192],[24, Section 9, pages 122-123]for a proof of this assertion,and of theanalogousones,usedinthisalgorithm). Withthisinmind,weconsider thefunctionQ obtainedbyrestrictingQ totheelementsu Z4 suchthat 2 1 there exists a matrix of GL(4,Z) having the first two rows e∈qual to u and 1 u, respectively. Let u = (n ,n ,n ,n ) Z4 be a point in which Q 2 21 22 23 24 2 ∈ attains its strictly positive minimum value. Up to a change of sign, we can assume that u S tu 0. 1 0 2 ≥