PERFECT LATTICES IN EUCLIDEAN SPACES : REFERENCES

Last update: December 13th, 2022

In what follows, we first give an updated version of the reference list of the book “Perfect Lattices in Euclidean Spaces”; this occupies pages 2 to 15.

We next list (from page 16 onwards) some new references related to notions discussed in the book: perfection, eutaxy, minimal classes, density, kissing number, …

Note: this second part is permanently in construction.

References

[Ash1] A. Ash, On eutactic forms, Can. J. Math. 29 (1977), 1040–1054.

[Ash2] A. Ash, On the existence of eutactic forms, Bull. London Math. Soc. 12 (1980), 192–196.

[A-MC] A. Ash, M. Mc Connel, Cohomology at infinity and the well-rounded

retract for general linear groups, Duke Math. J. 90 (1997), 549–576.

[ATLAS] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, ATLAS of Finite Groups, Oxford Univ. Press, Oxford (1985).

[Bc1] R. Bacher, Dense lattices in dimensions 27, 28 and 29, Invent. Math. 130 (1997), 153–158.

[Bc2] R. Bacher, Tables de réseaux entiers unimodulaires construits comme k-voisins de ℤ^{n}, J. Th. Nombres de Bordeaux 9 (1997), 479–497.

[Bc-V1] R. Bacher, B. Venkov, Lattices and association schemes: a unimodular example without roots in dimension 28, Ann. Inst. Fourier 45 (1995), 1163–1176.

[Bc-V2] R. Bacher, B. Venkov, Réseaux entiers unimodulaires sans racines en

dimension 27 et 28, [Mar’] (2001), 212–267.

[Bac1] C. Bachoc, Voisinages au sens de Kneser pour les réseaux quaternioniens, Comm. Math. Helvet. 70 (1995), 350–374.

[Bac2] C. Bachoc, Applications of coding theory to the construction of modular lattices, J. Comb. Th., A 78-1 (1997), 92–119.

[Bac-Bt] C. Bachoc, C. Batut, Étude algorithmique de réseaux construits avec

la forme trace, J. Exp. Math. 1 (1992), 183–190.

[Bac-C-N] C. Bachoc, R. Coulangeon, G. Nebe, Designs in Grassmannian spaces

and lattices, J. Algebraic Combinatorics 16 (2002), 5–19.

[Bac-Ne1] C. Bachoc, G. Nebe, Classification of Two Genera of 32-Dimensional Lattices of Rank 8 over the Hurwitz Order, Exp. Math. 6 (1997), 151–162.

[Bac-Ne2] C. Bachoc, G. Nebe, Extremal lattices of minimum 8 related to the

Mathieu group M_{22}, J. reine angew. Math. 494 (1998), 155–171.

[Bac-V] C. Bachoc, B. Venkov, Modular forms, lattices and spherical designs,

[Mar’] (2001), 87–111.

[B-I] R. Baeza, M.I. Icaza, On Humbert-Minkowski constant for a number

field, Proc. Amer. Math. Soc. 125 (1997), 3195–3202.

[B-C-I-O] R. Baeza, R. Coulangeon, M.I. Icaza, M. O’Ryan, Hermite’s constant for quadratic number fields, Exp. Math. 10 (2001), 543-550. E. Bannai. See [C-S], Chapter 14, and the complementary bibliography.

[Br-R] E.P. Baranovskii, S.S. Ryshkov, Derivation of perfect lattices from admissible centerings, Russian Math. Survey 40, 4 (1985), 155–156 ; Russian original: 1985.

[Bari] J-L. Baril, Autour de l’algorithme de Voronoï : construction de réseaux euclidiens, Thèse, Bordeaux (1996).

[Bar1] E.S. Barnes, Note on extreme forms, Canad. J. Math 7 (1955), 150– [Bar2] E.S. Barnes, On a theorem of Voronoï, Proc. Cambridge Phil. Soc. 53

(1957), 537–539.

[Bar3] E.S. Barnes, The perfect and extreme senary forms, Canad. J. Math. 9 (1957), 235–242.

[Bar4] E.S. Barnes, The complete enumeration of extreme senary forms, Phil. Trans. Roy. Soc. London (A), 249 (1957), 461–506.

[Bar5] E.S. Barnes, The construction of perfect and extreme forms I, II , Acta Arith. 5 (1959), 57–79, 461–506.

[Bar6] E.S. Barnes, Criteria for extreme forms, J. Austral. Math. Soc. 1 (1959), 17–20.

[B-W] E.S. Barnes, G.E. Wall, Some extreme forms defined in terms of

abelian groups, J. Austral. Math. Soc. 1 (1959), 47–63.

[Bt] C. Batut, Classification of quintic eutactic forms, Math. Comp. 70 (2001), 395–417.

[Bt-M] C. Batut, J. Martinet, Radiographie des réseaux parfaits, Exp. Math.

3 (1994), 39–49.

[Bt-M’] C. Batut, J. Martinet, A Catalogue of Perfect Lattices,

http://www.math.u-bordeaux.fr/ martinet/ .

[Bt-Q-S] C. Batut, H-G. Quebbemann, R. Scharlau, Computations of cyclotomic lattices, Exp. Math. 4 (1995), 175–179. C. Batut. See also C. Bachoc and the appendix to [Mar5].

[Bav1] C. Bavard, Systole et invariant d’Hermite, J. reine angew. Math. 482 (1997), 93–120.

[Bav2] C. Bavard, Classes minimales des réseaux et rétractions géométriques équivariantes dans les espaces symétriques, J. London Mat. Soc. 64 (2001), 275–286.

[Bav3] C. Bavard, Familles hyperboliques de réseaux symplectiques, Math. Ann. 320 (2001), 799–833.

[Bav4] C. Bavard, Théorie de Voronoï géométrique. Propriétés de finitude pour les familles de réseaux et analogues, Bull. Soc. Math. France 133 (2005), 205–257.

[Bav5] C. Bavard, Invariant d’Hermite isotrope et densité des réseaux orthogonaux lorenziens, Comment. Math. Helvetici 82 (2007), 39–60.

[Bay1] E. Bayer-Fluckiger, Definite unimodular lattices having an automorphism of given characteristic polynomial, Comm. Math. Helvet. 59 (1984), 509–538.

[Bay2] E. Bayer-Fluckiger, Réseaux unimodulaires, Sém. Th. Nombres Bordeaux (=J. Th. Nombres de Bordeaux) 1 (1989), 189–196.

[Bay3] E. Bayer-Fluckiger, Lattices and number fields, Contemporary Math. 241 (1999), 69–84.

[Bay4] E. Bayer-Fluckiger, Cyclotomic modular lattices, J. Th. Nombres de Bordeaux 12 (2000), 273–280.

[Bay5] E. Bayer-Fluckiger, Ideal lattices, Cambridge Univ. Press, Proc. of the conference in honor of Alan Baker, “Number Theory and Diophantine Geometry”, Cambridge Univ. Press (2002), pp. 168-184.

[Bay-M] E. Bayer-Fluckiger, J. Martinet, Formes quadratiques liées aux algèbres semi-simples, J. reine angew. Math. 451 (1994), 51–69.

[Be-B] E.F. Beckenbach, R. Bellman, Inequalities, Springer-Verlag, Ergebnisse 30, Heidelberg (1961). R. Bellman. See E.F. Beckenbach. C.T. Benson. See L.C. Grove.

[Ber1] A.-M. Bergé, Minimal vectors of pairs of dual lattices, J. Number Theory 52 (1995), 284–298.

[Ber2] A.-M. Bergé, Classification of positive forms having prescribed automorphisms, Contemporary Math. 249 (1999), 199–204.

[Ber3] A.-M. Bergé, Symplectic Lattices, Contemporary Math. 272 (2000), 9–22.

[Ber4] A.-M. Bergé, Une forme duale-extrême irrationnelle, J. Th. Nombres de Bordeaux 12 (2000), 281–291.

[B-M1] A.-M. Bergé, J. Martinet, Sur un problème de dualité lié aux sphères

en géométrie des nombres, J. Number Theory 32 (1989), 14–42.

[B-M2] A.-M. Bergé, J. Martinet, Sur les minorations géométriques des régulateurs, in Sém. Th. Nombres de Paris 1987–88, Prog. Math. 81, Birkhäuser, Basel (1989), 23–49.

[B-M3] A.-M. Bergé, J. Martinet, Réseaux extrêmes pour un groupe d’auto-

morphismes, Astérisque 198–200 (1992), 41–66.

[B-M4] A.-M. Bergé, J. Martinet, Densité dans des familles de réseaux. Ap-

plication aux réseaux isoduaux, L’Ens. Math. 41 (1995), 335–365.

[B-M5] A.-M. Bergé, J. Martinet, Sur la classification des réseaux eutactiques,

J. London Math. Soc. 53 (1996), 417–432.

[B-M6] A.-M. Bergé, J. Martinet, Symmetric groups and lattices, Monatshefte

Math. 140 (2003), 179–195.

[B-M-S] A.-M. Bergé, J. Martinet, F. Sigrist, Une généralisation de l’algorithme

de Voronoï, Astérisque 209 (1992), 137–158.

[Be-S] N. Bernstein, N.J.A. Sloane, Some Lattices Obtained from Riemann

Surfaces, Contemporary Math. 201 (2002), 29–32.

[Bl1] H.F. Blichfeldt, The minimum value of quadratic forms and the closest packing of spheres, Math. Ann. 101 (1929), 605–608.

[Bl2] H.F. Blichfeldt, The minimum value of positive quadratic forms in six, seven and eight variables, Math. Z. 39 (1935), 1–15.

[Bor1] R. Borcherds, The Leech lattice and other lattices, Ph. D. Thesis, Cambridge, U.K. (1984).

[Bor2] R. Borcherds, Classification of positive definite lattices, Duke Math J. 105 (2000), 525-567.

[Borel] A. Borel, Introduction aux groupes arithmétiques, Hermann, Paris (1969).

[Bou1] N. Bourbaki, Topologie Generale II, Chap. 5–10 , Hermann, Paris (1974).

[Bou2] N. Bourbaki, Algèbre, chapitres I–X, Masson, Paris (1980).

[Bou3] N. Bourbaki, Groupes et algèbres de Lie, Masson (now Dunod), Paris (1981).

[B-C] R. Brauer, H.S.M. Coxeter, A generalization of theorems of Schönhardt and Mehmke on polytopes, Trans. Roy. Soc. Canada III (3), 34 (1940), 29–34. [= Richard Brauer, Coll. papers III, The M.I.T. Press, Cambridge, Massachusetts (1980), 653–658.] [B-E] M. Broué, M. Enguehard, Une famille infinie de formes quadratiques

entières ; leurs groupes d’automorphismes, Ann. Sc. E.N.S. 6 (1973), 17–53.

[Bu-Sa] P. Buser, P. Sarnak, On the period matrix of a Riemann surface of large genus (with an Appendix by J.H. Conway and N.J.A. Sloane), Invent. Math. 117 (1994), 27–56.

[Cas1] J.W.S. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, Tract 45, Cambridge, U.K. (1965; first edition: 1957).

[Cas2] J.W.S. Cassels, An Introduction to the Geometry of Numbers, Springer-Verlag, Grundlehren 99, Heidelberg (1959).

[Cas3] J.W.S. Cassels, Rational Quadratic Forms, Academic Press, London (1978).

[C-F] J.W.S. Cassels, A. Fröhlich, ed., Algebraic Number Theory, Academic

Press, New–York (1967).

[Chn] T.W. Chaundy, The arithmetic minima of positive quadratic forms (1), Quart. J. Math. Oxford 17 (1946), 166–192.

[Coh] H. Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Math. 138, Springer-Verlag, Heidelberg (1993).

[C-S] J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups,

Grundlehren 290, Springer-Verlag, Heidelberg (1988).

[C-S’] J.H. Conway, N.J.A. Sloane, second edition of [C-S] (1993). [C-S”] J.H. Conway, N.J.A. Sloane, third edition of [C-S] (1999). [C-S1] J.H. Conway, N.J.A. Sloane, Complex and integral laminated lattices,

Trans. Amer. Math. Soc. 280 (1983), 463–490.

[C-S2] J.H. Conway, N.J.A. Sloane, The Coxeter–Todd lattice, the Mitchell group, and related sphere packings, Math. Proc. Cambridge Phil. Soc. 93 (1983), 421–440.

[C-S3] J.H. Conway, N.J.A. Sloane, Low-Dimensional Lattices. I. Quadratic Forms of Small Determinant, Proc. Royal Soc. London A 418 (1988), 17–41.

[C-S4] J.H. Conway, N.J.A. Sloane, Low-Dimensional Lattices. II. Subgroups

of GL(n, ℤ), Proc. Royal Soc. London A 419 (1988), 29–68.

[C-S5] J.H. Conway, N.J.A. Sloane, Low-Dimensional Lattices. III. Perfect

Forms, Proc. Royal Soc. London A 418 (1988), 43–80.

[C-S6] J.H. Conway, N.J.A. Sloane, Low-Dimensional Lattices. IV. The Mass

Formula, Proc. Royal Soc. London A 419 (1988), 259–286.

[C-S7] J.H. Conway, N.J.A. Sloane, Low-Dimensional Lattices V. Integral Coordinates for Integral Lattices, Proc. Royal Soc. London A 426 (1989), 211–232.

[C-S8] J.H. Conway, N.J.A. Sloane, Low-Dimensional Lattices. VI. Voronoi Reduction of Three-Dimensional Lattices, Proc. Royal Soc. London A 436 (1992), 55–68.

[C-S9] J.H. Conway, N.J.A. Sloane, On Lattices Equivalent to Their Duals,

J. Number Theory 48 (1994), 373–382.

[C-S10] J.H. Conway, N.J.A. Sloane, A Lattice Without a Basis of Minimal Vectors, Mathematika 42 (1995), 175–177.

[C-S11] J.H. Conway, N.J.A. Sloane, Low-Dimensional Lattices. VII. Coordi-

nation Sequences, Proc. Royal Soc. London A 453 (1997), 2369–2389.

[C-H-S] J.H. Conway, R. H. Hardin, N. J. A. Sloane, Packing Lines, Planes, etc.: Packings in Grassmannian Space, Experimental Math. 5 (1996), 139–159. J.H. Conway. See also [ATLAS] and Appendix 2 to [Bu-Sa].

[Cou1] R. Coulangeon, Réseaux quaternioniens et invariant de Venkov, Manuscripta Math. 82 (1994), 44–50.

[Cou2] R. Coulangeon, Réseaux unimodulaires quaternioniens en dimension ≤ 32, Acta Arith. 70 (1995), 9–24.

[Cou3] R. Coulangeon, Réseaux k-extrêmes, Proc. London Math. Soc 73 (1996), 555–574.

[Cou4] R. Coulangeon, Minimal vectors in the second exterior power of a lattice, J. Algebra 194 (1997), 467–476.

[Cou5] R. Coulangeon, Tensor products of Hermitian lattices, Acta Arith. 92 (2000), 115–130.

[Cou6] R. Coulangeon, Voronoi theory over algebraic number fields, [Mar’] (2001), 147–162. R. Coulangeon. See also Bachoc, Baeza, and the appendix to [Mar-V1].

[Cox1] H.S.M. Coxeter, Integral Cayley Numbers, Duke Math. J. 13 (1946), 561–578.

[Cox2] H.S.M. Coxeter, Extreme forms, Canad. J. Math. 3 (1951), 391–441. [C-T] H.S.M. Coxeter, J.A. Todd, An extreme duodenary form, Canad.

J. Math. 5 (1953), 384–392. H.S.M. Coxeter. See also R. Brauer.

[Cra1] M. Craig, A characterization of certain extreme forms, Illinois J. Math. 20 (1976), 706–717.

[Cra2] M. Craig, Extreme forms and cyclotomy, Mathematika 25 (1978), 44– 56.

[Cra3] M. Craig, A cyclotomic construction for Leech’s lattice, Mathematika 25 (1978), 236–241.

[Cs1] G. Csóka, Three theorems on the minimal vectors of a perfect lattice, Ann. Univ. Sc. Budapest Eőtvős, Sect. Math. 23 (1986), 283–291.

[Cs2] G. Csóka, There exists a basis of minimal vectors in every n ≤ 7 dimensional perfect lattice, Ann. Univ. Sc. Budapest Eőtvős, Sect. Math. 30 (1987), 245–258. R.T. Curtis. See [ATLAS].

[De] M. Deuring, Algebren, Chelsea Publishing Company, New York (1948); Springer-Verlag, Heidelberg (1968). [Original edition: Julius Springer, Berlin (1935).]

[Die] J. Dieudonné, Sur les groupes classiques (third edition), Hermann, Paris (1967).

[D-T] N. Dummigan, P.H. Tiep, Lower bounds for the minima of certain symplectic and unitary group lattices, Amer. J. Math. 121 (1999), 889–918.

[Eb] W. Ebeling, Lattices and codes, Vieweg, Wiesbaden (1994).

[E-R-S] Y. Edel, E.M. Rains, N.J.A. Sloane, On kissing numbers in dimensions

32 to 128 , Elect. J. Combinatorics 5 (1998), paper R22.

[El1] N. Elkies, Mordell-Weil lattices in characteristic 2, I: Construction and first properties, Internat. Math. Res. Notice 8 (1994), 343–361.

[El2] N. Elkies, A characterization of the ℤ^{n} lattice, Math. Res. Lett. 2 (1995), 321–326.

[El3] N. Elkies, Lattices and codes with long shadows, Math. Res. Lett. 2 (1995), 643–651.

[El4] N. Elkies, Mordell-Weil lattices in characteristic 2, II: The Leech lattice as a Mordell-Weil lattice, Invent. Math. 128 (1997), 1–8.

[El5] N. Elkies, The Klein quartic in number theory, in The Eightfold Way, S. Levy ed., Math. Sc. Research Inst. Pub. 35, Cambridge University Press, Cambridge, U.K. (1999).

[El6] N. Elkies, Lattices, linear codes, and invariants, Part I and Part II , Notices of the A.M.S. 47 (2000), 1238–1245 and 1382–1391.

[El7] N. Elkies, Mordell-Weil lattices in characteristic 2, III: A Mordell-Weil lattice of rank 128, Exp. Math. 10 (2001), 467–473. M. Enguehard. See M. Broué.

[Fe1] W. Feit, On integral representations of finite groups, J. London Math. Soc. 29 (1974), 633–683.

[Fe2] W. Feit, Some lattices over ℚ(\sqrt{−3}), J. Algebra 52 (1978), 248–263.

[Gi] M. Gindraux, after a talk at the Bordeaux Number Theory Seminar, December 1st, 2002 (in preparation).

• see [Gi’] in the COMPLEMENTARY BIBLIOGRAPHY below.

[Gro] B. Gross, Group representation and lattices, J. Amer. Math. Soc 3 (1990), 929–960.

[G-B] L.C. Grove, C.T. Benson, Finite Reflection Groups, Graduate Texts

in Math. 99, Springer-Verlag, Heidelberg (1971, 1985).

[Ha] A.J. Hahn, The coset lattices of E.S. Barnes and G.E. Wall, J. Austral. Math. Soc. 49 (1990), 418–433.

[Hl] T.C. Hales, The Kepler Conjecture, http://www.math.lsa.umich.edu/ hales/. R.H. Hardin. See J.H. Conway.

[H-W] G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, Clarendon Press (Oxford Science Publication), Oxford (1979). [Fifth edition; first edition: 1938.]

[Hecke] E. Hecke, Mathematische Werke, VandenHoeck & Ruprecht, Göttingen (1970).

[He1] Ch. Hermite, Sur la théorie des formes quadratiques ternaires, J. reine angew. Math 40 (1850), 173–177. [= [He3], tome I, pp. 94–99.]

[He2] Ch. Hermite, Lettres à Jacobi, J. reine angew. Math. 40 (1850), 261– 278. [= [He3], tome I, pp. 100–121.]

[He3] Ch. Hermite, Œuvres, Gauthier-Villars, Paris (1905).

[Hof] N. Hofreiter, Über Extremformen, Monatshefte Math. Phys. 40 (1933), 129–152.

[Hmb1] P. Humbert, Théorie de la réduction des formes quadratiques définies positives dans un corps algébrique K fini, Comment. Math. Helv. 12 (1940), 263-306.

[Hmb2] P. Humbert, Réduction de formes quadratiques dans un corps algébrique fini, Comment. Math. Helv. 23 (1949), 50–63.

[Hmp1] J.E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, Heidelberg (1980).

[Hmp2] J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, Cambridge, U.K. (1992). D. Husemoller. See J. Milnor.

[I] M.I. Icaza, Hermite constant and extreme forms for algebraic number fields, J. London Math. Soc. 55 (1977), 11–22. M.I. Icaza. See also R. Baeza, R. Coulangeon.

[Ja1] D-O. Jaquet, Domaines de Voronoï et algorithme de réduction des formes quadratiques définies positives, Sém. Th. Nombres de Bordeaux (= J. Th. Nombres de Bordeaux) 2 (1990), 163–215.

[Ja2] D-O. Jaquet, Énumération complète des classes de formes parfaites en dimension 7, Thèse, Neuchâtel (1991).

[Ja3] D-O. Jaquet, Classification des réseaux dans ℝ^{7} (à l’aide de la notion de formes parfaites), Astérisque 198–199–200 (1992), 177–185.

[Ja4] D-O. Jaquet-Chiffelle, Description des voisines de E_{7}, D_{7}, D_{8} et D_{9}, Sém. Th. Nombres de Bordeaux (= J. Th. Nombres de Bordeaux) 4 (1992), 273–374.

[Ja5] D-O. Jaquet-Chiffelle, Énumération complète des classes de formes parfaites en dimension 7, Ann. Inst. Fourier 43,1 (1993), 21–55.

[Ja6] D-O. Jaquet-Chiffelle, Trois théorèmes de finitude pour les G-formes, J. Th. Nombres de Bordeaux 7 (1995), 165–176.

[Ja7] D-O. Jaquet-Chiffelle, Mail to J. Martinet, June 14, 1993. [Ja-S1] D-O. Jaquet, F. Sigrist , Formes quadratiques contiguës à D_{7}, C. R.

Acad. Sc. Paris A 309 (1989), 641–644.

[Ja-S2] D-O. Jaquet-Chiffelle, F. Sigrist, Classification des formes quadratiques réelles : un contre-exemple à la finitude, Acta Arith. 68 (1994), 291–294.

[Ka-L] G.A. Kabatiansky, V.I. Levenshtein, Bounds for packings on a sphere

and in space, Problems of Information Transmission 14 (1978), 1–17.

[Kan] R. Kane, Reflection Groups and Invariant Theory, CMS Books in Mathematics, Springer-Verlag, Heidelberg (2001).

[Ker] M. Kervaire, Unimodular lattices with a complete root system, L’Ens. Math. 40 (1994), 59–104.

[Kg] O. King, A mass formula for unimodular lattices with no roots, Math. Comp. 72 (2003), 839–863; see also: Table of masses of 32-dimensional even unimodular lattices having any given root system (2 megabytes). Available at http://llama.mshri.on.ca/ king/

[Ki] Y. Kitaoka, Arithmetic of quadratic forms, Cambridge University Press, Tract 106, Cambridge, U.K. (1993).

[Kn1] M. Kneser, Two remarks on extreme forms, Canad. J. Math. 7 (1955), 145–149.

[Kn2] M. Kneser, Klassenzahlen definiter quadratischen Formen, Archiv der Math. 8 (1957), 241–250.

[K-M-R-T] M.-A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol, The Book of Invo-

lutions, American Mathematical Society (1998).

[K-V] H. Koch, B. Venkov, Über ganzzahlige unimodulare euklidische Gitter,

J. reine angew. Math. 398 (1989), 144–168.

[K-Z1] A. Korkine, G. Zolotareff, Sur les formes quadratiques positives qua-

ternaires, Math. Ann. 5 (1872), 581–583.

[K-Z2] A. Korkine, G. Zolotareff, Sur les formes quadratiques, Math. Ann. 6

(1873), 366–389.

[K-Z3] A. Korkine, G. Zolotareff, Sur les formes quadratiques positives, Math.

Ann. 11 (1877), 242–292.

[Ks-P] F.R. Kschischang, S. Pasupathy, Some ternary and quaternary codes and associated sphere packings, Trans. Information Theory 38 (1992), 227–246.

[Lah] M. Laïhem, Construction algorithmique de réseaux parfaits, Thèse, Bordeaux (1992).

[Lar] J. Larmouth, The enumeration of perfect forms, in Computers in Number Theory, A.O.L. Atkin and J. Birch ed., Academic Press, New– York, 237–239.

[Le-Sl] J. Leech, N.J.A. Sloane, Sphere packing and error-correcting codes,

Canad. J. Math. 23 (1971), 718–745. [= Chap. 5 of [C-S].]

[Le-Se ] P.W.H. Lemmens and J.J. Seidel, Equiangular lines, J. Algebra 24

(1973), 494–512.

[L-S-T] W. Lempken, B. Schröder, P.H. Tiep, Symmetric squares, spherical designs, and lattice minima (with an appendix by Christine Bachoc and Tiep), J. Algebra 240 (2001), 185–208.

[LLL] A.K. Lenstra, H.W. Lenstra, L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515–534. V.I. Levenshtein. See G.A. Kabatiansky.

[vL] J.H. van Lint, Introduction to coding theory, Graduate Texts in Math. 86, Springer-Verlag, Heidelberg (1999). [Third edition; first edition: 1982.]

[Lo-So] P. Loyer, P. Solé, Les réseaux BW_{32} et U_{32} sont équivalents, J. Th. Nombres de Bordeaux 6 (1994), 359–362. M. Mc Connel. See A. Ash.

[McW-S] F.J. MacWilliams, N.J.A. Sloane, The Theory of Error Correcting

Codes, North Holland, Amsterdam (2nd printing, 1978).

[Mar] J. Martinet, Les réseaux parfaits des espaces euclidiens, Masson (now Dunod), Paris (1996).

[Mar’] J. Martinet, editor (writer for 60%), Réseaux euclidiens, designs sphériques et formes modulaires, L’Ens. Math., Monographie 37, Genève (2001).

[Mar1] J. Martinet, Structures algébriques sur les réseaux, in Number Theory, (Sém. Th. Nombres de Paris 1992–93), S. David ed., Cambridge University Press, Cambridge, U.K. (1995), 167–186.

[Mar2] J. Martinet, Une famille de réseaux dual-extrêmes, J. Th. Nombres de Bordeaux 9 (1997), 169–181.

[Mar3] J. Martinet, Algebraic constructions of lattices; isodual lattices, Acts of the Eger congress (Hungary, 1996), Walter de Gruyter (1998), 349– 360.

[Mar4] J. Martinet, Perfect and eutactic lattices: extensions of Voronoï’s theory, in Voronoï’s Impact on Modern Science, P. Engel, H. Syta ed., Proc. Inst. Math. National Acad. Sc. Ukrain 21, book 1, Kyiv (1998), 186–198.

[Mar5] J. Martinet, Sur la classification des réseaux parfaits de dimension 5, J. Th. Nombres de Bordeaux 11 (Actes des Journées Arithmétiques de Limoges de 1997) (1999), 149–159.

[Mar6] J. Martinet, Sur l’indice d’un sous-réseau (with an appendix by C. Batut), [Mar’] (2001), 163–211.

[Mar7] J. Martinet, Sur certains designs sphériques liés à des réseaux entiers, [Mar’] (2001), 135–146.

[Mar8] J. Martinet, Reduction Modulo 2 and 3 of Euclidean Lattices, J. Algebra 251 (2002), 864–887.

[Mar-V1] J. Martinet, B. Venkov, Les réseaux fortement eutactiques (with an appendix by R. Coulangeon), [Mar’] (2001), 112–134. J. Martinet. See also C. Batut, E. Bayer-Fluckiger, A.-M. Bergé.

[Mic] L. Michel, E-mail to the author (September 12, 1997). [M-H] J. Milnor, D. Husemoller, Symmetric Bilinear Forms, Springer-Verlag,

Ergebnisse 73, Heidelberg (1973).

[Min1] H. Minkowski, Gesammelte Abhandlungen, Chelsea, New York (1967; first edition: Leipzig, 1911).

[Min2] H. Minkowski, Geometrie der Zahlen, Johnson reprint Co., New York (1968; first edition: Leipzig, 1896).

[Mor] L.J. Mordell, Observation on the minimum of a positive definite quadratic form in eight variables, J. London Math. Soc. 9 (1944), 3–6.

[Nap1] H. Napias, Sur quelques réseaux contenus dans les réseaux de Leech et de Quebbemann, Comptes Rendus Acad. Sc. Paris 319, série I (1994), 653–658.

[Nap2] H. Napias, Étude expérimentale et algorithmique des réseaux euclidiens, Thèse, Bordeaux (1996).

[Ne1] G. Nebe, Endliche rationale Matrixgruppen vom Grad 24, Aachener Beiträge zur Mathematik, Verlag der Augustinus Buchhandlung (1995).

[Ne2] G. Nebe, Finite subgroups of GL_{n}(ℚ) for 25 ≤ n ≤ 31, Comm. Alg. 24 (1996), 2341–2397.

[Ne3] G. Nebe, Finite subgroups of GL_{24}(ℚ), Exp. Math. 5 (1996), 163–195.

[Ne4] G. Nebe, The normalizer action and strongly modular lattices, L’Ens. Math. 43 (1997), 67–76.

[Ne5] G. Nebe, Some Cyclo-quaternionic Lattices, J. Algebra 199 (1998), 472–498.

[Ne-Pl] G. Nebe, W. Plesken, Finite rational matrix groups, Memoirs A.M.S. 116, no. 556 (1995), 1–144.

[Ne-R-S] G. Nebe, E. Rains, N.J.A. Sloane, A Simple Construction for the Barnes–Wall Lattices, in The Forney Festschrift, R. Blahut ed., in Codes, Graphs, and Systems: A Celebraton of the Life and Career of G. David Forney, Jr. on the Occasion of his Sixtieth Birthday, edited by R. E. Blahut and R. Koetter, Kluwer (2002), 333-342.

[Ne-Sl] G. Nebe, N.J.A. Sloane, A Catalogue of Lattices, http://www.math.rwth-aachen.de/ Gabriele.Nebe/LATTICES/

[Formerly: http://www.research.att.com/ njas/lattices/index.html.]

[Ne-V1] G. Nebe, B. Venkov, Nonexistence of extremal lattices in certain genera

of modular lattices, J. Number Theory 60 (1996), 310–317.

[Ne-V2] G. Nebe, B. Venkov, The strongly perfect lattices in dimension 10, J. Th. Nombres de Bordeaux 12 (2000), 503–518. G. Nebe. See also C. Bachoc.

[Nie] H-V. Niemeier, Definite Quadratischen Formen der Discriminante 1 und der Dimension 24, J. Number Theory 5 (1973), 141–178.

[No] J. Nottebaum, Sphärische 4-designs in Gittern, Diplomarbeit, Universität Oldenburg (1995). S.P. Norton. See [ATLAS]. A. M. Odlyzko. See [C-S], Chapters 13 and 19.

[Oe1] J. Oesterlé, Le théorème des classes jumelles de Zimmert et les formules explicites de Weil, in Sém. Th. Nombres de Paris 1983–84, Prog. Math. 59, Birkhäuser, Basel (1985), 181–197.

[Oe2] J. Oesterlé, Empilements de sphères, in Séminaire Bourbaki 1989-1990, Astérisque 189–190 (1990), 375–397.

[Ogg] A. Ogg, Modular forms and Dirichlet series, Benjamin, New York (1969).

[O-W] S. Ohno, T. Watanabe, Estimates of Hermite Constants for Algebraic

Number Fields, Comm. Math. Univ. Sancti Pauli 50 (2001), 53–63.

[O’M] O.T. O’Meara, Introduction to Quadratic Forms, Springer-Verlag, Grundlehren 117, Heidelberg (1963). M. O’Ryan. See R. Baeza. R.A. Parker. See [ATLAS]. S. Pasupathy. See F.R. Kschischang.

[Pl1] W. Plesken, Finite modular groups of prime degree and circulants, J. Algebra 97 (1985), 286–312.

[Pl2] W. Plesken, Lattices of covariant quadratic forms, L’Ens. Math. 47 (2001), 21–56.

[Pl-P1] W. Plesken, M. Pohst, Constructing Integral Lattices with Prescribed

Minimum. I , Math. Comp. 45 (1985), 209–221 and S5–S16.

[Pl-P2] W. Plesken, M. Pohst, Constructing integral alttices with prescribed

minimum. II , Math. Comp. 60 (1993), 817–825 .

[Pl-S] W. Plesken, B. Souvignier, Computing isometries of lattices, J. Symbolic Computation 24 (1997), 327–334. W. Plesken. See also G. Nebe. M. Pohst. See Plesken (above), Wagner (second part).

[Q1] H.-G. Quebbemann, Zur Klassification unimodularer Gitter mit Isometrie von Primzahlordnung, J. reine angew. Math. 326 (1981), 158–170.

[Q2] H.-G. Quebbemann, An application of Siegel’s formula over quaternion orders, Mathematika 31 (1984), 12–16.

[Q3] H.-G. Quebbemann, A construction of integral lattices, Mathematika 31 (1984), 138–141.

[Q4] H.-G. Quebbemann, Lattices with theta-functions for G(\sqrt{2}) and linear codes, J. Algebra 105 (1987), 443–450.

[Q5] H.-G. Quebbemann, Unimodular lattices with isometries of large prime order. II , Math. Nachr. 156 (1992), 219–224.

[Q6] H.-G. Quebbemann, Modular lattices in Euclidean spaces, J. Number Theory 54 (1995), 190–202.

[Q7] H.-G. Quebbemann, Atkin-Lehner eigenforms and strongly modular lattices, L’Ens. Math 43 (1997), 55–65. H.-G. Quebbemann. See also Batut.

[Qi-Z] J.R. Quine, P. L. Zhang, Extremal symplectic lattices, Israel J. Math.

108 (1998), 237–251.

[R-S] E. Rains, N.J.A. Sloane, Shadow theory of modular and unimodular

lattices, J. Number Theory 73 (1998), 359–389.

[Ran1] R.A. Rankin, On positive definite quadratic forms, J. London Math. Soc. 28 (1953), 309–314.

[Ran2] R.A. Rankin, On the minimal points of a perfect quadratic form, Math. Z. 84 (1964), 228–232.

[Rei] I. Reiner, Maximal orders, London Math. Soc. Monographs 5, London (1975).

[Rie] C. Riener, On extreme forms in dimension 8 , J. Th. Nombres de Bordeaux 18 (2006), 677-682.

[Rog] C.A. Rogers, Packing and covering, Cambridge University Press, Cambridge, U.K. (1964). S.S. Ryshkov. See E.P. Baranovskii and part 2.

[Sch1] R. Scharlau, Unimodular lattices over real quadratic fields, Math. Zeitschrift 216 (1994), 437–452.

[Sch-SP] R. Scharlau, R. Schulze-Pillot, Extremal lattices, in Algorithmic Algebra and Number Theory, B.H. Matzat, G.-M. Greul, G. Hiss ed., Springer-Verlag, Heidelberg (1999), 139–170.

[Sch-V] R. Scharlau, B.B. Venkov, The genus of the Barnes–Wall lattice, Comment. Math. Helvetici 69 (1994), 322–333. R. Scharlau. See also C. Batut.

[SchW] W. Scharlau, Quadratic and Hermitian Forms, Grundlehren 270, Springer-Verlag, Heidelberg (1985).

[Scho] B. Schoeneberg, Das Verhalten von mehrfachen Thetareihen bei Modulsubstitutionen, Math. Ann. 116 (1939), 511–523.

[S-W] R. Schoof, L. Washington, Quintic polynomials and real cyclotomic fields with large class numbers, Math. Comp. 50 (1988), 543–556. B. Schröder. See W. Lempken.

[Schu] T. Schulz, Konstruktion rationaler Darstellungen endlicher Gruppen, Thesis, Aachen, Lehrstuhl B (2002). R. Schulze-Pillot. See R. Scharlau.

[Sco1] P.R. Scott, On perfect and extreme forms, J. Austral. Math. Soc. 4 (1964), 56–77.

[Sco2] P.R. Scott, The construction of perfect and extreme forms, Canad. J. Math. 18 (1966), 147–158. J.J. Seidel. See P.W.H. Lemmens.

[Se1] J-P. Serre, Corps locaux, second edition, Hermann, Paris (1968), translated into English as:

[Se’1] J-P. Serre, Local Fields, Springer-Verlag, Heidelberg (1995).

[Se2] J-P. Serre, Cours d’arithmétique, P.U.F., Paris (1970), translated into English as:

[Se’2] J-P. Serre, A Course in Arithmetic, Graduate Texts in Math. 7, Sprin-

ger-Verlag, Heidelberg (1973).

[Se3] J-P. Serre, Représentations linéaires des groupes finis, second edition, Hermann, Paris (1971); translated into English as:

[Se’3] J-P. Serre, Linear Representations of Finite groups, Graduate Texts

in Math. 42, Springer-Verlag, Heidelberg (1977).

[Se4] J-P. Serre, Œuvres, Springer-Verlag, Heidelberg (1986, 2000).

[Sh1] T. Shioda, Some remarks on elliptic curves over function fields, Astérisque 209 (1992), 99–114.

[Sh2] T. Shioda, A uniform construction of the root lattices E_{6}, E_{7}, E_{8} and their dual lattices , Proc. Japan Academy 71(A) (1995), 140–143.

[Sig1] F. Sigrist, E-mail to J. Martinet (September 11, 1990).

[Sig2] F. Sigrist, Formes Quadratiques encapsulées, J. Th. Nombres de Bordeaux 2 (1990), 425–429.

[Sig3] F. Sigrist, Cyclotomic quadratic forms, J. Th. Nombres de Bordeaux 12 (2000), 519–530. F. Sigrist. See also A.-M. Bergé, D-O. Jaquet.

[Sl1] N.J.A. Sloane, Letter to J. Martinet (May 11, 1992); now expanded in [Ne-Sl] as A table of the highest kissing numbers presently known.

[Sl2] N.J.A. Sloane, The Sphere packing problem, in Proceedings Internat. Congress Math. Berlin 1998, Documenta Mathematika, III (1998), pp. 387–396. N.J.A. Sloane. See also N. Bernstein, J.H. Conway, Y. Edel, F.J. MacWilliams, G. Nebe, E. Rains, and Appendix 2 to [Bu-Sa]. P. Solé. See P. Loyer above; R. Chapman, Ph. Gaborit, below.

[Sou1] B. Souvignier, Die irreduziblen Bravaisgruppen in GL_{8}(ℤ), Diplomarbeit, RWTH Aachen (1991).

[Sou2] B. Souvignier, Irreducible finite integral matrix groups of degree 8 and 10 , Math. Comp. 63 (1994), 335–350. B. Souvignier. See also W. Plesken.

[Sta1] K. C. Stacey, The enumeration of perfect septenary forms, J. London Math. Soc. 10 (1975), 97–104.

[Sta2] K. C. Stacey, The perfect septenary forms with D_{4} = 2, J. Austral. Math. Soc. 22 (1976), 144–164.

[Sti] E. Stiemke, Über positiven Lösungen homogener linearer Gleichungen, Math. Ann. 76 (1915), 340–342.

[St] M.I. Štogrin, Locally quasi-densest lattice packings of spheres, Soviet Math. Dokl. 15 (1974), 1288–1292. P.H. Tiep. See N. Dummigan, W. Lempken.

[Ti] J. Tits, Quaternions over ℚ(\sqrt{5}), Leech’s lattice and the sporadic group of Hall–Janko, J. Algebra 63 (1980), 56–75. J.A. Todd. See H.S.M. Coxeter.

[Var] A. Vardy, A new sphere packing in 20 dimensions, Invent. Math. 121 (1995), 119–133.

[Ven1] B.B. Venkov, On even unimodular Euclidean lattices of dimension 32, J. Soviet Math. 26 (1984), 1860–1867 ; Russian original: 1982.

[Ven2] B.B. Venkov, Even unimodular extremal lattices, Proc. Steklov Inst. Math. 165 (1984), 47–52 ; Russian original: 1984.

[Ven3] B.B. Venkov, Réseaux et “designs” sphériques, (Notes by J. Martinet), [Mar’] (2001), 10–86.

(Referred to erroneously as a future [Bc-V3] in the French edition.)

[Ven4] B.B. Venkov, On even unimodular Euclidean lattices of dimension 32, II, J. Soviet Math. 36 (1987), 21–38 ; Russian original: 1984.

[Ven5] B.B. Venkov, Unimodular lattices and strongly regular graphs, J. Soviet Math. 29 (1985), 1121–1127 ; Russian original: 1983. B. Venkov. See also R. Bacher, C. Bachoc, J. Martinet, G. Nebe, H. Koch, R. Scharlau.

[Vet] N.M. Vetchinkin, Uniqueness of the classes of positive quadratic forms on which the values of Hermite constants are attained for 6 ≤ n ≤ 8, Proc. Steklov Inst. Math. 152, 34–86.

[Vi ] M-F. Vignéras, Arithmétique des Algèbres de Quaternions, Lecture

Notes 800, Springer-Verlag, Heidelberg (1980).

[Vo1] G. Voronoï, Nouvelles applications des paramètres continus à la théorie des formes quadratiques : 1 Sur quelques propriétés des formes quadratiques positives parfaites, J. reine angew. Math 133 (1908), 97–178.

[Vo2] G. Voronoï, Nouvelles applications des paramètres continus à la théorie des formes quadratiques : 2 Recherches sur les paralléloèdres primitifs, première partie, J. reine angew. Math 134 (1908), 198–287.

[Vo3] G. Voronoï, Nouvelles applications des paramètres continus à la théorie des formes quadratiques : 2 Recherches sur les paralléloèdres primitifs, deuxième partie, J. reine angew. Math 136 (1909), 67–181.

[vdW] B.L. van der Waerden, Die Reduktionstheorie der positiven quadratischen Formen, in Studien zur Theorie der quadratischen Formen, B.L. van der Waerden and H. Gross ed., Birkhäuser, Basel (1968), 17–44. G.E. Wall. See E.S. Barnes.

[Was] L.C. Washington, Introduction to Cyclotomic Fields, Graduate Texts 83, Springer-Verlag, Heidelberg (1982).

L. Washington. See also R. Schoof.

[Wata1] T. Watanabe, On an analog of Hermite’s constant, J. Lie Theory 10 (2000), 33–52.

[Wata2] T. Watanabe, Upper bounds for Hermite’s constants for orthogonal groups, Comm. Math. Univ. Sancti Pauli 48 (1999), 25–33. T. Watanabe. See also S. Ohno.

[Wat1] G.L. Watson, Transformations of a quadratic form which do not increase the class-number, Proc. London Math. Soc. 12 (1962), 577–587.

[Wat2] G.L. Watson, On the minimum of a positive quadratic form in n(≤ 8) variables (verification of Blichfeldt’s calculation), Proc. Camb. Phil. Soc 62 (1966), 719.

[Wat3] G.L. Watson, On the minimal points of a perfect septenary quadratic form, Mathematika 16 (1969), 170–177.

[Wat4] G.L. Watson, On the minimum points of a positive quadratic form, Mathematika 18 (1971), 60–70.

[Wat5] G.L. Watson, The number of minimum points of a positive quadratic form, Dissertationes Math. 84 (1971), 1–46.

[Wat6] G.L. Watson, The least common denominator of the coefficient of a perfect quadratic form, Acta Arith. 18 (1971), 29–36.

[Wat7] G.L. Watson, The number of minimum points of a positive quadratic form having no perfect binary section with the same minimum, Proc. London Math. Soc. 24 (1972), 625–646.

[Obituary of Watson: J.V. Armitage, Bull. london Math. Soc 45 (2013), 432–445.]

R.A. Wilson. See [ATLAS].

[Zah] N.V. Zahareva, Centerings of 8-dimensional lattices that preserve a frame of successive minima, Proc. Steklov Inst. Math. 152 (1982), 107–134 ; Russian original: 1980.

[Zas] H. Zassenhaus, The Theory of Groups (second edition), Chelsea, New York (1958). P.L. Zhang. See J.R. Quine.

[Zi] R. Zimmert, Ideale kleiner Norm in Idealklassen und eine Regulator Abschätzung, Invent. Math. 62 (1981), 367–380. G. Zolotareff . See A. Korkine.

COMPLEMENTARY BIBLIOGRAPHY

General information.

• The National Academy of Sciences of Ukraine organizes every five years in Kyiv the congress Voronoï’s impact on modern science. Congresses were held in 1993, 1998, 2003. The next one was planed to 2008, and a further is planed in 2013, September 16-20 (announcement on May, 22nd on the Number Theory Web). Acts for the last two congresses have been published: Book 1, Book 2 (1998), relative to the 1998 congress; Book3 (2005), relative to the 2003 congress.

• The book [Ne-R-S2] (see below) has been published in 2006 by Springer-Verlag.

• The book [Schr1] (see below) has been published in 2009 by the Amer. Math. Soc.

• The book [C-F-S-V], edited by W.K. Chan, L. Kukshansky, R. Schulze-Pillot and J.D. Vaaler, Diophantine Methods, Lattices, and Arithmetic Theory of Quadratic Forms, Contemporary Math. 587 (2013), contains several papers connected with the subject of my book.

References.

[Abd] K. Abdukhalikov, Unimodular Hermitian lattices in dimension 13 , J. Algebra 272 (2004), 186–190.

[Abd-S] Abdukhalikov, R. Scharlau, Unimodular lattices in dimensions 14 and 15 over the Eisenstein integers, Math. Comp. 78 (2009), 387–403; arXiv:0710.3664v1, 19 Oct. 2007.

[Anz] M.M. Anzin, On variation of positive quadratic forms (with application to the study of perfect forms), Proc. Steklov Inst. Math. 4 (1992), 11– 27 ; Russian original: 196 (1991).

[AMRT] A. Ash, D. Mumford, M. Rapoport, Y. Tai, Smooth compactification of locally symmetric varieties. Lie Groups: History, Frontiers and Applications, Vol. IV. Math. Sci. Press, Brookline, Mass. (1975), 335 pp. MR0457437 (56#15642); Reviewer: Ichiro Satake

[Bc3] R. Bacher, A new Inequality for the Hermite Constants, Int. J. Numbrer Th. 4 (2008), 363–386.

[Bc4] R. Bacher, Construction of some perfect integral lattices with minimum 4 , J. Th. Nombres de Bordeaux 27 (2015), 655–687 ; preprint at arXiv:1401.3601 (15. Jan. 2014).

[Bc5] R. Bacher, On the number of perfect lattices, J. Th. Nombres de Bordeaux 30 (2018), 917-945; preprint at arXiv:1704.02234 (7 April 2017).

[Bac4] C. Bachoc, Designs, groups, and Lattices, J. Th. Nombres de Bordeaux

(issue dedicated to the Journées Arithmétiques of Graz, 2003) 17 (2005), 25–44.

[Bac-Gb] C. Bachoc, Ph. Gaborit, Designs and self-dual codes with long shad-

ows, J. Comb. Th., A 105 (2004), 15–34.

[Bac-V1] C. Bachoc, F. Vallentin, New upper bounds for kissing numbers from semidefinite programming, J. Amer. Math. Soc. 21 (2008), 909-–924.

[Bac-V2] C. Bachoc, F. Vallentin, Semidefinite programming, multivariate orthogonal polynomials, and codes in spherical caps, European J. Combin. 30 (2009), 625—637.

[Ban1] E. Bannai, On some spherical t-designs, Geom. Dedicata 43 (1992), 167–179.

[Ban2] E. Bannai, Orthogonal polynomials, algebraic combinatorics and spherical t-designs, Proc. Sympos. Pure Math. 37 (1980), 465–468.

[Ban3] E. Bannai, Spherical t-designs which are orbits of finite groups, J. Math. Soc. Japan 36 (1984), 341–354.

[Ban4] E. Bannai, Spherical designs and group representations, Combinatorics and Algbra, C. Greene ed., Contemporary Math.34 (1984), 95–107.

[Bav5’] C. Bavard, Voronoï theory for hyperbolic families of symplectic lattices, Proceedings of the ”Third Voronoi Conference on Analytic Number Theory and Spatial Tessellations, Kyiv, September 22-28, 2003 (2005),

79–85. [This is an account for [Bav5] (2007).]

[Bav6] C. Bavard, Isodualité des réseaux euclidiens en petite dimension, submitted; prprint at arXiv (8 Nov 2014), 67 pp.

[Bay6] E. Bayer-Fluckiger, Determinants of integral ideal lattices and automorphisms with given characteristic polynomial, J. Algebra 257 (2002), 215-221.

[Bay7] E. Bayer-Fluckiger, Upper bounds for Euclidean minima of algebraic number fields, J. Number Theory 121 (2006), 305-323.

[Bay-Ne] E. Bayer-Fluckiger, G. Nebe, On the Euclidean minimum of some real

number fields, J. Th. Nombres Bordeaux 17 (2005), 437–454.

[Bay-S1] E. Bayer-Fluckiger, I. Suarez, Modular lattices over cyclotomic fields,

J. Number Theory 114 (2005), 394–411.

[Bay-S2] E. Bayer-Fluckiger, I. Suarez, Ideal lattices over totally real number

fields and Euclidean minima, Archiv Math. 86 (2006), 217-225.

[Ber5] A.-M. Bergé, On certain Coxeter lattices without perfect sections, J. Alg. Combinatorics 20 (2004), 5–16.

[Ber6] A.-M. Bergé, On lattices of maximal index two, arXiv:0806.0724v1, 4 June 2008.

[Anne-Marie Bergé passed away on September 20th, 2008; the obituary by J. Martinet, Anne-Marie Bergé — In Memoriam, J. Th. Nombres Bordeaux 20, fasc. 2 (2008), i–xi, can be read at the end of the main part of my homepage.]

[B-M7] A.-M. Bergé, J. Martinet, A generalization of some lattices of Coxeter,

Mathematika 51 (2004), 49–61.

[B-M8] A.-M. Bergé, J. Martinet, On weakly eutactic forms, J. London Math. Soc. 75 (2007), 187-198; see also Voronoi graphs, cells, and spherical designs, Oberwolfach Reports 1/2005, 20–23.

[B-M9] A.-M. Bergé, J. Martinet, Sublattices of certain Coxeter lattices (A paper in the honour of Georges Gras), J. Th. Nombres Bordeaux 17 (2005), 455–465.

[B-M10] A.-M. Bergé, J. Martinet, On Perfection Relations in Lattices, Contemporary Math. 493 (2009), 29–49; arXiv:math.NT/0611220 (8 Nov. 2006).

[B-F-G-M] A. Böttcher, L. Fukshansky, S.R. Garcia, H. Maharaj, On lattices generated by finite Abelian groups, preprint at arXiv:1406.7595 (1. July, 2014).

[B-Cou] O. Braun, R. Coulangeon, Perfect Lattices over Imaginary Quadratic Number Fields, Math. Comp. 84 (2015), 1451–1467; preprint at arXiv:1304.0559 (2 Apr 2013).

[B-C-N-S] O.Braun, R. Coulangeon, G. Nebe, S. Schoennenbeck, Computing in arithmetic groups with Voronoi’s algorithm, J. Algebra 435 (2015), 263–285; preprint at arXiv:1407.6234.

[C-I-L] W.K. Chan, M. I. Icaza, E.A. Lauret, A generalized Hermite constant and its computations for imaginary quadratic fields, arXiv:1307.3714 (2013), 11 pp.

[C-D-G-S] R. Chapman, S.T. Dougherty, P. Gaborit, P. Solé, 2-modular lattices

from ternary codes, J. Th. Nombres Bordeaux 14 (2002), 73–85.

[Chen] Hao Chen, On a generalization of Craig lattices, J. Th. Nombres Bordeaux 25 (2013), 59–70.

[Chng] S. Cheng [Shentian Cheng], Dense Packings from Algebraic Number Fields and Codes, arXiv:1506.00419 [math.NT], June 1st, 2015.

[Cn1] H. Cohn, New upper bounds on sphere packings II , Geometry & Topology 6 (2002), 329–353.

[Cn2] H. Cohn, A Conceptual Breakthrough in Sphere Packing, Prepreint, arXiv 1611.01685v1, 5 Nov. 2016.

[Cn-El] H. Cohn, N. Elkies, New upper bounds on sphere packings I , Ann.

Math. 157 (2003), 689-714.

[Cn-Km1] H. Cohn, A. Kumar, Optimality and uniqueness of the Leech lattice among lattices, Ann. Math. 170 (2009), 1003–1050; arXiv:math.MG/0403263.

[Cn-Km2] H. Cohn, A. Kumar, Uniqueness of the (22,891,1/4) spherical code, New York J. Math. 13 (2007), 147–157 (electronic); arXiv:math.MG/0607448.

[C-K-V-al] H. Cohn, A. Kumar, S. Miller, D. Radchenko, M. Viazovska, The sphere packing problem in dimension 24 , Ann. Math. 185 (2017), 1017–1033; preprint, arXiv:1603.06518 (March 21., 2016). [See also

[Cn2] above.]

[Cou7] R. Coulangeon, Spherical Designs and Zeta Functions of Lattices, Int. Math. Res. Not. (2006), Art. ID 49620, 16 pp.

[Cou8] R. Coulangeon, On Epstein zeta function of Humbert forms, Inter. J, Number Theory 4 (2008), 387–401.

[Cou-I-O] R. Coulangeon, M.I. Icaza, M. O’Ryan, Lenstra’s constant and

extreme forms in number fields, Exp. Math. 16 (2007), 455–462.

[Cou-L] R. Coulangeon, G. Lazzarini, Spherical designs and heights of Euclidean lattices, J. Number Theory 141 (2014), 288–315; preprint at arXiv:1401.2891 (13 Jan. 2014).

[Cou-Ne1] R. Coulangeon, G. Nebe, Dense lattices as Hermitian tensor

products, (Proceedings of a 2011 Banff workshop on lattices),

Contemporary Math. 587 (2013), 47–57; preprint at arXiv:1201.1832

(2012), under the title The unreasonable effectiveness of the tensor product.

[Cou-Ne2] R. Coulangeon, G. Nebe, Maximal finite subgroups and minimal classes, L’Ens. Math. 60 (2014), no. 3-4, 231–246. preprint (2013) at arXiv:1304.2597.

[Cou-Schr] Energy minimization, periodic sets and spherical designs, Int. Math.

Res. Not. 4 (2012), 829—848.

[Cou-Wt] R. Coulangeon, T. Watanabe, Hermite constant and Voronoi theory over a quaternion skew field. Osaka J. Math. 43 (2006), 517-556. R. Coulangeon, see also O. Braun. E. Dable-Heath. See C. Porter. S.T. Dougherty. See R. Chapman.

[D-S-V1] M. Dutour Sikirić, A. Schürmann, F. Vallentin, Classification of eight dimensional perfect forms, Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 21–32; preprint at arXiv:math/0609388.

[D-S-V2] M. Dutour Sikirić, A. Schürmann, F. Vallentin, A generalization of Voronoi’s reduction theory and its application, Duke Math. J. 142 (2008), 127-164; arXiv:math/0601084

[D-S-V3] M. Dutour Sikirić, A. Schürmann, F. Vallentin, Inhomogeneous extreme forms, Ann. Inst. Fourier 62 (2012), 2227–2255 (2013); arXiv:1008.4751.

[D-E-K-M] M. Dutour Sikirić, Ph. Elbaz-Vincent, A. Kupers, J. Martinet, Voronoi complexes in higher dimensions, cohomology of GL_{N}(ℤ) for N ≥ 8 and the triviality of K_{8}(ℤ), submitted; preprint at arXiv:1910.11598, 25 Oct. 2019.

[E-G-S1] P. Elbaz-Vincent, H. Gangl, C. Soulé, Quelques calculs de la cohomologie de GL_{N}(ℤ) et de la K-théorie de ℤ, C.R. Acad. Sci. Paris 335 (2002), 321–324.

[E-G-S2] P. Elbaz-Vincent, H. Gangl, C. Soulé, Perfect forms and the cohomology of modular groups, Advances in Math. 245 (2013), 587–624; arXiv:math/1001.0789v1.

replaces Perfect Lattices, Homology of Modular Groups and Algebraic K-theory, Oberwolfach Reports 1/2005 (2005), 36–39.

Ph. Elbaz-Vincent, see also M. Dutour Sikirić. S. Ferguson. See T.C. Hales.

[F-I-P] A.L. Flores, J.C. Interlando, T. Pires da Nóbrega Neto, An Extension

of Craig’s Family of Lattices, Canad. Math. Bull. 54 (2011), 645–653.

[FIPLO] A.L. Flores, J.C. Interlando, T. Pires da Nóbrega Neto, D. Lopes, J. Othon, On a refinement of Craig’s lattices, J. Pure Appl. Algebra 215 (2011), 1440–1442. L. Fukshansky. See A. Böttcher. H. Gangl. See P. Elbaz-Vincent.

S.R. Garcia. See A. Böttcher.

[Gb] Ph. Gaborit, Construction of new extremal unimodular lattices, European J. Combin. 25 (2004), 549–564.

[Gb-N-S] Ph. Gaborit, A.M. Natividad, P. Solé, Eisenstein lattices, Galois rings and quaternary codes, Int. J. Number Theory 2 (2006), 289–303.

[Gb-Zm] Ph. Gaborit, G. Zémor, On the construction of dense lattices with a given automorphism group, Ann. Inst. Fourier 57 (2007), 1051–1062. Ph. Gaborit. See also C. Bachoc, R. Chapman.

[Gi’] Marc Gindraux, Lamination et antilamination des réseaux euclidiens,

J. Th. Nombres Bordeaux 21 (2009), 535–557. (Replaces [Gi].)

[Gr1] R.L. Griess, Positive definite lattices of rank at most 8, J. Number Theory 103 (2003), 77–84.

[Gr2] R.L. Griess, An introduction to groups and lattices: finite groups and positive definite rational lattices, Advanced Lectures in Mathematics nu. 15, International Press (2011).

[Hl1] T.C. Hales, see http://www.math.pitt.edu/ thales/ .

M. Harada. Numerous papers often with (various ) co-authors deal with self-dual codes and applications to unimodular lattices. One example is referred below:

[Hr-Kt-Oz] M. Harada, M. Kitazume, M. Ozeki, Ternary code construction of unimodular lattices and self-dual codes over ℤ_{6}, J. Algebraic Combin. 16 (2002), 209–223.

[H-Pa1] P. de la Harpe, C. Pache, Spherical designs and finite group representations (some results of E. Bannai), European J. Combi. 25 (2004), 213–227.

[H-Pa2] P. de la Harpe, C. Pache, Cubature formulas, geometrical designs, reproducing kernels, and Markov operators, Acts of the International Conference on Group Theory (Gaeta, June 2003); Progr. Math., 248, Birkhäuser, Basel (2005), 219–267; arXiv:math/0502312.

[H-Pa-V] P. de la Harpe, C. Pache, B. Venkov, Construction of spherical cubature formulas using lattices, Algebra i Analiz 18 (2006), 162–186; translation in St. Petersburg Math. J. 18 (2007), no. 1, 119–139; arXiv:math/0505510.

[H-O] X. Hou, F. Oggier, Modular lattices from a variation of construction A over number fields, Adv. Math. Commun. 11 (2017), 719-–745 ; arXiv:math1604.01583e (6 Apr 2016), 26 pp.

[Hu-Ne1] S. Hu, G. Nebe, Strongly perfect lattices sandwiched between Barnes-Wall lattices, J. London Math. Soc. 101 (2020), 1068–1089; preprint at arXiv:math1805.01196 (3 May 2018).

[Hu-Ne2] S. Hu, G. Nebe, Low dimensional strongly perfect lattices IV: The dual strongly perfect lattices of dimension 16 , J. Number Theory 208 (2020), 262–294; preprint at arXiv:math1905.07307 (17 May 2019).

[Hu-Y] C. Hurlburt, S. Yazdani, Hermite’s Constant for Number Fields, Canad. J. Math. 64 (2012), 301–317. M.I. Icaza. See also W.K. Chan, R. Coulangeon J.C. Interlando. See A.L. Flores.

[K-M-S] W. Keller, J. Martinet, A. Schürmann, On classifying Minkowskian sublattices, Math. Comp. 81 (2012), 1063-1092; arXiv:0904.3110v3. M. Kitazume. See M. Harada. A. Kumar. See H. Cohn.

A. Kupers, see M. Dutour Sikirić.

[L-L-S] J.C. Lagarias, H.W. Lenstra, C.P. Schnorr, Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice, Combinatorica 10 (1990), 333-348. E.A. Lauret. See W.K.Chan. G. Lazzarini. See R. Coulangeon. H.W. Lenstra, see J.C. Lagarias. C. Ling. See Porter. D. Lopes. See A.T. Flores. H.Maharaj. See A. Böttcher.

[Mar9] J. Martinet, The Perfection Default Module of a Euclidean Lattice, preprint, Université Bordeaux 1 (2000), http://www.math.ubordeaux.fr/ martinet/ .

[Mar10] J. Martinet, Reduction Modulo 2 and 3 of Euclidean Lattices, II , Acts of the Conference on Quadratic Forms, Talca, Chile, December 2002; Contemporary Math. 344 (2004), 265–273.

[Mar11] J. Martinet, Bases of minimal vectors in Euclidean lattices, I Archiv Math. 89 (2007), 404–410.

[Mar12] J. Martinet, Bases of minimal vectors in Euclidean lattices, II Archiv Math.89 (2007), 541–551.

[Mar13] J. Martinet, On the index system of well-rounded lattices, arXiv:1202.2295v1 [math.NT] (10 Feb 2012).

[Mar14] J. Martinet, Hermite versus Minkowski, preprint at arXiv:1403.1457v1 [math.NT], March 6th, 2014

[Mar-Sch] Former reference for [K-M-S] above. [Mar-Schr1] J. Martinet, A. Schürmann, Bases of minimal vectors in Eu-

clidean lattices, III, Inter. J. Number T. 08, 2 (2012), 551–567. arXiv:1105.5889v1

[Mar-V2] J. Martinet, B. Venkov, On integral lattices having an odd minimum, Algebra and Analysis (Saint-Petersburg) 16, 3 (2004), 99–142. J. Martinet. See also A.-M. Bergé, M. Dutour Sikirić, W. Keller.

[Mey] B. Meyer, Generalised Hermite constants, Voronoi theory and heights on flag varieties, Bull. S.M.F. 137 (2009), 127–158. S. Miller. See also H.Cohn.

[McM] C.T. McMullen, Minkowski’s conjecture, well-rounded lattices and topological dimension, J. Amer. Math. Soc 18 (2005), 711–734. D. Mumford, see A. Ash. A.M. Natividad. See Ph. Gaborit.

[Ne6] G. Nebe, An even unimodular 72-dimensional lattice of minimum 8, J. reine angew. Math. 673 (2012), 237–247; arXiv:1008.2862.

[Ne7] G. Nebe, Boris Venkov’s Theory of Lattices and Spherical Designs, Proceedings of a 2011 Banff workshop on lattices, Contemporary Math. 587 (2013), 1-19; arXiv:1201.1834.

[Ne8] G. Nebe, On automorphisms of extremal even unimodular lattices, Int. J. Number Theory 9 (2013), 1933-1959; arXiv:1212.0865 (2012).

[Ne9] G. Nebe, A new extremal even unimodular lattice of dimension 48 , arXiv:1312.3780 (2013), 7 pp.

[Ne-No-V] , G. Nebe, E. Nossek, B. Venkov, Low dimensional strongly perfect lattices. II: Dual strongly perfect lattices of dimension 13 and 15, J. Th. Nombres Bordeaux 25 (2013), 147–161.

[Ne-P] G. Nebe, R. Parker, On extremal even unimodular 72-dimensional lat-

tices/. Math. Comp., 83 (2014), 1489-1494.

[Ne-R-S1] G. Nebe, E.Rains, N.J.A. Sloane, The invariants of the Clifford groups,

Designs, Codes and Cryptography 24,1 (2001), 99–122.

[Ne-R-S2] G. Nebe, E.Rains, N.J.A. Sloane, Self-Dual Codes and Invariant Theory (vol. 17 of “Algorithms and Computations in Mathematics”), Springer-Verlag, Heidelberg, 2006.

[Ne-Schi] G. Nebe, K. Schindelar, S-extremal strongly modular lattices, J. Th.

Nombres Bordeaux, 19 (2007), 683–701.

[Ne-V3] G. Nebe, B. Venkov, Unimodular lattices with long shadow, J. Number

Theory 99 (2003), 307-317.

[Ne-V4] G. Nebe, B. Venkov, Low dimensional strongly perfect lattices. I: The 12-dimensional case, L’Ens. Math. 51 (2005), 129–163; arXiv:math/0503446.

[Ne-V5] G. Nebe, B. Venkov, On lattices whose minimal vectors form a 6design, preprint (2006); European J. Combin. 30 (2009), 716–724. (Special issue dedicated to Eiichi Bannai’s 60th birthday.)

[Ne-V6] G. Nebe, B. Venkov, Low dimensional strongly perfect lattices. III: Dual strongly perfect lattices in dimension 14 , Inter. J. Number T. 6 (2010), 387–409. G. Nebe. See also E. Bayer-Fluckiger, O. Braun, R. Coulangeon, S. Hu.

[Nos1] E. Nossek, Spherical designs and lattices, Beitr. Algebra Geom. 55 (2014), 25—31; preprint, arXiv:1111.00772 [Math.NT] (3. Nov. 2011). E. Nossek. See also G. Nebe.

[Nov] N. V. Novikova, Three missed centering the 8-dimensional lattices, keeping a consecutive minima base, preprint, Ivanovo (1981), 8 pages.

[Oe3] J. Oesterlé, Densité maximale des empilements de sphères en dimension 3 [d’après T. Hales et J. Ferguson], in Séminaire Bourbaki 1998-1999, exp. 863, Astérisque 266 (2000), 405–413.

[Oe4] J. Oesterlé, Densité maximale des empilements de sphères en dimensions 8 et 24 [d’après Viazovska et al.], in Séminaire Bourbaki 2016-2017 (June17th, 2017 ; preprint). F. Oggier. See X. Hou. K. Okuda. See M. Harada, and K. Sawatami below.

[Op] J. Opgenorth, Dual cones and the Voronoi algorithm, Exp. Math. 10 (2004), 500–608. M. O’Ryan. See R. Coulangeon. J. Othon. See A.T. Flores.

[Oz] M. Ozeki, On a problem posed by R. Salvati Manni, Acta Arith. 150 (2011), 1–22. M. Ozeki, see also M. Harada.

[Pa1] C. Pache, Espaces polynomiaux et formules de cubature, Thèse, Univ. de Genève (2006).

[Pa2] C. Pache, Shells of selfdual lattices viewed as spherical designs, preprint at arXiv:math/0502313.

[Pa3] C. Pache, Extremality and designs in spaces of quadratic forms, Internat. J. Algebra Comput. 15 (2005), 1085–1127; arXiv:0804.0173. C. Pache. See also P. de la Harpe. R. Parker, See G. Nebe and first part. T. Pires. See A.L. Flores.

[Pl3] W. Plesken, Solving XX^{tr} = A over the integers, Linear Algebra Appl. 226/228 (1995), 331–344.

[Pe-vZ] R.A. Pendavingh, S.H.M. van Zam, New Korkin-Zolotarev inequali-

ties, SIAM J. Optim. 18,1 (2007), 363–378.

[Pf-Zg] F. Pfender, G.M. Ziegler, Kissing numbers, sphere packings, and some unexpected proofs, Notices Amer. Math. Soc.51 (2004), 873-–883. M. Pohst. See Plesken (first part), Wagner (below).

[P-Y1] C. Poor, D.S. Yuen, Linear dependence among Siegel Modular form,

Math. Ann. 318 (2000), 205–234.

[P-Y2] C. Poor, D.S. Yuen, Slopes of integral lattices, J. Number Theory 100

(2003), 363–380.

[P-Y3] C. Poor, D.S. Yuen, The extreme core, Abh. Math. Seminar Hamburg,

75 (2005), 1–25.

[P-Y4] C. Poor, D.S. Yuen, The Bergé-Martinet constant and slopes of Siegel

cusp forms, Bull. London Math. Soc. 38 (2006), 913–924.

[P-D-L] C. Porter, E. Dabl-Heath, C. Ling, New Bound for the Orthogonality Defect of HKZ Reduced Lattices, arXiv:2208.11092 (11 Aug 2022). D. Radchenko, see H. Cohn. E.Rains, see G. Nebe. M. Rapoport , see A. Ash.

[Reg] S. Regavim (with an appendix joint with L. Hadassi), Minkowski bases, Korkin-Zolotarev bases and successive minima, arXiv:2106.03183 (6 Jun 2021).

[Ry] S. S. Ryshkov, On the problem of the determination of quadratic forms in many variables, Proc. Steklov Inst. Math. 142 (1979), 233–259 ; Russian original: 1976.

[S-Wt] K. Sawatami, T. Watanabe, Type one functions and Voronoi’s theo-

rem, J. Convex Analysis 18, 2 (2011), 341–351.

[S-Wt-O] K. Sawatami, T. Watanabe, K. Okuda, A note on the Hermite-Rankin

constant, J. Th. Nombres Bordeaux 22 (2010), 209–217.

[Sch2] R. Scharlau, The extremal lattice of dimension 14, level 7 and its genus, Contemporary Math. 587 (2013), 167–183. R. Scharlau. See also Abdukhalikov. A. Schindelar . See G. Nebe C.P. Schnorr, see J.C. Lagarias.

[Schr1] A. Schürmann, Computational geometry of positive definite quadratic forms, University Lecture Series 48, AMS, Providence, 2008.

[Schr2] A. Schürmann, Perfect, strongly eutactic lattices are periodic extreme, Adv. Math. 225 (2010), no. 5, 2546–2564; arXiv:0808.2013v1.

[Schr3] A. Schürmann, Enumerating Perfect Forms, Contemporary Math. 493 (2009), 359–377; arXiv:0901.1587.

[Schr-V1] A. Schürmann, F. Vallentin, Methods in the local theory of packing and covering lattices, preprint, http://fma2.math.uni-magde-burg.de/ latgeo/index/

[Schr-V2] A. Schürmann, F. Vallentin, Computational Approaches to Lattice Packing and Covering Problems, Discrete Comp. Geom. 35 (2006), 73–116; arXiv:Math/0403272. A. Schürmann . See also R. Coulangeon, M. Dutour Sikirić, W. Keller, J. Martinet. N.J.A. Sloane, see G. Nebe. P. Solé. See R. Chapman, Ph. Gaborit. C.. Soulé. See P. Elbaz-Vincent. I. Suarez, see E. Bayer.

[Ste-Wak] D. Stehlé, M. Watkins, On the extremality of an 80-dimensional lattice, Algorithmic number theory, Springer Lecture Notes in Comput. Sci. 6197 (Berlin, 2010), 340–356. T. Tai, see A. Ash. F. Vallentin. See C. Bachoc, M. Dutour Sikirić, A. Schürmann.

[Van1] S. Vance, Mordell’s inequality for Eisenstein and Hurwitz lattices, Trans. Amer. Math. Soc. 362 (2010), 3827–3839; arXiv:0810.2336v4 (April 1st, 2009).

[Van2] S. Vance, New lower bounds for the spheere packing density in high dimensions, Advances in Mathematics 227 (2011), 2144–2156; arXiv:1105.3779. S.H.M. van Zam, , see R.A. Pendavingh.

[Vnk] A. Venkatesh, A Note on Sphere Packings in High Dimension, Inter. Math. Research Notices 2013, 7 (2013), 1628–1642. B. Venkov. See also P. de la Harpe, G. Nebe, J. Martinet.

[Boris B. Venkov passed away on November 10th, 2011; see [Ne7].]

[Viaz] M. Viazovska, The sphere packing problem in dimension 8 , Ann. Math. 185 (2017), 991–1015; preprint at arXiv:1603.04246v1 (14 Mar 2016). M. Viazovska, see also H. Cohn.

B. L. van der Waerden. Here is an alternative reference for [vdW]: [vdW’] Die Reduktionstheorie der positiven quadratischen Formen, Acta

Math. 96 (1956), 265–309.

[W-P] M. Wagner, M. Pohst, On the Computation of Hermite-Humbert Constants for Algebraic Number Fields, J. Th. Nombres de Bordeaux 17 (2005), 905–920. T. Watanabe. See R. Coulangeon, K. Sawatami.

[Watk] M. Watkins, Another 80-dimensional extremal lattice, J. Th.. Nombres Bordeaux 24 (2012), 237–255. M. Watkins, see also D. Stehlé.

[vW1] W.P.J. van Woerden, Perfect quadratic forms: an upper bound and challenges in enumeration, Master’s thesis, Leiden University, (2018).

[vW2] W.P.J. van Woerden, An upper bound on the number of perfect quadratic forms, preprint at arXiv:1901.04807v1 (16 Jan. 2019)

[Xing] C. Xing [Chaoping Xing], Dense Packings from Quadratic Fields and Codes, J. Combin. Theory 115 (2008), 1021–1035. S. Yazdani, see C. Hurlburt. D.S. Yuen. See C. Poor.

[Zah1] See [Nov] above. Maxim Anzin, in an e-mail dated March 23rd, 2004, pointed out to me that the three possible structures which were forgotten in [Zah] (related to my paper [Mar6]) were corrected by the author in a preprint in Russian written under the name of N. V. Novikova (see above), a preprint that I have never seen. Note also that in Math. Reviews, [Zah] reads Zaharova.

G. Zémor. See Ph. Gaborit. G.M. Ziegler. See F. Pfender