The characteristic masses of Niemeier lattices
Journal de théorie des nombres de Bordeaux, Volume 32 (2020) no. 2, pp. 545-583.

Let L be an integral lattice in an n-dimensional Euclidean space E and W an irreducible representation of the orthogonal group of E. We give an implemented algorithm computing the dimension of the subspace of invariants in W under the isometry group O(L) of L. A key step is the determination, for any polynomial P, of the proportion of elements in O(L) with characteristic polynomial P, a collection of rational numbers that we call the characteristic masses of L. As an application, we determine the characteristic masses of all the Niemeier lattices, and more generally of any even lattice of determinant 2 in dimension n25.

For Niemeier lattices, as a verification, we provide an alternative (human) computation of the characteristic masses. The main ingredient is the determination, for each Niemeier lattice L with non-empty root system R, of the G(R)-conjugacy classes of the elements of the “umbral” subgroup O(L)/W(R) of G(R), where G(R) is the automorphism group of the Dynkin diagram of R, and W(R) its Weyl group.

These results have consequences for the study of the spaces of automorphic forms of the definite orthogonal groups in n variables over . As an example, we provide concrete dimension formulas in the level 1 case, as a function of the weight W, up to n=25.

Soient L un réseau entier d’un espace euclidien E de dimension n et W une représentation irréductible du groupe orthogonal de E. Nous donnons un algorithme calculant la dimension du sous-espace des éléments de W invariants par le groupe O(L) des isométries de L. Une étape clef est de déterminer, pour tout polynôme P, la proportion des éléments de O(L) de polynôme caractéristique P, une collection de nombres rationnels que nous appelons les masses caractéristiques de L. En guise d’application, nous déterminons les masses caractéristiques de tous les réseaux de Niemeier, et plus généralement de tous les réseaux pairs de déterminant 2 en dimension n25.

Pour les réseaux de Niemeier, en guise de vérification, nous donnons une méthode alternative (et humaine) pour calculer leurs masses caractéristiques. L’ingrédient principal est la détermination, pour chaque réseau de Niemeier L de système de racines R non vide, des G(R)-classes de conjugaison d’éléments du sous-groupe « ombral » O(L)/W(R) de G(R), où G(R) est le groupe des automorphismes du diagramme de Dynkin de R, et W(R) son groupe de Weyl.

Ces résultats ont des applications à l’étude des espaces de formes automorphes des groupes orthogonaux de formes quadratiques sur définies positives : nous donnons des formules concrètes pour la dimension de ces espaces en niveau 1, comme fonction du poids W, en tout rang n25.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1134
Classification: 11F, 11F55, 11H55, 11H56, 11H71, 20D08, 22C05
Keywords: Euclidean lattices, isometry groups, Niemeier lattices, automorphic forms
Chenevier, Gaëtan 1

1 CNRS, Université Paris-Saclay Laboratoire de mathématiques d’Orsay 91405 Orsay, France
@article{JTNB_2020__32_2_545_0,
     author = {Chenevier, Ga\"etan},
     title = {The characteristic masses of {Niemeier} lattices},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {545--583},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {32},
     number = {2},
     year = {2020},
     doi = {10.5802/jtnb.1134},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.1134/}
}
TY  - JOUR
AU  - Chenevier, Gaëtan
TI  - The characteristic masses of Niemeier lattices
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2020
SP  - 545
EP  - 583
VL  - 32
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - http://www.numdam.org/articles/10.5802/jtnb.1134/
DO  - 10.5802/jtnb.1134
LA  - en
ID  - JTNB_2020__32_2_545_0
ER  - 
%0 Journal Article
%A Chenevier, Gaëtan
%T The characteristic masses of Niemeier lattices
%J Journal de théorie des nombres de Bordeaux
%D 2020
%P 545-583
%V 32
%N 2
%I Société Arithmétique de Bordeaux
%U http://www.numdam.org/articles/10.5802/jtnb.1134/
%R 10.5802/jtnb.1134
%G en
%F JTNB_2020__32_2_545_0
Chenevier, Gaëtan. The characteristic masses of Niemeier lattices. Journal de théorie des nombres de Bordeaux, Volume 32 (2020) no. 2, pp. 545-583. doi : 10.5802/jtnb.1134. http://www.numdam.org/articles/10.5802/jtnb.1134/

[1] Bayer-Fluckiger, Eva Definite unimodular lattices having an automorphism of given characteristic polynomial, Comment. Math. Helv., Volume 59 (1984), pp. 509-538 | DOI | MR | Zbl

[2] Bayer-Fluckiger, Eva; Taelman, Lenny Automorphisms of even unimodular lattices and equivariant Witt groups (2017) (https://arxiv.org/abs/1708.05540, to appear in J. Eur. Math. Soc.)

[3] Borcherds, Richard E. Classification of positive definite lattices, Duke Math. J., Volume 105 (2000) no. 3, pp. 525-567 | DOI | MR | Zbl

[4] Bourbaki, Nicolas Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Éléments de mathématique, Masson, 1981 | Zbl

[5] Carter, Roger W. Conjugacy classes in the Weyl group, Compos. Math., Volume 25 (1972), pp. 1-59 | Numdam | MR | Zbl

[6] Chenevier, Gaëtan Characteristic masses of lattices (http://gaetan.chenevier.perso.math.cnrs.fr/charmasses)

[7] Chenevier, Gaëtan; Clozel, Laurent Corps de nombres peu ramifiés et formes automorphes autoduales, J. Am. Math. Soc., Volume 22 (2009) no. 2, pp. 467-519 | DOI | Zbl

[8] Chenevier, Gaëtan; Lannes, Jean Automorphic forms and even unimodular lattices, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 69, Springer, 2019 | MR | Zbl

[9] Chenevier, Gaëtan; Renard, David Level one algebraic cusp forms of classical groups of small rank, Memoirs of the American Mathematical Society, 1121, American Mathematical Society, 2015 | Zbl

[10] Chenevier, Gaëtan; Taïbi, Olivier Siegel modular forms of weight 13 and the Leech lattice (2019) (https://arxiv.org/abs/1907.08781)

[11] Chenevier, Gaëtan; Taïbi, Olivier Discrete series multiplicities for classical groups over Z and level 1 algebraic cusp forms, Publ. Math., Inst. Hautes Étud. Sci., Volume 131 (2020), pp. 261-323 | DOI | MR | Zbl

[12] Cheng, Miranda; Duncan, John F. R.; Harvey, Jeffrey A. Umbral moonshine and the Niemeier lattices, Res. Math. Sci., Volume 1 (2014), 3, 81 pages | MR | Zbl

[13] Cohen, Arjeh M. Finite complex reflection groups, Ann. Sci. Éc. Norm. Supér., Volume 9 (1976), pp. 379-436 erratum in ibid. 11 (1978), no. 4, p. 613 | DOI | Numdam | MR | Zbl

[14] Cohen, Arjeh M. Finite quaternionic reflection groups, J. Algebra, Volume 64 (1980), pp. 293-324 | DOI | MR | Zbl

[15] Conway, John H. A group of order 8,315,553,613,086,720,000, Bull. Lond. Math. Soc., Volume 1 (1979), pp. 79-88 | DOI | MR | Zbl

[16] Conway, John H.; Curtis, Robert T.; Norton, Simon P.; Parker, Richard A.; Wilson, Robert A. Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups, Oxford University Press, 1985, xxxiv+252 pages (with computational assistance from J. G. Thackray) | Zbl

[17] Conway, John H.; Sloane, Neil J. A. Low-dimensional lattices. IV. The mass formula, Proc. R. Soc. Lond., Ser. A, Volume 419 (1988) no. 1857, pp. 259-286 | MR | Zbl

[18] Conway, John H.; Sloane, Neil J. A. Sphere packings, lattices and groups, Grundlehren der Mathematischen Wissenschaften, 290, Springer, 1999 | MR | Zbl

[19] Dembélé, Lassina On the computation of algebraic modular forms on compact inner forms of GSp 4 , Math. Comput., Volume 83 (2014) no. 288, pp. 1931-1950 | DOI | MR | Zbl

[20] Dummigan, Neil A simple trace formula for algebraic modular forms, Exp. Math., Volume 22 (2013) no. 2, pp. 123-131 | DOI | MR | Zbl

[21] Ebeling, Wolfgang Lattices and Codes. A course partially based on lectures by F. Hirzebruch, Advanced Lectures in Mathematics, Vieweg, 2002 | Zbl

[22] Erokhin, V. A. Groups of automorphisms of 24-dimensional even unimodular lattices, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova, Volume 116 (1982), pp. 68-73 | MR | Zbl

[23] Fincke, Ulrich; Pohst, Michael Improved methods for calculating vectors of short length in a lattice, including a complexity analysis, Math. Comput., Volume 44 (1985), pp. 463-471 | DOI | MR | Zbl

[24] Frobenius, Georg Über die Charaktere der mehrfach transitiven Gruppen, Berl. Ber., Volume 1904 (1904), pp. 558-571 | Zbl

[25] Fulton, William; Harris, Joe Representation theory. A first course, Graduate Texts in Mathematics, 129, Springer, 1991 | Zbl

[26] The GAP Group GAP — Groups, Algorithms, and Programming, Version 4.10.2, 2019 (http://www.gap-system.org)

[27] Greenberg, Matthew; Voight, John Lattice methods for algebraic modular forms on classical groups, Computations with modular forms (Contributions in Mathematical and Computational Sciences), Volume 6, Springer, 2014, pp. 147-179 | DOI | MR | Zbl

[28] Gross, Benedict; McMullen, Curtis Automorphisms of even unimodular lattices and unramified Salem numbers, J. Algebra, Volume 257 (2002) no. 2, pp. 265-290 | DOI | MR | Zbl

[29] Hall, Marshall Note on the Mathieu group M 12 , Arch. Math., Volume 13 (1962), p. 334-240 | DOI | MR

[30] Hulpke, Alexander Conjugacy classes algorithms in finite permutation groups via homomorphic images, Math. Comput., Volume 69 (2000) no. 232, pp. 1633-1651 | DOI | MR | Zbl

[31] Ingraham, Mark H A note on determinants, Bull. Am. Math. Soc., Volume 43 (1937), pp. 579-580 | DOI | MR | Zbl

[32] Kneser, Martin Klassenzahlen definiter quadratischer formen, Arch. Math., Volume 8 (1957), pp. 241-250 | DOI | MR | Zbl

[33] Koike, Kazuhiko; Terada, Itaru Young-diagrammatic methods for the representation theory of the classical groups of type B n , C n , D n , J. Algebra, Volume 107 (1987), pp. 466-511 | DOI | Zbl

[34] Kostant, Bertram Lie Algebra Cohomology and the Generalized Borel-Weil Theorem, Ann. Math., Volume 74 (1961), pp. 320-387 | MR | Zbl

[35] Lansky, Joshua; Pollack, David Hecke algebras and automorphic forms, Compos. Math., Volume 130 (2002) no. 1, pp. 21-48 | DOI | MR | Zbl

[36] Loeffler, David Explicit Calculations of Automorphic Forms for Definite Unitary Groups, LMS J. Comput. Math., Volume 11 (2010), pp. 326-342 | DOI | MR | Zbl

[37] Nebe, Gabriele On automorphisms of extremal even unimodular lattices, Int. J. Number Theory, Volume 9 (2013) no. 8, pp. 1933-1959 | DOI | MR | Zbl

[38] Niemeier, Hans-Volker Definite quadratische Formen der Dimension 24 und Diskriminante 1, J. Number Theory, Volume 5 (1973), pp. 142-178 | DOI | MR

[39] The PARI Group PARI/GP version 2.11.0, 2014 (available from http://pari.math.u-bordeaux.fr/)

[40] Plesken, Wilhelm; Souvignier, Bernd Computing isometries of lattices, J. Symb. Comput. (1997), pp. 327-334 | DOI | MR

[41] Schur, J. Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene linear Substitutionen, J. für Math., Volume 139 (1911), pp. 155-250 | MR | Zbl

[42] Sloane, Neil J. A. The On-Line Encyclopedia of Integer Sequences, 2010 (http://oeis.org) | Zbl

[43] Taïbi, Olivier Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula, Ann. Sci. Éc. Norm. Supér., Volume 50 (2017) no. 2, pp. 269-344 | DOI | MR | Zbl

[44] Venkov, Boris B. On the classification of integral even unimodular 24-dimensional quadratic forms, 1999 (Chapter 18 in [18])

[45] Wendt, Robert Weyl’s character formula for non-connected Lie groups and orbital theory for twisted affine Lie algebras, J. Funct. Anal., Volume 180 (2001) no. 1, pp. 31-65 | DOI | MR | Zbl

[46] Weyl, Hermann The Classical Groups, Princeton Mathematical Series, 1, Princeton University Press, 1946 | Zbl

Cited by Sources: