×

Group representations and lattices. (English) Zbl 0745.11035

Various authors have constructed and classified \(G\)-stable lattices in higher-dimensional Euclidean space using sufficiently restrictive hypotheses on the action of a finite group \(G\). The present paper contains a particularly deep application of this principle.
The author considers lattices \(L\) for representations \(V\) of \(G\) over \(\mathbb{Q}\) which satisfy the following conditions of “global irreducibility”: \(V\otimes\mathbb{R}\) is irreducible, there is a maximal order \(R\) in \(K=\text{End}_ G V\) such that \(L\) is an \(RG\)-lattice, and for any maximal two-sided ideal \({\mathfrak p}\) of \(R\) the reduction \(V_{\mathfrak p}=L/{\mathfrak p}L\) is irreducible. These conditions ensure that, after some normalization of the inner product, the class group of \(R\) acts simply transitively on the \(\mathbb{Z} G\)-isometry classes of the lattices (so this is the classification, although not yet over \(\mathbb{Z}\)). To find all admissible pairs \((G,V)\) remains an open problem. For the case \(K=\mathbb{Q}\), studied earlier by J. G. Thompson, there are almost no examples (and no new ones) known. But when \(K\) is imaginary quadratic or a definite quaternion algebra over \(\mathbb{Q}\), many examples are given here. Some of the pairs \((G,L)\) have been known before, e.g., for the Weil representation of \(G=\text{Sp}_{2n}(q)\) by R. Gow (his lattices should have been referred too).
It is an important aspect of the paper that in several cases the lattices arising from representation theory are also described as a Mordell-Weil group modulo torsion for some elliptic curve \(E\) over a global function field \(F\). Such Mordell-Weil lattices have been studied by T. Shioda and N. Elkies; the author especially refers to Elkies’ (unpublished) work. In the last example, \(E\) is defined over the field \(k\) with \(q^ 2\) elements such that \(E(k)\) has order \((q+1)^ 2\), \(F\) is the function field of the Fermat curve with exponent \(q+1\) over \(k\), and \(G=PU_ 3(q^ 2)\). Here the global irreducibility condition is satisfied only when \(q=p\) (a prime) or \(q=p^ 2\), but for higher \(q\) the author calculates part of the Tate-Shafarevich group which gives an upper bound for the determinant of the Mordell-Weil lattice.

MSC:

11H56 Automorphism groups of lattices
20C15 Ordinary representations and characters
11G05 Elliptic curves over global fields
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Allan Adler, Some integral representations of \?\?\?\(_{2}\)(\?_{\?}) and their applications, J. Algebra 72 (1981), no. 1, 115 – 145. · Zbl 0479.20017
[2] N. Bourbaki, Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Actualités Scientifiques et Industrielles, No. 1364. Hermann, Paris, 1975 (French). · Zbl 0329.17002
[3] Michel Broué and Michel Enguehard, Une famille infinie de formes quadratiques entières; leurs groupes d’automorphismes, Ann. Sci. École Norm. Sup. (4) 6 (1973), 17 – 51 (French). · Zbl 0261.20022
[4] Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. · Zbl 0567.20023
[5] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. · Zbl 0568.20001
[6] J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1988. With contributions by E. Bannai, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. · Zbl 0634.52002
[7] M. Eichler, Über die Klassenzahl total definiter quaternionenalgebren, Math. Z. 43 (1937), 102-109. · JFM 63.0093.02
[8] N. Elkies, Letters to N. J. Sloane, 15 August 1989, 15 September 1989.
[9] Walter Feit, The computations of some Schur indices, Israel J. Math. 46 (1983), no. 4, 274 – 300. · Zbl 0528.20009
[10] Walter Feit, The representation theory of finite groups, North-Holland Mathematical Library, vol. 25, North-Holland Publishing Co., Amsterdam-New York, 1982. · Zbl 0493.20007
[11] Paul Gérardin, Weil representations associated to finite fields, J. Algebra 46 (1977), no. 1, 54 – 101. · Zbl 0359.20008
[12] R. Gow, Schur indices of some groups of Lie type, J. Algebra 42 (1976), no. 1, 102 – 120. · Zbl 0352.20013
[13] R. Gow, On the Schur indices of characters of finite classical groups, J. London Math. Soc. (2) 24 (1981), no. 1, 135 – 147. · Zbl 0468.20034
[14] Robert L. Griess Jr., Automorphisms of extra special groups and nonvanishing degree 2 cohomology, Pacific J. Math. 48 (1973), 403 – 422. · Zbl 0283.20028
[15] E. Hecke, Über ein fundamentalproblem aus der theorie der Elliptischen modulfunktionen, Abh. Math. Sem. Univ. Hamburg 6 (1928), 235-257 (= Werke 28). · JFM 54.0405.02
[16] Ryoshi Hotta and Kiyoshi Matsui, On a lemma of Tate-Thompson, Hiroshima Math. J. 8 (1978), no. 2, 255 – 268. · Zbl 0404.14001
[17] J. E. Humphreys, Representations of \?\?(2,\?), Amer. Math. Monthly 82 (1975), 21 – 39. · Zbl 0296.20020
[18] Gregory Karpilovsky, The Schur multiplier, London Mathematical Society Monographs. New Series, vol. 2, The Clarendon Press, Oxford University Press, New York, 1987. · Zbl 0619.20001
[19] Vicente Landazuri and Gary M. Seitz, On the minimal degrees of projective representations of the finite Chevalley groups, J. Algebra 32 (1974), 418 – 443. · Zbl 0325.20008
[20] G. Lusztig, Coxeter orbits and eigenspaces of Frobenius, Invent. Math. 38 (1976/77), no. 2, 101 – 159. · Zbl 0366.20031
[21] Barry Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 2, 207 – 259. · Zbl 0593.14021
[22] J. S. Milne, The Tate-Šafarevič group of a constant abelian variety, Invent. Math. 6 (1968), 91 – 105. · Zbl 0159.22402
[23] John Milnor and Dale Husemoller, Symmetric bilinear forms, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73. · Zbl 0292.10016
[24] Daniel Quillen, The \?\?\? 2 cohomology rings of extra-special 2-groups and the spinor groups, Math. Ann. 194 (1971), 197 – 212. · Zbl 0225.55015
[25] Bruno Schoeneberg, Elliptic modular functions: an introduction, Springer-Verlag, New York-Heidelberg, 1974. Translated from the German by J. R. Smart and E. A. Schwandt; Die Grundlehren der mathematischen Wissenschaften, Band 203. · Zbl 0285.10016
[26] J.-P. Serre, Complex multiplication, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) Thompson, Washington, D.C., 1967, pp. 292 – 296.
[27] J.-P. Serre, A course in arithmetic, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French; Graduate Texts in Mathematics, No. 7. · Zbl 0256.12001
[28] Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott; Graduate Texts in Mathematics, Vol. 42. · Zbl 0355.20006
[29] Tetsuji Shioda, Mordell-Weil lattices and Galois representation. I, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no. 7, 268 – 271. · Zbl 0715.14015
[30] -, Mordell-Weil lattices and sphere packings, preprint, 1989.
[31] Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. · Zbl 1196.22001
[32] Robert Steinberg, Representations of algebraic groups, Nagoya Math. J. 22 (1963), 33 – 56. · Zbl 0271.20019
[33] John Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134 – 144. · Zbl 0147.20303
[34] -, On the conjecture of Birch and Swinnerton-Dyer and a geometric analog, Sém. Bourbaki 306 1965-1966. · Zbl 1334.81106
[35] J. G. Thompson, A simple subgroup of \( {E_8}(3)\), Finite Groups , Tokyo, 1976, pp. 113-116.
[36] J. G. Thompson, Finite groups and even lattices, J. Algebra 38 (1976), no. 2, 523 – 524. · Zbl 0344.20001
[37] Marie-France Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980 (French). · Zbl 0422.12008
[38] G. van der Geer and M. van der Vlugt, Reed-Muller codes and supersingular curves, Ann. École. Norm. Sup. (to appear). · Zbl 0804.14014
[39] A. E. Zalesskiĭ and I. D. Suprunenko, Representations of dimension (\?\(^{n}\)\pm 1)/2 of the symplectic group of degree 2\? over a field of characteristic \?, Vestsī Akad. Navuk BSSR Ser. Fīz.-Mat. Navuk 6 (1987), 9 – 15, 123 (Russian, with English summary). · Zbl 0708.20011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.