Lower bounds for the minima of certain symplectic and unitary group lattices

American Journal of Mathematics

Volumn 121, Issue 4, 1999, Pages 889-918

  Dummigan, Neil ^a   Tiep, Pham Huu ^a  

^a NONE

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[No Author keywords available]

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EID: 0002990959     PISSN: 00029327     EISSN: None     Source Type: Journal    
DOI: 10.1353/ajm.1999.0027     Document Type: Article

References (32)

1. E. S. Barnes and N. J. A. Sloane, New lattice packings of spheres, Canad. J. Math. 35 (1983), 117-130.
2. E. S. Barnes and G. E. Wall, Some extreme forms defined in terms of abelian groups, J. Austral. Math. Soc. 1 (1959), 47-63.
3. H. Cohn, A Classical Invitation to Algebraic Numbers and Class Fields, Springer-Verlag. New York, 1978.
4. J. H. Conway and N. J. A. Sloane, On lattices equivalent to their duals, J. Number Theory 48 (1994), 373-382.
5. C. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Pure Appl. Math., vol. 11, Wiley Interscience, New York, 1962.
6. N. Dummigan, Symplectic group lattices as Mordell-Weil sublattices, J. Number Theory 61 (1996), 365-387.
7. _, Algebraic cycles and even unimodular lattices, J. London Math. Soc. (2) 56 (1997), 209-221.
8. N. Dummigan and Pham Huu Tiep, Congruences for certain theta series, J. Number Theory 71 (1998), 86-105.
9. P. Gérardin, Weil representations associated to finite fields, J. Algebra 46 (1977), 54-101.
10. R. Gow, On the Schur indices cf characters of finite classical groups, J. London Math. Soc. (2) 24 (1981), 135-147.
11. _, Even unimodular lattices associated with the Weil representation of the finite symplectic group, J. Algebra 122 (1989), 510-519.
12. B. H. Gross, Group representations and lattices. J. Amer. Math. Soc. 3 (1990), 929-960.
13. A. Grothendieck, Standard conjectures on algebraic cycles, Algebraic Geometry, Bombay 1968, Oxford University Press, Oxford, 1969, pp. 193-199.
14. R. M. Guralnick and Pham Huu Tiep, Low-dimensional representations of special linear groups in cross characteristics, Proc. London Math. Soc. 78 (1999), 116-138.
15. R. Hotta and K. Matsui, On a lemma of Tate-Thompson, Hiroshima Math. J. 8 (1978), 255-268.
16. T. Katsura and T. Shioda, On Fermat varieties, Tohoku Math. J. 31 (1979), 97-115.
17. V. Landazuri and G. Seitz, On the minimal degrees of projective representations of finite Chevalley groups, J. Algebra 32 ( 974), 418-443.
18. J. S. Milne, On a conjecture of Artin and Tate, Ann. of Math. 102 (1975), 517-533.
19. _, Values of zeta functions of varieties over finite fields, Amer. J. Math. 108 (1986), 297-360.
20. H.-G. Quebbemann, Modular latices in Euclidean spaces, J. Number Theory 54 (1995), 190-202.
21. R. Scharlau and Pham Huu Tiep, Symplectic groups, symplectic spreads, codes and unimodular lattices, J. Algebra 194 (1997), 113-156.
22. _, Symplectic group lattices, Trans. Amer. Math. Soc. (to appear).
23. T. Shioda, Some observations on Jacobi sums, Galois Representations and Arithmetic Algebraic Geometry, Adv. Stud. Pure Math., vol. 12. Academic Press, Boston, MA, 1987, pp. 119-135.
24. J. Tate, Algebraic cycles and poles of zeta functions, Arithmetical Algebraic Geometry (O. F. G. Schilling, ed.), Harper and Row, New York, 1965, pp. 93-110.
25. _, On the conjecture of Birch and Swinnerton-Dyer and a geometric analog. Séminaire Bourbaki 306 (1995), 415-440.
26. J. G. Thompson, Finite groups and even lattices, J. Algebra 38 (1976), 523-524.
27. Pham Huu Tiep, Globally irreducible representations of finite groups and integral lattices, Geom. Dedicata 64 (1997), 85-123.
28. _, Weil representations as globally irreducible representations, Math. Nachr. 184 (1997), 313-327.
29. Pham Huu Tiep and A. E. Zalesskii, Minimal characters of the finite classical groups, Comm. Algebra 24 (1996), 2093-2167.
30. _, Some characterizations of the Weil representations of the symplectic and unitary groups, J. Algebra 192 (1997), 130-165.
31. k), Topics in Algebra, vol. 26, Banach Center Publications, Warsaw, 1990, pp. 491-500.
32. K. Zsigmondy, Zur Theorie der Potenzreste, Monath. Math. Phys. 3 (1892), 265-284.