Slopes of Euclidean lattices, tensor product and group actions

  • Renaud CoulangeonEmail author
  • Gabriele Nebe


We study the behavior of the minimal slope of Euclidean lattices (or more generally Ok-lattices) under tensor product. A general conjecture predicts that
$${\mu _{\min }}\left( {L \otimes M} \right) = {\mu _{\min }}\left( L \right){\mu _{\min }}\left( M \right)$$
for all lattices L and M. We prove that this is the case under the additional assumptions that L and M are acted on multiplicity-free by their automorphism group, such that one of them has at most 2 irreducible components.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Y. André, Slope filtrations, Confluentes Mathematici 1 (2009), 1–85.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Y. André, On nef and semistable Hermitian lattices, and their behavior under tensor product, Tohoku Mathematical Journal 63 (2011), 629–649.CrossRefGoogle Scholar
  3. [3]
    J.-B. Bost, Periodes et isogenies des varietes abeliennes sur les corps de nombres (d’apres D. Masser et G. Wu¨stholz), Asterisque 237 (1996), 115–161.zbMATHGoogle Scholar
  4. [4]
    J.-B. Bost and H. Chen, Concerning the semistability of tensor products in Arakelov geometry, Journal de Mathematiques Pures et Appliquees 99 (2013), 436–488.MathSciNetCrossRefGoogle Scholar
  5. [5]
    B. Casselman, Stability of lattices and the partition of arithmetic quotients, The Asian Journal of Mathematics 8 (2004), 607–637.MathSciNetCrossRefGoogle Scholar
  6. [6]
    H. Chen, Harder-Narasimhan categories, Journal of Pure and Applied Algebra 214 (2010), 187–200.MathSciNetCrossRefGoogle Scholar
  7. [7]
    C. W. Curtis and I. Reiner, Methods of Representation Theory. Vol. I, John Wiley &Sons, New York, 1981.zbMATHGoogle Scholar
  8. [8]
    É. Gaudron and G. Rémond, Minima, pentes et algèbre tensorielle, Israel Journal of Mathematics 195 (2013), 565–591.MathSciNetCrossRefGoogle Scholar
  9. [9]
    D. R. Grayson, Reduction theory using semistability, Commentarii Mathematici Helvetici 59 (1984), 600–634.MathSciNetCrossRefGoogle Scholar
  10. [10]
    D. R. Grayson, Reduction theory using semistability. II, Commentarii Mathematici Helvetici 61 (1986), 661–676.MathSciNetCrossRefGoogle Scholar
  11. [11]
    G. Harder and M. S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves, Mathematische Annalen 212 (1974/75), 215–248.MathSciNetCrossRefGoogle Scholar
  12. [12]
    Y. Kitaoka, Scalar extension of quadratic lattices. II, Nagoya Mathematical Journal 67 (1977), 159–164.MathSciNetCrossRefGoogle Scholar
  13. [13]
    Y. Kitaoka, Arithmetic of Quadratic Forms, Cambridge Tracts in Mathematics, Vol. 106, Cambridge University Press, Cambridge, 1993.Google Scholar
  14. [14]
    J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 73 Springer, New York-Heidelberg, 1973.Google Scholar
  15. [15]
    D. Mumford, Projective invariants of projective structures and applications, in Proceedings of the International Congress of Mathematicians (Stockholm, 1962), Institut Mittag-Leffler, Djursholm, 1963, pp. 526–530.Google Scholar
  16. [16]
    M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Annals of Mathematics 82 (1965), 540–567.MathSciNetCrossRefGoogle Scholar
  17. [17]
    O. T. O’Meara, Introduction to Quadratic Forms, Classics in Mathematics, Springer-Verlag, Berlin, 2000.Google Scholar
  18. [18]
    U. Stuhler, Eine Bemerkung zur Reduktionstheorie quadratischer Formen, Archiv der Mathematik 27 (1976), 604–610.MathSciNetCrossRefGoogle Scholar
  19. [19]
    U. Stuhler, Zur Reduktionstheorie der positiven quadratischen Formen. II, Archiv der Mathematik 28 (1977), 611–619.MathSciNetCrossRefGoogle Scholar

Copyright information

© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Institut de Mathématiques de BordeauxUniversité de Bordeaux CNRS, IMB, UMR 5251Talence cedexFrance
  2. 2.Lehrstuhl D für MathematikRWTH AachenAachenGermany

Personalised recommendations