Abstract
Slopes of an adelic vector bundle exhibit a behaviour akin to successive minima. Comparisons between the two amount to a Siegel lemma. Here we use Zhang’s version for absolute minima over the algebraic numbers. We prove a Minkowski-Hlawka theorem in this context. We also study the tensor product of two hermitian bundles bounding both its absolute minimum and maximal slope, thus improving an estimate of Chen. We further include similar inequalities for exterior and symmetric powers, in terms of some lcm of multinomial coefficients.
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Références
Y. André, On nef and semistable hermitian lattices, and their behaviour under tensor product, The Tohoku Mathematical Journal 63 (2011), 629–649.
J.-B. Bost, Périodes et isogénies des variétés abéliennes sur les corps de nombres, (d’après D. Masser et G. Wüstholz), Séminaire Bourbaki, Vol. 237 de Astérisque, Société Mathématique de France, 1996, pp. 115–161.
J.-B. Bost et H. Chen, Concerning the semi-stability of tensor products in Arakelov geometry, Journal de Mathématiques Pures et Appliquées 99 (2013), 436–488.
H. Chen, Maximal slope of tensor product of Hermitian vector bundles, Journal of Algebraic Geometry 18 (2009), 575–603.
R. Coulangeon, Tensor products of Hermitian lattices, Acta Arithmetica 92 (2000), 115–130.
W. Fulton et J. Harris, Representation Theory, Graduate Texts in Mathematics, Vol. 129, Springer-Verlag, Berlin, 1991.
É. Gaudron, Pentes des fibrés vectoriels adéliques sur un corps global, Rendiconti del Seminario Matematico della Università di Padova 119 (2008), 21–95.
É. Gaudron, Géométrie des nombres adélique et lemmes de Siegel généralisés, Manuscripta mathematica 130 (2009), 159–182.
Y. Kitaoka, Arithmetic of Quadratic Forms, Cambridge Tracts in Mathematics, Vol. 106, Cambridge University Press, 1993.
J. Milnor et D. Husemoller, Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73, Springer-Verlag, Berlin, 1973.
M. Nair, On Chebyshev-type inequalities for primes, The American Mathematical Monthly 89 (1982), 126–129.
J. Rosser et L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois Journal of Mathematics 6 (1962), 64–94.
D. Roy et J. Thunder, An absolute Siegel’s lemma, Journal für die Reine und Angewandte Mathematik 476 (1996), 1–26. Addendum et erratum, ibid., 508 (1999), 47–51.
J. Thunder, An adelic Minkowski-Hlawka theorem and an application to Siegel’s lemma, Journal für die Reine und Angewandte Mathematik 475 (1996), 167–185.
J. Vaaler, The best constant in Siegel’s lemma, Monatshefte für Mathematik 140 (2003), 71–89.
I. Williams, On a problem of Kurt Mahler concerning binomial coefficients, Bulletin of the Australian Mathematical Society 14 (1976), 299–302.
S. Zhang, Positive line bundles on arithmetic varieties, Journal of the American Mathematical Society 8 (1995), 187–221.
S. Zhang, Heights and reductions of semi-stable varieties, Compositio Mathematica 104 (1996), 77–105.
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Gaudron, É., Rémond, G. Minima, pentes et algèbre tensorielle. Isr. J. Math. 195, 565–591 (2013). https://doi.org/10.1007/s11856-012-0109-x
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DOI: https://doi.org/10.1007/s11856-012-0109-x