Valerio Talamanca^[1]

On canonical heights on endomorphism rings over global function fields

Pages: 247-258
Received: 9 December 2015
Accepted in revised form: 10 March 2016
Mathematics Subject Classification (2010): 11G.
Keywords: Heights, adelic vector bundles, endomorphism rings.
Author address:
[1] : Università degli Studi Roma Tre, Largo San Leonardo Murialdo 1, Roma, 00185, Italy

Abstract: We present a construction of a canonical height on the endomorphism ring of a finite dimensional vector space over a global function field. We also prove a limit formula analogous to the Tate's formula defining the canonical heights on abelian varieties.

This research was partially supported by G.N.S.A.G.A of Istituto Nazionale di Alta Matematica and Prin2011 Geometria delle Varietà Algebriche

References

[1] E. Breuillard, A height gap theorem for finite subsets of \({\rm GL}_d(\overline{\Bbb Q})\) and nonamenable subgroups, Ann. of Math. (2) 174 (2011), no. 2, 1057-1110. [MR2831113]
[2] G. S. Call and J. H. Silverman, Canonical heights on varieties with morphisms, Compositio Math. 89 (1993), no. 2, 163-205. [MR1255693]
[3] L. Denis, Hauteurs canoniques et modules de Drinfel'd, Math. Ann. 294 (1992), no. 2, 213-223. [MR1183402]
[4] É. Gaudron, Pentes des fibrés vectoriels adéliques sur un corps global, Rend. Semin. Mat. Univ. Padova 119 (2008), 21-95. [MR2431505]
[5] É. Gaudron, Geómetrie des nombres adéliques et lemmes de Siegel généralisés, Manuscripta Math. 130 (2009), 159-182. [MR2545512]
[6] A. Néron, Quasi-fonctions et hauteurs sur les variétés abéliennes, Ann. of Math. (2) 82 (1965), 249-331. [MR0179173]
[7] Ju. I. Manin, The Tate height of points on an Abelian variety, its variants and applications (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1363-1390. [MR0173676]
[8] J.-P. Serre, Lectures on the Mordell-Weil theorem, Vieweg & Sohn, Braunshweig 1989. [MR1002324]
[9] V. Talamanca, A Gelfand-Beurling type formula for heights on endomorphism rings, J. Number Theory 83 (2000), no. 1, 91-105. [MR1767654]
[10] J. D. Vaaler, Small zeros of quadratic forms over number fields, Trans. Amer. Math. Soc. 302 (1987), 281-296. [MR0887510]
[11] A. Weil, Basic Number Theory, second edition, Springer Verlag, New York 1973. [Zbl 0267.12001]