Equidistribution and integral points for Drinfeld modules
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- by D. Ghioca and T. J. Tucker PDF
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Abstract:
We prove that the local height of a point on a Drinfeld module can be computed by averaging the logarithm of the distance to that point over the torsion points of the module. This gives rise to a Drinfeld module analog of a weak version of Siegel’s integral points theorem over number fields and to an analog of a theorem of Schinzel’s regarding the order of a point modulo certain primes.References
- Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990. MR 1070709, DOI 10.1090/surv/033
- M. Baker, S. I. Ih, and R. Rumely, A finiteness property of torsion points, 2005, preprint, available at arXiv:math.NT/0509485, 30 pages.
- Vincent Bosser, Minorations de formes linéaires de logarithmes pour les modules de Drinfeld, J. Number Theory 75 (1999), no. 2, 279–323 (French, with English summary). MR 1681633, DOI 10.1006/jnth.1998.2336
- —, Transcendance et approximation diophantienne sur les modules de Drinfeld, Ph.D. thesis, Université Paris 6, 2000.
- Vincent Bosser, Hauteurs normalisées des sous-variétés de produits de modules de Drinfeld, Compositio Math. 133 (2002), no. 3, 323–353 (French, with English summary). MR 1930981, DOI 10.1023/A:1020072121370
- Matthew H. Baker and Robert Rumely, Equidistribution of small points, rational dynamics, and potential theory, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 3, 625–688 (English, with English and French summaries). MR 2244226
- Hans Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103–144 (1965). MR 194595, DOI 10.1007/BF02591353
- Gregory S. Call and Susan W. Goldstine, Canonical heights on projective space, J. Number Theory 63 (1997), no. 2, 211–243. MR 1443758, DOI 10.1006/jnth.1997.2099
- Antoine Chambert-Loir, Mesures et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math. 595 (2006), 215–235 (French). MR 2244803, DOI 10.1515/CRELLE.2006.049
- L. Denis, Géométrie diophantienne sur les modules de Drinfel′d, The arithmetic of function fields (Columbus, OH, 1991) Ohio State Univ. Math. Res. Inst. Publ., vol. 2, de Gruyter, Berlin, 1992, pp. 285–302 (French, with French summary). MR 1196525
- Laurent Denis, Hauteurs canoniques et modules de Drinfel′d, Math. Ann. 294 (1992), no. 2, 213–223 (French). MR 1183402, DOI 10.1007/BF01934322
- Ricardo Mañé, On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), no. 1, 27–43. MR 736567, DOI 10.1007/BF02584743
- Charles Favre and Juan Rivera-Letelier, Équidistribution quantitative des points de petite hauteur sur la droite projective, Math. Ann. 335 (2006), no. 2, 311–361 (French, with English and French summaries). MR 2221116, DOI 10.1007/s00208-006-0751-x
- Charles Favre and Juan Rivera-Letelier, Théorème d’équidistribution de Brolin en dynamique $p$-adique, C. R. Math. Acad. Sci. Paris 339 (2004), no. 4, 271–276 (French, with English and French summaries). MR 2092012, DOI 10.1016/j.crma.2004.06.023
- D. Ghioca, The Mordell-Lang theorem for Drinfeld modules, Int. Math. Res. Not. (2005), no. 53, 3273–3307.
- Dragos Ghioca, Equidistribution for torsion points of a Drinfeld module, Math. Ann. 336 (2006), no. 4, 841–865. MR 2255176, DOI 10.1007/s00208-006-0017-7
- Dragos Ghioca, The local Lehmer inequality for Drinfeld modules, J. Number Theory 123 (2007), no. 2, 426–455. MR 2301224, DOI 10.1016/j.jnt.2006.07.006
- Dragos Ghioca, The Tate-Voloch conjecture for Drinfeld modules, J. Number Theory 125 (2007), no. 1, 85–94. MR 2333120, DOI 10.1016/j.jnt.2006.09.014
- David Goss, Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 35, Springer-Verlag, Berlin, 1996. MR 1423131, DOI 10.1007/978-3-642-61480-4
- D. Ghioca and T. J. Tucker, Siegel’s theorem for Drinfeld modules, Math. Ann. 339 (2007), no. 1, 37–60. MR 2317762, DOI 10.1007/s00208-007-0105-3
- L.-C. Hsia, On the reduction of a non-torsion point of a Drinfeld module, preprint.
- M. Ju. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3 (1983), no. 3, 351–385. MR 741393, DOI 10.1017/S0143385700002030
- K. Mahler, An application of Jensen’s formula to polynomials, Mathematika 7 (1960), 98–100. MR 124467, DOI 10.1112/S0025579300001637
- Ricardo Mañé, The Hausdorff dimension of invariant probabilities of rational maps, Dynamical systems, Valparaiso 1986, Lecture Notes in Math., vol. 1331, Springer, Berlin, 1988, pp. 86–117. MR 961095, DOI 10.1007/BFb0083068
- Charles F. Osgood, Effective bounds on the “Diophantine approximation” of algebraic functions over fields of arbitrary characteristic and applications to differential equations, Nederl. Akad. Wetensch. Proc. Ser. A 78=Indag. Math. 37 (1975), 105–119. MR 0387204
- Richard Pink, On the order of the reduction of a point on an abelian variety, Math. Ann. 330 (2004), no. 2, 275–291. MR 2089426, DOI 10.1007/s00208-004-0548-8
- Bjorn Poonen, Local height functions and the Mordell-Weil theorem for Drinfel′d modules, Compositio Math. 97 (1995), no. 3, 349–368. MR 1353279
- Jorge Pineiro, Lucien Szpiro, and Thomas J. Tucker, Mahler measure for dynamical systems on ${\Bbb P}^1$ and intersection theory on a singular arithmetic surface, Geometric methods in algebra and number theory, Progr. Math., vol. 235, Birkhäuser Boston, Boston, MA, 2005, pp. 219–250. MR 2166086, DOI 10.1007/0-8176-4417-2_{1}0
- K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1–20; corrigendum, 168. MR 72182, DOI 10.1112/S0025579300000644
- Thomas Scanlon, Diophantine geometry of the torsion of a Drinfeld module, J. Number Theory 97 (2002), no. 1, 10–25. MR 1939133, DOI 10.1006/jnth.2001.2751
- A. Schinzel, Primitive divisors of the expression $A^{n}-B^{n}$ in algebraic number fields, J. Reine Angew. Math. 268(269) (1974), 27–33. MR 344221, DOI 10.1515/crll.1974.268-269.27
- Jean-Pierre Serre, Lectures on the Mordell-Weil theorem, 3rd ed., Aspects of Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1997. Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt; With a foreword by Brown and Serre. MR 1757192, DOI 10.1007/978-3-663-10632-6
- C. L. Siegel, Über einige anwendungen diophantisher approximationen, Abh. Preuss. Akad. Wiss. Phys. Math. Kl. (1929), 41–69.
- Joseph H. Silverman, Integer points, Diophantine approximation, and iteration of rational maps, Duke Math. J. 71 (1993), no. 3, 793–829. MR 1240603, DOI 10.1215/S0012-7094-93-07129-3
- L. Szpiro and T. J. Tucker, Equidistribution and generalized Mahler measures, preprint, Available at arXiv:math.NT/0603438, 29 pages.
- L. Szpiro, E. Ullmo, and S. Zhang, Équirépartition des petits points, Invent. Math. 127 (1997), no. 2, 337–347 (French). MR 1427622, DOI 10.1007/s002220050123
- Yuichiro Taguchi, Semi-simplicity of the Galois representations attached to Drinfel′d modules over fields of “infinite characteristics”, J. Number Theory 44 (1993), no. 3, 292–314. MR 1233291, DOI 10.1006/jnth.1993.1055
- J. F. Voloch, Explicit $p$-descent for elliptic curves in characteristic $p$, Compositio Math. 74 (1990), no. 3, 247–258. MR 1055695
- J. F. Voloch, On the conjectures of Mordell and Lang in positive characteristics, Invent. Math. 104 (1991), no. 3, 643–646. MR 1106753, DOI 10.1007/BF01245094
- José Felipe Voloch, Diophantine geometry in characteristic $p$: a survey, Arithmetic geometry (Cortona, 1994) Sympos. Math., XXXVII, Cambridge Univ. Press, Cambridge, 1997, pp. 260–278. MR 1472501
Additional Information
- D. Ghioca
- Affiliation: Department of Mathematics, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4K1
- Address at time of publication: Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta, Canada T1K 3M4
- MR Author ID: 776484
- Email: dghioca@math.mcmaster.ca, dragos.ghioca@uleth.ca
- T. J. Tucker
- Affiliation: Department of Mathematics, Hylan Building, University of Rochester, Rochester, New York 14627
- MR Author ID: 310767
- ORCID: 0000-0002-8582-2198
- Email: ttucker@math.rochester.edu
- Received by editor(s): September 5, 2006
- Published electronically: April 16, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 4863-4887
- MSC (2000): Primary 11G50; Secondary 11J68, 37F10
- DOI: https://doi.org/10.1090/S0002-9947-08-04508-X
- MathSciNet review: 2403707