A combinatorial Li–Yau inequality and rational points on curves

被引:0
|
作者
Gunther Cornelissen
Fumiharu Kato
Janne Kool
机构
[1] Universiteit Utrecht,Mathematisch Instituut
[2] Kumamoto University,Department of Mathematics
[3] Max-Planck-Institut für Mathematik,undefined
来源
Mathematische Annalen | 2015年 / 361卷
关键词
05C50; 11G09; 11G18; 11G30; 14G05; 14G22; 14H51;
D O I
暂无
中图分类号
学科分类号
摘要
We present a method to control gonality of nonarchimedean curves based on graph theory. Let k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document} denote a complete nonarchimedean valued field. We first prove a lower bound for the gonality of a curve over the algebraic closure of k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document} in terms of the minimal degree of a class of graph maps, namely: one should minimize over all so-called finite harmonic graph morphisms to trees, that originate from any refinement of the dual graph of the stable model of the curve. Next comes our main result: we prove a lower bound for the degree of such a graph morphism in terms of the first eigenvalue of the Laplacian and some “volume” of the original graph; this can be seen as a substitute for graphs of the Li–Yau inequality from differential geometry, although we also prove that the strict analogue of the original inequality fails for general graphs. Finally, we apply the results to give a lower bound for the gonality of arbitrary Drinfeld modular curves over finite fields and for general congruence subgroups Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma $$\end{document} of Γ(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma (1)$$\end{document} that is linear in the index [Γ(1):Γ]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[\varGamma (1):\varGamma ]$$\end{document}, with a constant that only depends on the residue field degree and the degree of the chosen “infinite” place. This is a function field analogue of a theorem of Abramovich for classical modular curves. We present applications to uniform boundedness of torsion of rank two Drinfeld modules that improve upon existing results, and to lower bounds on the modular degree of certain elliptic curves over function fields that solve a problem of Papikian.
引用
收藏
页码:211 / 258
页数:47
相关论文
共 50 条
  • [41] RATIONAL POINTS OF 3 ELLIPTICAL CURVES
    WOHLFAHR.K
    JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, 1974, 268 : 348 - 359
  • [42] On the distribution of integer points of rational curves
    Dimitrios Poulakis
    Evaggelos Voskos
    Periodica Mathematica Hungarica, 2003, 46 (1) : 89 - 101
  • [43] Trinomial curves with many rational points
    Beelen, P
    Pellikaan, R
    1998 INFORMATION THEORY WORKSHOP - KILLARNEY, IRELAND, 1998, : 38 - 39
  • [44] Rational points on Jacobians of hyperelliptic curves
    Mueller, Jan Steffen
    ADVANCES ON SUPERELLIPTIC CURVES AND THEIR APPLICATIONS, 2015, 41 : 225 - 259
  • [45] Counting rational points on cubic curves
    Heath-Brown, Roger
    Testa, Damiano
    SCIENCE CHINA-MATHEMATICS, 2010, 53 (09) : 2259 - 2268
  • [46] Mirror congruence for rational points on Calabi-Yau varieties
    Fu, Lei
    Wan, Daqing
    ASIAN JOURNAL OF MATHEMATICS, 2006, 10 (01) : 1 - 10
  • [47] A logarithmic Sobolev form of the Li-Yau parabolic inequality
    Bakry, Dominique
    Ledoux, Michel
    REVISTA MATEMATICA IBEROAMERICANA, 2006, 22 (02) : 683 - 702
  • [48] Rational curves with many rational points over a finite field
    Fukasawa, Satoru
    Homma, Masaaki
    Kim, Seon Jeong
    ARITHMETIC, GEOMETRY, CRYPTOGRAPHY AND CODING THEORY, 2012, 574 : 37 - +
  • [49] K3 surfaces, rational curves, and rational points
    Baragar, Arthur
    McKinnon, David
    JOURNAL OF NUMBER THEORY, 2010, 130 (07) : 1470 - 1479
  • [50] ON UNIFORM BOUNDS FOR RATIONAL POINTS ON RATIONAL CURVES AND THIN SETS
    Rault, Patrick X.
    JP JOURNAL OF ALGEBRA NUMBER THEORY AND APPLICATIONS, 2011, 23 (02): : 171 - 185