staple
With cedram.org

Search the site

Table of contents for this issue | Previous article | Next article
Xavier Xarles
Trivial points on towers of curves
Journal de théorie des nombres de Bordeaux, 25 no. 2 (2013), p. 477-498, doi: 10.5802/jtnb.845
Article PDF | Reviews MR 3228317 | Zbl 1294.11109
Class. Math.: 11G30, 11G20, 11B39, 11D45, 14G25

Résumé - Abstract

In order to study the behavior of the points in a tower of curves, we introduce and study trivial points on towers of curves, and we discuss their finiteness over number fields. We relate the problem of proving that the only rational points are the trivial ones at some level of the tower, to the unboundeness of the gonality of the curves in the tower, which we show under some hypothesis.

Bibliography

[1] Abramovich. D., A linear lower bound on the gonality of modular curves, International Math. Res. Notices 20 (1996), 1005–1011.  MR 1422373 |  Zbl 0878.14019
[2] Baker, M., Specialization of Linear Systems from Curves to Graphs, Algebra and Number Theory 2, no. 6 (2008), 613–653.  MR 2448666 |  Zbl 1162.14018
[3] Baker, M., Norine, S., Riemann-Roch and Abel-Jacobi Theory on a Finite Graph, Advances in Mathematics, 215 (2007), 766–788.  MR 2355607 |  Zbl 1124.05049
[4] Bekka, B., de la Harpe, P., Valette, A., Kazhdan’s property (T), New Mathematical Monographs, 11, Cambridge University Press, 2008.  MR 2415834 |  Zbl 1146.22009
[5] Bourgain, J., Gamburd, A. Expansion and random walks in $SL_d ({\mathbb{Z}}/p^n{\mathbb{Z}})$:I, J. Eur. Math. Soc. 10, (2008), 987–1011.  MR 2443926 |  Zbl 1193.20059
[6] Bourgain, J., Gamburd, A. Expansion and random walks in $SL_d ({\mathbb{Z}}/p^n{\mathbb{Z}})$: II, J. Eur. Math. Soc. 11, 1057–1103 (2009)  MR 2538500 |  Zbl 1193.20060
[7] Brooks, R., On the angles between certain arithmetically defined subspaces of ${\mathbb{C}}^n$, Annales Inst. Fourier 37 (1987), 175–185. Cedram |  MR 894565 |  Zbl 0611.15003
[8] Burger, M., Estimations de petites valeurs propres du laplacien d’un revêtement de variétés riemanniennes compactes, C.R. Acad. Sc. Paris 302 (1986), 191–194.  MR 832070 |  Zbl 0585.53035
[9] Cadoret, A., Tamagawa, A., Uniform boundedness of p-primary torsion of abelian schemes, Invent. Math., 188 (2012), 83–125.  MR 2897693 |  Zbl pre06037979
[10] Çiperiani, M., Stix, J., Weil-Châtelet divisible elements in Tate-Shafarevich groups, arXiv:1106.4255.
[11] Diaconis, P., Saloff-Coste, L., Comparison Techniques for Random Walk on Finite Groups, Ann. Probab. 21, no. 4 (1993), 2131–2156.  MR 1245303 |  Zbl 0790.60011
[12] Dinai, O., Poly-log diameter bounds for some families of finite groups, Proc. Amer. Math. Soc. 134 (2006), 3137–3142.  MR 2231895 |  Zbl 1121.05058
[13] Dinai, O. Diameters of Chevalley groups over local rings, arXiv:1201.4686.  MR 3000421
[14] Ellenberg, J., Hall, C., Kowalski, E., Expander graphs, gonality and variation of Galois representations, Duke Math. J. 161, no. 4 (2012), 1233–1275.  MR 2922374 |  Zbl 1262.14021
[15] Faltings, G., Endlichkeitssätze für abelsche Variatäten über Zahlkörpern, Invent. math. 73 (1983), 349–366.  MR 718935 |  Zbl 0588.14026
[16] Faltings, G., Diophantine approximation on abelian varieties, Annals of Math. 133 (1991), 549–576.  MR 1109353 |  Zbl 0734.14007
[17] Frey, G., Curves with infinitely many points of fixed degree, Israel J. Math. 85 (1994), no. 1-3, 79–83.  MR 1264340 |  Zbl 0808.14022
[18] Fried, M. D., Introduction to modular towers: generalizing dihedral group-modular curve connections, in Recent developments in the inverse Galois problem (Seattle, WA, 1993), 111-171, Contemp. Math., 186, Amer. Math. Soc., Providence, RI, 1995.  MR 1352270 |  Zbl 0957.11047
[19] González-Jiménez, E., Xarles, X., On symmetric square values of quadratic polynomials, Acta Arithmetica 149, 145–159 (2011).  Zbl 1243.11046
[20] González-Jiménez, E., Xarles, X., Five squares in arithmetic progression over quadratic fields, to appear in Rev. Mat. Iberoamericana.  MR 2590593
[21] Hindri, M., Silverman, J.H., Diophantine Geometry, An introduction. Graduate Texts in Mathematics 201. Springer-Verlag, New York, 2000.  MR 1745599 |  Zbl 0948.11023
[22] Lang, S., Tate, J., Principal homogeneous spaces over abelian varieties, Amer. J. Math. 80, (1958), 659–684.  MR 106226 |  Zbl 0097.36203
[23] Lazarsfeld, R., Lectures on Linear Series, With the assistance of Guillermo Fernández del Busto. IAS/Park City Math. Ser., 3, Complex algebraic geometry (Park City, UT, 1993), 161-219, Amer. Math. Soc., Providence, RI, 1997.  MR 1442523 |  Zbl 0906.14002
[24] Li, P. and Yau, S.T., A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces Invent. math. 69 (1982), 269–291.  MR 674407 |  Zbl 0503.53042
[25] Lubotzky, A., Discrete groups, expanding graphs and invariant measures, Progress in Math. 125, Birkaüser 1994.  MR 1308046 |  Zbl 0826.22012
[26] Manin, Y., A uniform bound for p-torsion in elliptic curves, Izv. Akad. Nauk. CCCP 33, (1969), 459–465.  Zbl 0191.19601
[27] Merel, L., Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), no. 1-3, 437–449.  MR 1369424 |  Zbl 0936.11037
[28] Mohar, B., Eigenvalues, diameter, and mean distance in graphs, Graphs Combin. 7 (1991) 53–64.  MR 1105467 |  Zbl 0771.05063
[29] Poonen, B. Gonality of modular curves in characteristic p, Math. Res. Lett. 14 (2007), no. 4, 691–701.  MR 2335995 |  Zbl 1138.14016
[30] Silverman, J.H., The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 106, Springer-Verlag, 1986.  MR 817210 |  Zbl 0585.14026
[31] Silverman, J.H., The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics 241, Springer-Verlag, 2007.  MR 2316407 |  Zbl 1130.37001
[32] Xarles, X. Squares in arithmetic progression over number fields, J. Number Theory 132 (2012) 379–389.  MR 2875345 |  Zbl pre06005605
[33] Zograf, P. Small eigenvalues of automorphic Laplacians in spaces of cusp forms, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 134 (1984), 157-168; translation in Journal of Math. Sciences 36, Number 1, 106–114.  MR 741858 |  Zbl 0536.10018