Trivial points on towers of curves
Journal de théorie des nombres de Bordeaux, Tome 25 (2013) no. 2, pp. 477-498.

Afin d’étudier le comportement des points dans une tour de courbes, nous introduisons et étudions les points triviaux sur les tours de courbes, et nous discutons de leur finitude sur les corps de nombres. Nous relions le problème de prouver que les seuls points rationnels sont les points triviaux à un certain niveau de la tour à la non-existence d’une borne de la gonalité des courbes de la tour, que nous démontrons sous certaines hypothèses.

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.

DOI : 10.5802/jtnb.845
Classification : 11G30, 11G20, 11B39, 11D45, 14G25
Xavier Xarles 1

1 Departament de Matemàtiques Universitat Autònoma de Barcelona 08193 Bellaterra, Barcelona, Spain
@article{JTNB_2013__25_2_477_0,
     author = {Xavier Xarles},
     title = {Trivial points on towers of curves},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {477--498},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {25},
     number = {2},
     year = {2013},
     doi = {10.5802/jtnb.845},
     mrnumber = {3228317},
     zbl = {1294.11109},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.845/}
}
TY  - JOUR
AU  - Xavier Xarles
TI  - Trivial points on towers of curves
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2013
SP  - 477
EP  - 498
VL  - 25
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.845/
DO  - 10.5802/jtnb.845
LA  - en
ID  - JTNB_2013__25_2_477_0
ER  - 
%0 Journal Article
%A Xavier Xarles
%T Trivial points on towers of curves
%J Journal de théorie des nombres de Bordeaux
%D 2013
%P 477-498
%V 25
%N 2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.845/
%R 10.5802/jtnb.845
%G en
%F JTNB_2013__25_2_477_0
Xavier Xarles. Trivial points on towers of curves. Journal de théorie des nombres de Bordeaux, Tome 25 (2013) no. 2, pp. 477-498. doi : 10.5802/jtnb.845. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.845/

[1] Abramovich. D., A linear lower bound on the gonality of modular curves, International Math. Res. Notices 20 (1996), 1005–1011. | MR | Zbl

[2] Baker, M., Specialization of Linear Systems from Curves to Graphs, Algebra and Number Theory 2, no. 6 (2008), 613–653. | MR | Zbl

[3] Baker, M., Norine, S., Riemann-Roch and Abel-Jacobi Theory on a Finite Graph, Advances in Mathematics, 215 (2007), 766–788. | MR | Zbl

[4] Bekka, B., de la Harpe, P., Valette, A., Kazhdan’s property (T), New Mathematical Monographs, 11, Cambridge University Press, 2008. | MR | Zbl

[5] Bourgain, J., Gamburd, A. Expansion and random walks in SL d (/p n ):I, J. Eur. Math. Soc. 10, (2008), 987–1011. | MR | Zbl

[6] Bourgain, J., Gamburd, A. Expansion and random walks in SL d (/p n ): II, J. Eur. Math. Soc. 11, 1057–1103 (2009) | MR | Zbl

[7] Brooks, R., On the angles between certain arithmetically defined subspaces of n , Annales Inst. Fourier 37 (1987), 175–185. | Numdam | MR | Zbl

[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 | Zbl

[9] Cadoret, A., Tamagawa, A., Uniform boundedness of p-primary torsion of abelian schemes, Invent. Math., 188 (2012), 83–125. | MR

[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 | Zbl

[12] Dinai, O., Poly-log diameter bounds for some families of finite groups, Proc. Amer. Math. Soc. 134 (2006), 3137–3142. | MR | Zbl

[13] Dinai, O. Diameters of Chevalley groups over local rings, arXiv:1201.4686. | MR

[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 | Zbl

[15] Faltings, G., Endlichkeitssätze für abelsche Variatäten über Zahlkörpern, Invent. math. 73 (1983), 349–366. | MR | Zbl

[16] Faltings, G., Diophantine approximation on abelian varieties, Annals of Math. 133 (1991), 549–576. | MR | Zbl

[17] Frey, G., Curves with infinitely many points of fixed degree, Israel J. Math. 85 (1994), no. 1-3, 79–83. | MR | Zbl

[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 | Zbl

[19] González-Jiménez, E., Xarles, X., On symmetric square values of quadratic polynomials, Acta Arithmetica 149, 145–159 (2011). | Zbl

[20] González-Jiménez, E., Xarles, X., Five squares in arithmetic progression over quadratic fields, to appear in Rev. Mat. Iberoamericana. | MR

[21] Hindri, M., Silverman, J.H., Diophantine Geometry, An introduction. Graduate Texts in Mathematics 201. Springer-Verlag, New York, 2000. | MR | Zbl

[22] Lang, S., Tate, J., Principal homogeneous spaces over abelian varieties, Amer. J. Math. 80, (1958), 659–684. | MR | Zbl

[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 | Zbl

[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 | Zbl

[25] Lubotzky, A., Discrete groups, expanding graphs and invariant measures, Progress in Math. 125, Birkaüser 1994. | MR | Zbl

[26] Manin, Y., A uniform bound for p-torsion in elliptic curves, Izv. Akad. Nauk. CCCP 33, (1969), 459–465. | Zbl

[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 | Zbl

[28] Mohar, B., Eigenvalues, diameter, and mean distance in graphs, Graphs Combin. 7 (1991) 53–64. | MR | Zbl

[29] Poonen, B. Gonality of modular curves in characteristic p, Math. Res. Lett. 14 (2007), no. 4, 691–701. | MR | Zbl

[30] Silverman, J.H., The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 106, Springer-Verlag, 1986. | MR | Zbl

[31] Silverman, J.H., The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics 241, Springer-Verlag, 2007. | MR | Zbl

[32] Xarles, X. Squares in arithmetic progression over number fields, J. Number Theory 132 (2012) 379–389. | MR

[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 | Zbl

Cité par Sources :