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    ARAI, KEISUKE 2018. RATIONAL POINTS ON SHIMURA CURVES AND THE MANIN OBSTRUCTION. Nagoya Mathematical Journal, Vol. 230, Issue. , p. 144.
    Gillibert, Florence 2017. Surface abéliennes à multiplication quaternionique munie d'une structure de niveau Γ0(N). International Journal of Number Theory, Vol. 13, Issue. 05, p. 1317.
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Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves

  • Victor Rotger (a1) and Carlos de Vera-Piquero (a1)
Abstract

The purpose of this note is to introduce a method for proving the non-existence of rational points on a coarse moduli space X of abelian varieties over a given number field K in cases where the moduli problem is not fine and points in X(K) may not be represented by an abelian variety (with additional structure) admitting a model over the field K. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired by the work of Ellenberg and Skinner on the modularity of ℚ-curves, is that one may still attach a Galois representation of Gal(/K) with values in the quotient group GL(Tℓ(A))/ Aut(A) to a point P = [A] ∈ X(K) represented by an abelian variety A/, provided Aut(A) lies in the centre of GL(Tℓ(A)). We exemplify our method in the cases where X is a Shimura curve over an imaginary quadratic field or an Atkin–Lehner quotient over ℚ.

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COPYRIGHT: © Canadian Mathematical Society 2014
References
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[BFGR06] Bruin, N., Flynn, V., Gonzàlez, J., and Rotger, V., On finiteness conjectures for endomorphism algebras of abelian surfaces. Math. Proc. Cambridge Philos. Soc. 141(2006), no. 3, 383–408 http://dx.doi.org/10.1017/S0305004106009613.
[Cla03] Clark, P. L., Rational points on Atkin–Lehner quotients of Shimura curves. Thesis (Ph.D.)–Harvard University, ProQuest LLC, Ann Arbor, MI, 2003.
[ES01] Ellenberg, J. S. and Skinner, C., On the modularity of Q-curves. Duke Math. J. 109(2001), no. 1, 97–122 http://dx.doi.org/10.1215/S0012-7094-01-10914-9.
[Gil10] Gillibert, F., Points rationnels sur les quotients d'Atkin–Lehner de courbes de Shimura de discriminant pq. arxiv:1012.3414v1, 2010.
[GR06]González, J. and Rotger, V., Non elliptic Shimura curves of genus one. J. Math. Soc. Japan 58(2006), no. 4, 927–948 http://dx.doi.org/10.2969/jmsj/1179759530.
[Jor81] Jordan, B.W., On the Diophantine arithmetic of Shimura curves. Thesis (Ph.D.)–Harvard University, Proquest LLC, Ann Arbor, MI, 1981.
[Jor86] Jordan, B.W., Points on Shimura curves rational over number fields. J. Reine Angew. Math. 371(1986), 92–114.
[JL85] Jordan, B.W. and Livné, R. A., Local Diophantine properties of Shimura curves. Math. Ann. 270(1985), no. 2, 235–248 http://dx.doi.org/10.1007/BF01456184.
[Me90] Mestre, J.-F., Construction de courbes de genre 2 à partir de leurs modules. In: Effective methods in algebraic geometry (Castiglioncello, 1990), Progr. Math., 94, Birkhäuser Boston, Boston, MA, 1991, pp. 313–334.
[Mil79] Milne, J. S., Points on Shimura varieties mod p. In: Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., 33, American Mathematical Society, Providence, RI, 1979, pp. 165–184.
[Mil86] Milne, J. S., Abelian varieties. In: Arithmetic geometry (Storrs, Conn, 1984), Springer, New York, 1986, pp. 103–150.
[Mil04] Milne, J. S., Introduction to Shimura varieties. http://www.jmilne.org/math/xnotes.
[Mor81] Morita, Y., Reduction modulo β of Shimura curves. HokkaidoMath. J. 10(1981), no. 2, 209–238.
[Ogg83] Ogg, A. P., Real points on Shimura curves. In: Arithmetic and geometry, Vol. 1, Progr. Math., 35, Birkäuser Boston, Boston, MA, 1983, pp. 277–307.
[Ogg85] Ogg, A. P., Mauvaise réduction des courbes de Shimura. Séminaire de théorie des nombres, Paris 1983–84, Progr. Math., 59, Birkäuser Boston, MA, 1985, pp. 199–217.
[Oht64] Ohta, M., On ladic representations of Galois groups obtained from certain two-dimensional abelian varieties. J. Fac. Sci. Univ. Tokyo IA Math. 21(1974), 299–308.
[PY07] Parent, P. and Yafaev, A., Proving the triviality of rational points on Atkin–Lehner quotients of Shimura curves. Math. Ann. 339(2007), no. 4, 915–935 http://dx.doi.org/10.1007/s00208-007-0136-9.
[Rot03] Rotger, V., Quaternions, polarizations and class numbers. J. Reine Angew. Math. 561(2003), 177–197.
[Rot04] Rotger, V., Modular Shimura varieties and forgetful maps. Trans. Amer. Math. Soc. 356(2004), no. 4, 1535–1550 http://dx.doi.org/10.1090/S0002-9947-03-03408-1.
[Rot08] Rotger, V., Which quaternion algebras act on a modular abelian variety? Math. Res. Lett. 15(2008), no. 2, 251–263.
[RSY05] Rotger, V., Skorobogatov, A., and Yafaev, A., Failure of the Hasse principle for Atkin–Lehner quotients of Shimura curves over Q. Moscow Math. J. 5(2005), no. 2, 463–476, 495.
[Ser72] Serre, J.-P., Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. Invent. Math. 15(1972), no. 4, 259–331 http://dx.doi.org/10.1007/BF01405086.
[ST68] Serre, J.-P. and Tate, J. , Good reduction of abelian varieties. Ann. of Math. 88(1968), 492–517 http://dx.doi.org/10.2307/1970722.
[Shi63] Shimura, G., On analytic families of polarized abelian varieties and automorphic functions. Ann.of Math. 78(1963), 149–192 http://dx.doi.org/10.2307/1970507.
[Shi67] Shimura, G., Construction of class fields and zeta functions of algebraic curves. Ann. of Math. 85(1967), 58–159 http://dx.doi.org/10.2307/1970526.
[Shi75] Shimura, G., On the real points of an arithmetic quotient of a bounded symmetric domain. Math. Ann. 215(1975), 135–164 http://dx.doi.org/10.1007/BF01432692.
[Sko01] Skorobogatov, A., Torsors and rational points. Cambridge Tracts in Mathematics, 144, Cambridge University Press, Cambridge, 2001.
[Sko05] Skorobogatov, A., Shimura coverings of Shimura curves and the Manin obstruction. Math. Res. Lett. 12(2005), no. 5–6, 779–788.
[SY04] Skorobogatov, A. and Yafaev, A., Descent on certain Shimura curves. Israel J. Math. 140(2004).319–332 http://dx.doi.org/10.1007/BF02786638.
[dVP] de Vera-Piquero, C., The Shimura covering of a Shimura curve: automorphisms and étale subcoverings. J. Number Theory 133(2013), no. 10, 3500–3516 http://dx.doi.org/10.1016/j.jnt.2013.04.018.
[Vig80] Vignéras, M. F., Arithmétique des algébres de quaternions. Lecture Notes in Mathematics, 800, Springer, Berlin, 1980.
[Wei56] Weil, A., The field of definition of a variety. Amer. J. Math. 78(1956), 509–524 http://dx.doi.org/10.2307/2372670.
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Canadian Journal of Mathematics
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