Twists of Shimura Curves

Stankewicz, James

Canadian journal of mathematics, Tome 66 (2014) no. 4, pp. 924-960

Résumé

Consider a Shimura curve $X_{0}^{D}\left( N \right)$ over the rational numbers. We determine criteria for the twist by an Atkin–Lenher involution to have points over a local field. As a corollary we give a new proof of the theorem of Jordan and Livné on ${{\mathbf{Q}}_{p}}$ points when $p|D$ and for the first time give criteria for ${{\mathbf{Q}}_{p}}$ points when $p|N$ . We also give congruence conditions for roots modulo $p$ of Hilbert class polynomials.

Détail

DOI : 10.4153/CJM-2013-023-8

Mots-clés : 11G18, 14G35, 11G15, 11G10, Shimura curves, complex multiplication, modular curves, elliptic curves

Bibliographie

[BLR90] [BLR90] Bosch, Siegfried, Lütkebohmert, Werner, and Raynaud, Michel, Néron models. Ergeb. Math. Grenzgeb. 21, Springer-Verlag, Berlin, 1990. Google Scholar

[CES03] [CES03] Conrad, Brian, Edixhoven, Bas, and Stein, William, J1(p) has connected fibers. Doc. Math. 8(2003), 331–408 (electronic). Google Scholar

[Cla03] [Cla03] Clark, Pete L., Rational Points on Atkin–Lehner Quotients of Shimura Curves. Ph.D. thesis, Harvard, 2003. Google Scholar

[Cox89] [Cox89] Cox, David A., Primes of the form x2 + ny2. AWiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989. Google Scholar

[Del69] [Del69] Deligne, Pierre, Variétés abéliennes ordinaires sur un corps fini. Invent. Math. 8(1969), 238–243. Google Scholar | DOI

[Eic56] [Eic56] Eichler, Martin, Modular correspondences and their representations. J. Indian Math. Soc. (N.S.) 20(1956), 163–206. Google Scholar

[GR91] [GR91] Gonzàlez Rovira, Josep, Equations of hyperelliptic modular curves. Ann. Inst. Fourier (Grenoble) 41(1991), 779–795. Google Scholar | DOI

[GR04] [GR04] González, Josep and Victor Rotger, , Equations of Shimura curves of genus two. Int. Math. Res. Not. 14(2004), 661–674. Google Scholar | DOI

[Hel07] [Hel07] Helm, David, On maps between modular Jacobians and Jacobians of Shimura curves. Israel J. Math. 160(2007), 61–117. Google Scholar | DOI

[JL85] [JL85] Jordan, Bruce W. and LivnÉ, Ron A., Local Diophantine properties of Shimura curves. Math. Ann. 270(1985), 235–248. Google Scholar | DOI

[Kan11] [Kan11] Kani, Ernst, Products of CM elliptic curves. Collect. Math. 62(2011), 297–339. Google Scholar | DOI

[KR08] [KR08] Kontogeorgis, Aristides and Rotger, Victor, On the non-existence of exceptional automorphisms on Shimura curves. Bull. London Math. Soc. 40(2008), 363–374. Google Scholar | DOI

[Kur79] Kurihara, Akira, On some examples of equations defining Shimura curves and the Mumford uniformization. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 25(1979), 277–300. Google Scholar

[Lan87] [Lan87] Lang, Serge, Elliptic functions. Second edition, Graduate Texts in Math. 112, Springer-Verlag, New York, 1987. Google Scholar

[Liu02] [Liu02] Liu, Qing, Algebraic geometry and arithmetic curves. Oxford Graduate Texts in Math. 6, Oxford University Press, Oxford, 2002. Google Scholar

[Lor11] [Lor11] Lorenzini, Dino J., Wild Models of Curves. http://www.math.uga.edu/_lorenz/Paper2.pdf Google Scholar

[Mes72] [Mes72] Messing, William, The crystals associated to Barsotti–Tate groups: with applications to abelian schemes. Lecture Notes in Math. 264, Springer-Verlag, Berlin, 1972. Google Scholar

[Mil86] [Mil86] Milne, James S., Jacobian varieties. In: Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986, 167–212. Google Scholar | DOI

[Mol12] [Mol12] Molina, Santiago, Ribet bimodules and the specialization of Heegner points. Israel J. Math. 189(2012), 1–38. Google Scholar | DOI

[Neu99] [Neu99] Neukirch, Jürgen, Algebraic number theory. Grundlehren Math.Wiss. 322, Springer-Verlag, Berlin, 1999. Google Scholar

[Ogg74] [Ogg74] Ogg, Andrew P., Hyperelliptic modular curves. Bull. Soc. Math. France 102(1974), 449–462. Google Scholar | DOI

[Ogg83] [Ogg83] Ogg, Andrew P., Real points on Shimura curves. In: Arithmetic and geometry, Vol. I, Progr. Math. 35(1983), 277–307. Google Scholar

[Ogg85] [Ogg85] Ogg, Andrew P., Mauvaise réduction des courbes de Shimura. In: Séminaire de théorie des nombres (Paris 1983–84), Progr. Math. 59(1985), 199–217. Google Scholar

[Ozm12] [Ozm12] Ozman, Ekin, Local Points on Quadratic Twists of X(N). Acta Arith. 152(2012), 323–348. Google Scholar | DOI

[Piz76] [Piz76] Pizer, Arnold, On the arithmetic of quaternion algebras. Acta Arith. 31(1976), 61–89. Google Scholar | DOI

[Rib89] [Rib89] Ribet, Kenneth A., Bimodules and abelian surfaces. In: Algebraic number theory, Adv. Stud. Pure Math. 17(1989), 359–407. Google Scholar

[RS11] [RS11] Ribet, Ken and Stein, William, Lectures on Modular Forms and Hecke Operators. http://wstein.org/books/ribet-stein/main.pdf, 2011. Google Scholar

[RSY05] [RSY05] Rotger, Victor, Skorobogatov, Alexei, and Yafaev, Andrei, Failure of the Hasse principle for Atkin–Lehner quotients of Shimura curves over ℚ. Mosc. Math. J. 5(2005), 463–476, 495. Google Scholar

[S+] [S+] Stein, W. A. et al., Sage Mathematics Software (Version 4.8). The Sage Development Team, 2012. http://www.sagemath.org Google Scholar

[Sad10] [Sad10] Sadek, Mohammad, On Quadratic Twists of Hyperelliptic Curves. , 2010. Google Scholar | arXiv

[Shi67] [Shi67] Shimura, Goro, Construction of class fields and zeta functions of algebraic curves. Ann. of Math. (2) 85(1967), 58–159. Google Scholar | DOI

[Shi71] [Shi71] Shimura, Goro, Introduction to the arithmetic theory of automorphic functions. Kanô Memorial Lectures, No. 1, Publications of the Mathematical Society of Japan 11, Iwanami Shoten, Publishers, Tokyo, 1971. Google Scholar

[Shi79] [Shi79] Shioda, Tetsuji, Supersingular K3 surfaces. In: Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Math. 732(1979), 564–591. Google Scholar

[Vie77] [Vie77] Viehweg, Eckart, Invarianten der degenerierten Fasern in lokalen Familien von Kurven. J. Reine Angew. Math. 293/294(1977), 284–308. Google Scholar

[Vig80] [Vig80] France, Marie Vignéras, , Arithmétique des algèbres de quaternions. Lecture Notes in Math. 800, Springer, Berlin, 1980. Google Scholar