Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-28T10:22:48.025Z Has data issue: false hasContentIssue false
  • English
  • Français

Darmon’s Points and Quaternionic Shimura Varieties

Published online by Cambridge University Press:  20 November 2018

Jérôme Gärtner
Jérôme Gärtner*
Affiliation:
Insitut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 4 place Jussieu 75005, Paris, France email: jgartner@math.jussieu.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we generalize a conjecture due to Darmon and Logan in an adelic setting. We study the relation between our construction and Kudla's works on cycles on orthogonal Shimura varieties. This relation allows us to conjecture a Gross-Kohnen-Zagier theorem for Darmon's points.

Keywords

11G0514G3511F6711G40elliptic curvesStark-Heegner pointsquaternionic Shimura varieties
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 6 , 01 December 2012 , pp. 1248 - 1288
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Aflalo, E. and Nekovář, J., Non-triviality of CM points in ring class field towers. Israel J. Math. 175(2010), 225–284. http://dx.doi.org/10.1007/s11856-010-0011-3 Google Scholar
[2] Brylinski, J.-L. and Labesse, J.-P., Cohomologie d’intersection et fonctions L de certaines variétés de Shimura. Ann. Sci. École Norm. Sup. (4) 17(1984), no. 3, 361–412.Google Scholar
[3] Bruinier, J. H., Regularized theta lifts for orthogonal groups over totally real field. http://mathematik.tu-darmstadt.de/_bruinier/. Google Scholar
[4] Cornut, C. and Jetchev, D., Liftings of reduction maps for quaternion algebras. http://people.math.jussieu.fr/_cornut. Google Scholar
[5] Cornut, C. and Vatsal, V., Nontriviality of Rankin-Selberg L-functions and CM points. In: L-functions and Galois representations, London Math. Soc. Lecture Note Ser., 320, Cambridge University Press, Cambridge, 2007, pp. 121–186.Google Scholar
[6] Cornut, C. and Vatsal, V., CM points and quaternion algebras. Doc. Math. 10(2005), 263–309.Google Scholar
[7] Darmon, H., Rational points on modular elliptic curves. CBMS Regional Conference Series in Mathematics, 101, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004.Google Scholar
[8] Darmon, H. and Logan, A., Periods of Hilbert modular forms and rational points on elliptic curves. Int. Math. Res. Not. 40(2003), 2153–2180.Google Scholar
[9] Darmon, H. and Tornarıa, G., Stark-Heegner points and the Shimura correspondence. Compos.Math. 144(2008), no. 5, 1155–1175. http://dx.doi.org/10.1112/S0010437X08003552. Google Scholar
[10] Freitag, E., Hilbert modular forms. Springer-Verlag, Berlin, 1990.Google Scholar
[11] Gärtner, J., Points de Darmon et variétés de Shimura, Thèse de l’université Paris 6, 2011. http://tel.archives-ouvertes.fr/tel-00555470/fr/ Google Scholar
[12] Gross, B. H. and Zagier, D. B., Heegner points and derivatives of L-series. Invent. Math. 84(1986), no. 2, 225–320. http://dx.doi.org/10.1007/BF01388809 Google Scholar
[13] Jacquet, H., Automorphic forms on GL(2). Part II. Lecture Notes in Mathematics, 278, Springer-Verlag, Berlin, 1972.Google Scholar
[14] Jacquet, H. and Langlands, R. P., Automorphic forms on GL(2). Lecture Notes in Mathematics, 114, Springer-Verlag, Berlin-New York, 1970.Google Scholar
[15] Kudla, S. S., Algebraic cycles on Shimura varieties of orthogonal type, Duke Math. J. 86(1997), no. 1, 39–78. http://dx.doi.org/10.1215/S0012-7094-97-08602-6 Google Scholar
[16] Kudla, S. S., Special cycles and derivatives of Eisenstein series. In: Heegner points and Rankin L-series, Math. Sci. Res. Inst. Publ., 49, Cambridge University Press, Cambridge, 2004, pp. 243–270.Google Scholar
[17] Kudla, S. S. and Millson, J. J., Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables. Inst. Hautes Études Sci. Publ. Math. 71(1990), 121–172.Google Scholar
[18] Langlands, R. P., On the zeta functions of some simple Shimura varieties. Canad. J. Math. 31(1979), no. 6, 1121–1216. http://dx.doi.org/10.4153/CJM-1979-102-1 Google Scholar
[19] Matsushima, Y. and Shimura, G., On the cohomology groups attached to certain vector valued differential forms on the product of the upper half planes. Ann. of Math. (2), 78(1963), 417–449. http://dx.doi.org/10.2307/1970534 Google Scholar
[20] Milne, J. S., Canonical models of (mixed) Shimura varieties and automorphic vector bundles. In: Automorphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), Perspect. Math., 10, Academic Press, Boston, MA, 1990, pp. 283–414.Google Scholar
[21] Milne, J. S., Introduction to Shimura varieties. In: Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., 4, American Mathematical Society, Providence, RI, 2005, pp. 265–378.Google Scholar
[22] Nekovář, J., The Euler system method for CM points on Shimura curves, In: L-functions and Galois representations, London Math. Soc. Lecture Note Ser., 320, Cambridge University Press, Cambridge, 2007, pp. 471–547.Google Scholar
[23] Nekovář, J., Selmer complexes. Astérisque 310(2006).Google Scholar
[24] Prasad, D., Some applications of seesaw duality to branching laws. Math. Ann. 304(1996), no. 1, 1–20. http://dx.doi.org/10.1007/BF01446282 Google Scholar
[25] Reimann, H., The semi-simple zeta function of quaternionic Shimura varieties. Lecture Notes in Mathematics, 1657, Springer-Verlag, Berlin, 1997.Google Scholar
[26] Reimann, H. and Zink, T., The good reduction of Shimura varieties associated to quaternion algebras over a totally real number field. To appear, University of Toronto Press.Google Scholar
[27] Saito, H., On Tunnell's formula for characters of GL(2). Compositio Math. 85(1993), no. 1, 99–108.Google Scholar
[28] Tunnell, J. B., Local C-factors and characters of GL(2). Amer. J. Math. 105(1983), no. 6, 1277–1307. http://dx.doi.org/10.2307/2374441 Google Scholar
[29] Vignéras, M.-F., Arithmétique des algèbres de quaternions. Lecture Notes in Mathematics, 800, Springer, Berlin, 1980.Google Scholar
[30] Yoshida, H., On the zeta functions of Shimura varieties and periods of Hilbert modular forms, Duke Math. J. 75(1994), no. 1, 121–191. http://dx.doi.org/10.1215/S0012-7094-94-07505-4 Google Scholar
[31] Yuan, X., Zhang, S.-W., and Zhang, W., The Gross-Kohnen-Zagier theorem over totally real fields. Compos. Math. 145(2009), no. 5, 1147–1162. http://dx.doi.org/10.1112/S0010437X08003734 Google Scholar
[32] Zagier, D., Modular points, modular curves, modular surfaces and modular forms. In: Workshop Bonn 1984 (Bonn, 1984), Lecture Notes in Math., 1111, Springer, Berlin, 1985, pp. 225–248.Google Scholar
[33] Zhang, S.-W., Gross-Zagier formula for GL2. Asian J. Math. 5(2001), no. 2, 183–290.Google Scholar