Abstract
For a given order R in an imaginary quadratic field K, we study the specialization of the set CM(R) of Heegner points on the Shimura curve X = X 0(D, N) at primes p | DN. As we show, if p does not divide the conductor of R, a point P ∈ CM(R) specializes to a singular point (resp. a irreducible component) of the special fiber \(\tilde X\) of X at p if p ramifies (resp. does not ramify) in K. Exploiting the moduli interpretation of X 0(D, N) and K. Ribet’s theory of bimodules, we give a construction of a correspondence Φ between CM(R) and a set of conjugacy classes of optimal embeddings of R into a suitable order in a definite quaternion algebra that allows the explicit computation of these specialization maps. This correspondence intertwines the natural actions of Pic(R) and of an Atkin-Lehner group on both sides. As a consequence of this and the work of P. Michel, we derive a result of equidistribution of Heegner points in \(\tilde X\). We also illustrate our results with an explicit example.
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M. Bertolini and H. Darmon, A rigid analytic Gross-Zagier formula and arithmetic applications, Annals of Mathematics 146 (1997), 111–147. With an appendix by Bas Edixhoven.
J.-F. Boutot and H. Carayol, Uniformisation p-adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfeld, Astérisque 196–197 (1992), 45–158. Courbes modulaires et courbes de Shimura (Orsay, 1987/1988).
P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, in Modular Functions of One Variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Mathematics, vol. 349, Springer, Berlin, 1973, pp. 143–136.
M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197–272.
B. Edixhoven, Appendix of: A rigid analytic Gross-Zagier formula and arithmetic applications, Annals of Mathematics 146 (1997), 111–147.
M. Eichler, Über die Idealklassenzahl hyperkomplexer Systeme, Mathematische Zeitschrift 43 (1938), 481–494.
J. González and V. Rotger, Equations of Shimura curves of genus two, International Mathematics Research Notices 14 (2004), 661–674.
J. González and V. Rotger, Non-elliptic Shimura curves of genus one, Journal of the Mathematical Society of Japan 58 (2006), 927–948.
X. Guitart and S. Molina, Parametrization of abelian k-surfaces with quaternionic multiplication, Comptes rendus — Mathematique 347 (2009), 1325–1330.
Y. Ihara, Congruence relations and Shimura curves, in Automorphic Forms, Representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, American Mathematical Society, Providence, RI, 1979, pp. 291–311.
S. Johansson, A description of quaternion algebras, Preprint. http://www.math.chalmers.se/~sj/forskning.html.
B. W. Jordan, On the Diophantine arithmetic of Shimura curves, PH.D. thesis, Harvard University, IL, 1981.
A. Kontogeorgis and V. Rotger, On the non-existence of exceptional automorphisms on shimura curves, The Bulletin of the London Mathematical Society 40 (2008), 363–374.
K.-Z. Li and F. Oort, Moduli of Supersingular Abelian Varieties, Lecture Notes in Mathematics, vol. 1680, Springer-Verlag, Berlin, 1998.
M. Longo, On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields, Université de Grenoble. Annales de l’Institut Fourier 56 (2006), 689–733.
P. Michel, The subconvexity problem for Rankin-Selberg L-functions and equidistribution of Heegner points, Annals of Mathematics 160 (2004), 185–236.
S. Molina, Equations of hyperelliptic Shimura curves, submitted.
S. Molina and V. Rotger, Work in progress.
Y. Morita, Reduction modulo \(\mathfrak{P}\) of Shimura curves, Hokkaido Mathematical Journal 10 (1981), 209–238.
A. P. Ogg, Real points on Shimura curves in Arithmetic and Geometry, vol. I, Progress in Mathematics, vol. 35, Birkhäuser Boston, Boston, MA, 1983, pp. 277–307.
A. Ogus, Supersingular K3 crystals, in Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. II, Astérisque, vol. 64, Soc. Math. France, Paris, 1979, pp. 8–36.
K. A. Ribet, Endomorphism algebras of abelian varieties attached to newforms of weight 2, in Seminar on Number Theory, Paris 1979–80, Progress in Mathematics, vol. 12, Birkhäuser Boston, Boston, MA, 1981, pp. 263–276.
K. A. Ribet, Bimodules and abelian surfaces in Algebraic Number Theory, Advanced Studies in Pure Mathematics, vol. 17, Academic Press, Boston, MA, 1989, pp. 359–407.
G. Shimura, Construction of class fields and zeta functions of algebraic curves, Annals of Mathematics 85 (1967), 58–159.
T. Shioda, Supersingular K3 surfaces, in Algebraic Geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Mathematics, vol. 732, Springer, Berlin, 1979, pp. 564–591.
J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, vol. 106, second edn., Springer, Dordrecht, 2009.
M.-F. Vignéras, Arithmétique des alg`ebres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980.
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The research of the author is supported financially by DGICYT Grant MTM2009-13060-C02-01.
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Molina, S. Ribet bimodules and the specialization of Heegner points. Isr. J. Math. 189, 1–38 (2012). https://doi.org/10.1007/s11856-011-0172-8
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DOI: https://doi.org/10.1007/s11856-011-0172-8