Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T05:22:35.607Z Has data issue: false hasContentIssue false
  • English
  • Français

Twisted Gross–Zagier Theorems

Published online by Cambridge University Press:  20 November 2018

Benjamin Howard
Benjamin Howard*
Affiliation:
Department of Mathematics, Boston College, Chestnut Hill, MA 02467, USA, e-mail: howardbe@bc.edu
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.

The theorems of Gross–Zagier and Zhang relate the Néron–Tate heights of complex multiplication points on the modular curve ${{X}_{0}}\,(N)$ (and on Shimura curve analogues) with the central derivatives of automorphic $L$-function. We extend these results to include certain CM points on modular curves of the form $X({{\Gamma }_{0}}(M)\bigcap {{\Gamma }_{1}}(S))$ (and on Shimura curve analogues). These results are motivated by applications to Hida theory that can be found in the companion article “Central derivatives of $L$ -functions in Hida families”, Math. Ann. 399(2007), 803–818.

Keywords

11G1814G35
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 4 , 01 August 2009 , pp. 828 - 887
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Bertolini, M. and Darmon, H., The p-adic L-functions of modular elliptic curves. In: Mathematics Unlimited—2001 and Beyond, Springer, Berlin, 2001 pp. 109–170.Google Scholar
[2] Bertolini, M. and Darmon, H., Iwasawa's main conjecture for elliptic curves over anticyclotomic Zp-extensions. Ann. of Math. 162(2005), no. 1, 1–64.Google Scholar
[3] Carayol, H., Sur la mauvaise réduction des courbes de Shimura. Compositio Math. 59(1986), no. 2, 151–230.Google Scholar
[4] Cornut, C. and Vatsal, V., C M points and quaternion algebras. Doc. Math. 10(2005), 263–309 (electronic).Google Scholar
[5] Cornut, C. and Vatsal, V., Nontriviality of Rankin-Selberg L-functions and C M 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] Gelbart, S., Lectures on the Arthur-Selberg Trace Formula. University Lecture Series 9. American Mathematical Society, Providence, RI, 1996.Google Scholar
[7] Gelbart, S. and Jacquet, H., Forms of GL(2) from the analytic point of view. In: Automorphic Forms, Representations and L-Functions, Proc. Sympos. Pure Math. 33, American Mathematical Society, Providence, RI, 1979, pp. 213–251.Google Scholar
[8] Gillet, H. and Soulé, C., Arithmetic intersection theory. Inst. Hautes Études Sci. Publ. Math. 72(1990), 93–174.Google Scholar
[9] Gross, B., Heights and the special values of L-series. In: Number Theory, C MS Conf. Proc. 7, American Mathematical Society, Providence, RI, 1987, pp. 115–187.Google Scholar
[10] Gross, B., Local orders, root numbers, and modular curves. Amer. J. Math. 110(1988), no. 6, 1153–1182.Google Scholar
[11] Gross, B., Heegner points and representation theory. In: Heegner points and Rankin L-series, Math. Sci. Res. Inst. Publ. 49, Cambridge University Press, Cambridge, 2004, pp. 37–65.Google Scholar
[12] Gross, B. and Prasad, D., Test vectors for linear forms. Math. Ann. 291(1991), no. 2, 343–355.Google Scholar
[13] Gross, B. and Zagier, D., Heegner points and derivatives of L-series. Invent. Math. 84(1986), NO. 2, 225–320.Google Scholar
[14] Howard, B., Central derivatives of L-functions in Hida families. Math. Ann. 399(2007), no. 4, 803–818.Google Scholar
[15] Howard, B., Variation of Heegner points in Hida families. Invent. Math. 167(2007), no. 1, 91–129.Google Scholar
[16] Jacquet, H. and Langlands, R. P., Automorphic forms on GL(2). Lecture Notes in Mathematics 114, Springer-Verlag, Berlin, 1970.Google Scholar
[17] Jordan, B. and Livné, R., Integral Hodge theory and congruences between modular forms. Duke Math. J. 80(1995), no. 2, 419–484.Google Scholar
[18] Kudla, S., Tate's thesis. In: An Introduction to the Langlands Program, Birkhäuser Boston, Boston, MA, 2003, pp. 109–131.Google Scholar
[19] Lang, S., Introduction to Arakelov Theory. Springer-Verlag, New York, 1988.Google Scholar
[20] Lubotzky, A., Discrete Groups, Expanding Graphs and Invariant Measures. Progress in Mathematics 125, Birkhäuser Verlag, Basel, 1994.Google Scholar
[21] Mann, W. R., Appendix to B. Conrad, Gross-Zagier revisited. In: Heegner points and Rankin L-series, Math. Sci. Res. Inst. Publ. 49, Cambridge University Press, Cambridge, 2004, pp. 67–163.Google Scholar
[22] Mazur, B. and Wiles, A., Class fields of abelian extensions of Q. Invent. Math. 76(1984), no. 2, 179–330.Google Scholar
[23] Milne, J., Canonical models of (mixed) Shimura varieties and automorphic vector bundles. In: Automorphic Forms, Shimura Varieties, and L-Functions, Perspect. Math. 10, Academic Press, Boston, MA, 1990, pp. 283–414.Google Scholar
[24] Milne, J., 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
[25] Miyake, T., Modular Forms. Translated from the Japanese by Yoshitaka Maeda. Springer-Verlag, Berlin, 1989.Google Scholar
[26] Nekovář, J., Selmer Complexes. Astérisque, No. 310, 2006.Google Scholar
[27] Nekovář, J., The Euler system method for C M 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
[28] Shimura, G., Introduction to the Arithmetic Theory of Automorphic Functions. Publications of the Mathematical Society of Japan, No. 11, Iwanami Shoten, Tokyo, 1971.Google Scholar
[29] Soulé, C., Lectures on Arakelov geometry. Cambridge Studies in Advanced Mathematics 33, Cambridge University Press, Cambridge, 1992.Google Scholar
[30] Vatsal, V., Special values of anticyclotomic L-functions. Duke Math. J. 116(2003), no. 2. 219–261.Google Scholar
[31] Vatsal, V., Special value formulae for Rankin L-functions. In: Heegner Points and Rankin L-Series, Math. Sci. Res. Inst. Publ. 49, Cambridge University Press, Cambridge, 2004, pp. 165–190.Google Scholar
[32] Waldspurger, J.-L., Quelques propriétés arithmétiques de certaines formes automorphes sur GL(2). Compositio Math. 54(1985), no. 2, 121–171.Google Scholar
[33] Weil, A., Basic Number Theory. Reprint of the second (1973) edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.Google Scholar
[34] Zhang, S.-W., Gross-Zagier formula for GL2. Asian J. Math. 5(2001), no. 2, 183–290.Google Scholar
[35] Zhang, S.-W., Heights of Heegner points on Shimura curves. Ann. of Math. 153(2001), no. 1, 27–147.Google Scholar
[36] Zhang, S.-W., Gross-Zagier formula for GL(2). II. In: Heegner Points and Rankin L-Series, Math. Sci. Res. Inst. Publ. 49, Cambridge University Press, Cambridge, 2004, pp. 191–214.Google Scholar