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Gross-Zagier formula for $$\text{GL}_2$$. II. (English) Zbl 1126.11026
Darmon, Henri (ed.) et al., Heegner points and Rankin $$L$$-series. Papers from the workshop on special values of Rankin $$L$$-series, Berkeley, CA, USA, December 2001. Cambridge: Cambridge University Press (ISBN 0-521-83659-X/hbk; 0-511-20831-6/e-book). Mathematical Sciences Research Institute Publications 49, 191-241 (2004).
The author reviews the proofs in his previous papers: Gross-Zagier formula for $$\text{GL}_2$$ [Asian J. Math. 5, No. 2, 183–290 (2001; Zbl 1111.11030)] and Heights of Heegner points on Shimura curves [Ann. Math. (2) 153, No. 1, 27–147 (2001; Zbl 1036.11029)]. He also deduces a new formula for the derivative at $$s=\frac{1}{2}$$ of the Rankin $$L$$-series associated to a Hilbert newform over a totally real algebraic number field, in terms of heights of CM-points on appropriate Shimura varieties. These results should have applications to the Birch and Swinnerton-Dyer conjecture, $$p$$-adic $$L$$-series and Iwasawa theory.
For the entire collection see [Zbl 1051.11004].

##### MSC:
 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11G18 Arithmetic aspects of modular and Shimura varieties 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations
##### Keywords:
Rankin $$L$$-series; Shimura curves; CM-points; special values
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