an:02145117
Zbl 1126.11026
Zhang, Shou-Wu
Gross-Zagier formula for \(\text{GL}_2\). II
EN
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).
2004
a
11F67 11G18 11G40 11F41 11F66
Rankin \(L\)-series; Shimura curves; CM-points; special values
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].
Florin Nicolae (Berlin)
Zbl 1036.11029; Zbl 1111.11030