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Heights of Heegner cycles and derivatives of L-series. (English) Zbl 0882.11029

The author extends Gross and Zagier’s limit formula to high weight modular forms by using arithmetic intersection theory. More precisely, he defines the (global) height pairing between CM-cycles in certain Kuga-Sato varieties, and shows an identity between the height pairings of Heegner cycles and coefficients of certain cusp forms of high weights.
He gives some applications to the generalized Birch and Swinnerton-Dyer conjecture of Beilinson and Bloch, and to the algebraicity conjecture of Gross and Zagier. He also shows that the index conjecture of Gillet-Soulé and Beilinson-Bloch, and the Generalized Riemann Hypothesis have a common consequence: the positivity of the derivatives of certain L-series.

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11F11 Holomorphic modular forms of integral weight
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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