Faltings heights of CM cycles and derivatives of L-functions

2008/07/03 by Jan Hendrik Bruinier, Bruinier, Jan Hendrik, Tonghai Yang +1 · 3 citations
Mathematics · #11F67 #11G18 #14G40 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.0807.0502

openalex publication_date 2008/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Faltings height pairing of arithmetic Heegner divisors and CM cycles on Shimura varieties associated to orthogonal groups. We compute the Archimedian contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a Rankin L-function. We prove the conjecture in certain cases where the Shimura variety has dimension 0, 1, or 2. In particular, we obtain a new proof of the Gross-Zagier formula.

Citations

• Heegner points and derivatives ofL-series. II
• On two geometric theta lifts
• Modular Elliptic Curves and Fermat's Last Theorem

Cited by

• A regularized theta lift on the symmetric space of SL_N
• A new converse theorem for Borcherds products
• L-functions of elliptic curves in ring class extensions of real quadratic fields via regularized theta liftings

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• Heights of CM cycles and derivatives of L-series
• Faltings heights of abelian varieties with complex multiplication
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• Modularity of generating series of divisors on unitary Shimura varieties II: arithmetic applications