## Height pairings for algebraic cycles on Abelian varieties.(English)Zbl 1073.14527

Summary: Beilinson and Bloch have given conditional constructions of height pairings between algebraic cycles on smooth projective varieties over number fields. These pairings generalize the classical Néron-Tate height pairing between divisors and zero-cycles and give conjecturally a description of the behavior of motivic $$L$$-functions near the central point. We give an unconditional construction of height pairings for algebraic cycles on abelian varieties. This improves a previous result [Duke Math. J. 95, No.1, 161–212 (1998; Zbl 0955.14017)] where we have defined height pairings on abelian varieties which have totally degenerate reduction at all places of bad reduction. Our construction is based on the existence of projective regular models for abelian varieties over number fields and on the study of cycles on degenerate fibers in mixed characteristic initiated by Bloch, Gillet, and Soulé.

### MSC:

 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 11G10 Abelian varieties of dimension $$> 1$$ 11G50 Heights 14C25 Algebraic cycles 14K15 Arithmetic ground fields for abelian varieties

Zbl 0955.14017
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### References:

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