Height pairings for algebraic cycles on the product of a curve and a surface

For the product $X = C \times S$ of a curve and a surface over a number field, we prove Beilinson’s (1987) and Bloch’s (1984) conjecture about the existence of a height pairing between homologically trivial cycles. Then, for an embedding $f : C \to S$ , we construct an arithmetic diagonal cycle modified from the graph of $f$ and study its height. This work extends the previous work of Gross and Schoen (1995) to the product of three curves, and makes the Gan–Gross–Prasad conjecture unconditional for $O (1, 2) \times O (2, 2)$ and $U (1, 1) \times U (2, 1)$ .

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Keywords

arithmetic diagonal cycles, Beilinson–Bloch height pairing, Gan–Gross–Prasad conjecture

Mathematical Subject Classification

Primary: 14C25, 14G25, 14G40

Milestones

Received: 22 June 2023

Revised: 19 March 2024

Accepted: 30 April 2024

Published: 30 September 2024

Authors

Shou-Wu Zhang
Department of Mathematics Princeton University Princeton, NJ United States

Vol. 6, No. 3, 2024

Shou-Wu Zhang

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors