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

For the product X = C × 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 → 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) × O(2, 2) and U(1, 1) × U(2, 1).