For the product
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
,
we construct an arithmetic diagonal cycle modified from the graph of
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
and
.
