# Degree zero zero-cycles on the square of a curve

A well-known mathematician once explained the following conjecture to me, as "an example of how little we know about cycles of codimension $\geq 2$." Let $C$ be a curve defined over a number field $k$, and let $x_1,x_2 \in C(\overline{k})$ be distinct points. Then the zero-cycle $\gamma=(x_1,x_1)+(x_2,x_2)-(x_1,x_2)-(x_2,x_1)\subset C\times C$ has degree zero, and the conjecture is that this cycle is torsion in the Chow group $Ch^2(C\times C)_0$ of degree-zero zero-cycles modulo rational equivalence. In other words, there should be an integer $n\geq 1$, curves $C_i \subset C\times C$ and rational functions $f_i$ on $C_i$, such that

$n\gamma=\sum_{C_i}\mathrm{div}(f_i)$.

The significance of $k$ being a number field is that $\gamma$ maps to zero in the relevant intermediate Jacobian, so apparently some conjectures of Bloch on height pairings suggest that it is torsion. (My apologies for the vagueness here; I heard this years ago!) When $C$ has genus one, the conjecture is easy to prove. My questions: