Degree zero zero-cycles on the square of a curve

Question

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:

1. Is anything known about this conjecture in higher genus?

2. Is this conjecture related to, or implied by, any "mainstream" conjectures in arithmetic geometry?

3. Is there an in-print-somewhere reference for this conjecture? Should it be attributed to Bloch, or to somebody else?

Answer 1

I think this conjecture is normally attributed to Bloch and Beilinson, and it is a special case of their general conjecture that Albanese equivalence coincides with rational equivalence (up to torsion) on smooth projective varieties over number fields. (For varieties over any field --- of char. zero say --- the Chow groups are expected to have a filtration, whose first steps are homological equivalence, then albanese equivalence. The higher steps in the filtration are conjecturally related to how the two basic conditions --- homological or Albanese triviality --- interact with specialization of the variety. Since a variety over a number field can't be specialized, the filtration should stop after Albanese equivalence, hence the conjecture.)

As far as I know there is no literature on this question to speak of, and it is wide open. (I would love to hear something to the contrary!) There is literature on the conjectural filtration on Chow groups --- look in the motives volumes, at some of the papers of Green and Girffiths on arithmetic Hodge structures, and at some of Morihiko Saito's papers. (In both cases, the papers --- at least the ones I remember --- are from around ten years ago. I think there might be '98 ICM talk, for example.)