Note on curves in a Jacobian

Results of Beauville imply that Zα is finite dimensional (cf. 2.5 below). In case A = J(C), the jacobian of a curve C, Ceresa has shown that the cycle C−C := C−(−1)∗C ∈ ZC is not algebraically equivalent to zero for generic C of genus g ≥ 3, which implies that for such a curve dimQ ZC ≥ 2. In this note we investigate the subspace ZWm of CHm(J(C))Q, with Wm the image of the m-th symmetric power of C in J(C) (so W1 = C). To simplify matters we will actually work modulo algebraic equivalence (rather than linear equivalence, note that translates of a cycle are algebraically equivalent).