Some examples of torsion in the Griffiths group (Q1204234)

Let \(W\) be a smooth projective variety over a field, \(K\), with algebraic closure \(\overline K\). The Griffiths group of codimension \(r\) cycles on \(W_{\overline K}\) is the quotient of the group of nullhomologous cycles in codimension \(r\) by the subgroup of cycles which are algebraically equivalent to zero. This paper begins the investigation of \(Gr^ r(W_{\overline K})_{\text{tors}}\). Examples are produced with \(K=\mathbb{C}\), \(\mathbb{Q}\), and \(\mathbb{F}_ p\) and \(Gr^ r(W_{\overline K})_{\text{tors}} \neq 0\). When the base field has characteristic zero the varieties used in the examples are all quite special. When the base field is finite it seems to be easier to find torsion in \(Gr^ r(W_{\overline K})\). It is shown that there is no bound on the order of the group \((Gr^ 2(E^ 3_{\overline\mathbb{F}_ p})_{\text{tors}})^{\text{Gal} (\overline\mathbb{F}_ p/ \mathbb{F}_ p)}\) when \(p\) varies through all primes and \(E_{\mathbb{F}_ p}\) varies through the various reductions of a certain elliptic curve over \(\mathbb{Q}\).