The Griffiths-group rank prediction for bielliptic Picard curves

Let $C$ be a bielliptic Picard curve over a number field $k$, with Jacobian $J$ admitting an isogeny decomposition $J\sim P\times E$. Let $V$ be the transcendental part of $\operatorname H^2(P)$, and define

$$\mathfrak h_{\mathrm{Gr}}^3(J):=\mathfrak h_{\mathrm{tr}}^2(P)\otimes\mathfrak h^1(E).$$

Write ${\rm Gr}^2(J)$ for the Griffiths group of codimension-$2$ cycles on $J$.

Griffiths-group rank prediction. One expects

$$\operatorname{rk}{\rm Gr}^2(J)\leq\operatorname*{ord}_{s=2}L(\mathfrak h_{\mathrm{Gr}}^3(J),s),$$

with equality if $V\otimes\operatorname H^1(E)$ is an irreducible $\operatorname{Gal}_k$-representation.

This prediction comes from compatibility of the Beilinson–Bloch conjecture with coniveau filtrations and gives an arithmetic upper bound for the Griffiths-group rank, with equality under the stated irreducibility condition.