Thank you! Just checking, in your paper, you work with homogenous cochains, so a 2-cocycle would have 3 arguments instead of 2; does this change the formula?

@KasperAndersen: I don't think this is the problem, I can evaluate z([H.1,H.1]) for example. If I put in f(<G.1,G.1>) I get the same error.

@MikhailBorovoi Thanks for your comments, they are very helpful. Do you have a reference for this formula? One subtle feature is that Magma views $G$-modules as acting on the right (as opposed to the traditional action on the left), so this might affect the formula.

Is $t=s$ in the equation of $C$? Also what is a generic elliptic curve?

One thing to note is that if $X$ is normal and (geometrically) irreducible, then the codimension of $Y$ in $X$ must be $1$. Indeed, if the codimension of $Y$ is at least $2$, then the restriction map $k = \mathcal{O}(X)\rightarrow \mathcal{O}(U)$ is an isomorphism, so $U$ cannot be affine.

This would require knowing that the `generic fiber' homomorphism $\text{CH}^p(X)\rightarrow \text{CH}^p(X_K)$ is surjective when restricted to the subgroups of algebraically trivial cycle classes. I don't see an obvious reason why that's true, nor what the definition of such a subgroup of $\text{CH}^p(X)$ would actually be. (Fulton only considers algebraic triviality for varieties over a field.)

The note people.math.harvard.edu/~mmwood/Splitting.pdf might be helpful in adapting the proof in the Galois case to the non-Galois case.

I'm a bit confused about the sentence ' for any rational point s in S(Q), the pullback of AJ(Z) over s should match the usual AJ map in the absolute setting'. How it the pullback of an element of $H^1(\Gamma, R^{2d-1}\pi_*(\mathbb{Q}_l)^H)$ naturally an element of $H^1(G_{\mathbb{Q}}, H^{2d-1}(\bar{X},\mathbb{Q}_l))$? Also what do you mean by $Z\rightarrow S'$ having connected fibres? Each closed subscheme in the support of $Z$ has connected fibres?

No. Thanks for catching the typo!

Ah that's a great example. Do you have a reference for the statement with $\mathbb{Q}$-coefficients?

@JasonStarr if you have any thoughts on my followup question I would be very grateful if you would like to share them: mathoverflow.net/questions/462776/…