Setup: Let $k$ be a field of characteristic $0$, let $f(x) \in k[x]$ be a monic separable polynomial of degree $n \geq 4$, and let $\theta$ denote the image of $x$ under the map $k[x] \to K_f := k[x]/(f(x))$. Consider the hyperelliptic curve with affine equation $y^2 = f(x)$. The Cassels "$x - \theta$" map takes a point $(x,y) \in C(k)$ satisfying $y \neq 0$ to $[x - \theta] \in (K_f^{\times}/K_f^{\times 2})_{\mathrm{N} \equiv 1}$, where the subscript "$\mathrm{N}\equiv 1$" means "consider only those elements whose norm in $k$ is a square"; the map has a more complicated definition on Weierstrass points of $C$. This map can be thought of as a restriction to $C(k)$ of the composite map $$J(k) \to J(k)/2J(k) \to H^1(\operatorname{Gal}(\overline{k}/k), J[2]) \simeq (K_f^{\times}/K_f^{\times 2})_{\mathrm{N} \equiv 1}$$ obtained by taking Galois cohomology of the exact sequence $0 \to J[2](\overline{k}) \to J(\overline{k}) \to J(\overline{k}) \to 0$, where $J$ denotes the Jacobian of $C$, and $\overline{k}$ denotes the separable closure of $k$.
(Admittedly vague) Question: Is there a way to realize the Cassels map as an algebraic map of some sort?
What I know: A construction of Bhargava in https://arxiv.org/pdf/1308.0395.pdf (sections 2 and 3) can be summarized as follows. For even $n$, there is a representation $V$ of the algebraic group $\operatorname{SL}_n^{\pm}$ such that the following properties hold:
- The orbits of $\operatorname{SL}_n^{\pm}(k)$ on $V(k)$ are in bijection with $(K_f^{\times}/K_f^{\times 2})_{\mathrm{N} \equiv 1}$;
- There is a rational map of $k$-varieties $\phi \colon C -\to V$, defined away from the points at infinity, such that the $\operatorname{SL}_n^{\pm}(k)$-orbit of $\phi(x,y)$ is equal to the image of $(x,y)$ under the Cassels "$x - \theta$" map for any $(x,y) \in C(k)$ with $y \neq 0$.
I would like to show that the $\operatorname{SL}_n^{\pm}(k)$-orbit of $\phi(x,y)$ is equal to the image of $(x,y)$ under the Cassels "$x - \theta$" map for any $(x,y) \in C(k)$, not just for non-Weierstrass points (this fact is required to prove Theorem 3 in Bhargava's paper, although no proof is given). To do this, it would be helpful to know if the Cassels map had some algebraic realization, so that I could say that it must agree with $\phi$ because it agrees with $\phi$ on a dense set.
