of ${\mathbb Q}$, we let ${{\rm res}}_v\colon H\to H_v$ be the canonical restriction map induced by the natural inclusion of fields ${\mathbb Q}\hookrightarrow {\mathbb Q}_v$.

3.  Let ${{\rm Div}^{0}_-}(C)$ denote the group of degree-zero divisors on $C$ with support disjoint from the principal divisor ${{\rm div}}(y)$.

In our case, the curve is given by $C:y^2=f(x)$, and the support of ${{\rm div}}(y)$ is exactly the points with coordinates $(\alpha,0)$, where $\alpha$ is a root of $f$, and the unique point at infinity.

Now for each $K$, there is a homomorphism $$F_K\colon {{\rm Div}^{0}_-}(C)(K)\to L^\times_K, \quad \sum_{P} n_PP\mapsto \prod_P (x(P) - \theta )^{n_P},$$ and this homomorphism induces a homomorphism $\delta_K\colon J(K) \to H_K$ with kernel $2J(K)$ by [@stoll Lemma 4.1]. By abusing notation, we also use $\delta_K$ to denote the induced map $J(K)/2J(K) \to H_K$.

All of these facts, together with some category theory, give us the following characterization of the 2-Selmer group of $J$ over ${\mathbb Q}$.

\[prop-selmer\_characterization\] The $2$-Selmer group of $J$ over ${\mathbb Q}$ can be identified as follows: $${{\rm Sel}}^{(2)}({\mathbb Q},J) = \{\xi\in H \mid {{\rm res}}_v(\xi) \in\delta_v(J({\mathbb Q}_v))\hbox{ for all } v\}.$$

In order to take advantage of this description of the Selmer group, we need some 