additional facts about the 2-torsion of $J$ and the $\delta_K$ maps.

\[lem-deltak\_maps\] Let $K$ be a field extension of ${\mathbb Q}$.

1.  For a point $P\in C(K)$ not in the support of ${{\rm div}}(y)$, $\delta_K(P-\infty) = x(P) - \theta \bmod (L_K^\times)^2.$

2.  Let $f = f_1\cdots f_{m_K}$ be the factorization of $f$ over $K$ into monic irreducible factors. Then, to every factor $f_j$, we can associate an element $P_j\in J(K)[2]$ such that:

    1.  The points $\{P_j\}$ generate $J(K)[2]$ and satisfy $\sum_{j=1}^{m_K}P_j = 0.$

    2.  Let $h_j$ be the polynomial such that $f = f_jh_j$. Then $$\delta_K(P_j) = (-1)^{\deg(f_j)}f_j(\theta) + (-1)^{\deg(h_j)}h_j(\theta) \bmod (L_K^\times)^2.$$

3.  $\dim J(K)[2] = m_K -1.$

Let $I_v = \ker(N\colon I_v(L)/I_v(L)^2 \to I({\mathbb Q})/I({\mathbb Q})^2)$ and let ${{\rm val}}_v\colon H_v\to I_v$ be the map induced by the valuations on each component of $L_v$. Considering all primes at once, we get another map ${{\rm val}}\colon H \to I(L)/I(L)^2$. More specifically, the ${{\rm val}}$ map is the product of ${{\rm val}}_v({{\rm res}}_v)$ over all places $v$.

Next, the following lemma helps us compute the dimensions of various groups when $K$ is a local field or ${\mathbb R}$.

\[lem-local\_dims\] Let $K$ be a $v$-adic local field, and let $d_K=[K:{\mathbb Q}_2]$ if 