the ${\mathbb F}_2$-dimension of the 2-Selmer group (see Lemma 4.10 and the discussion under Step 4), as follows.

\[prop-Sel2\_UB\]\[stollbnd\] With notation as above, $$\dim_{{\mathbb F}_2}{{\rm Sel}}^{(2)}({\mathbb Q},J)\leq (m_\infty-1) + \dim_{{\mathbb F}_2}({{\rm Cl}}(L)[2]) + \dim_{{\mathbb F}_2}\ker\big(G\to{{\rm Cl}}(L)/2{{\rm Cl}}(L)\big).$$

In the next section, we modify the proof of the bound in Proposition \[prop-Sel2\_UB\] to allow for extra conditions at infinity, before we specialize to totally real, and cyclic extensions.

About the proof of Proposition \[prop-Sel2\_UB\] {#sec-betterbounds}
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The following commutative diagrams helps understand where the Selmer group fits: $$\xymatrix{
J({\mathbb Q})/2J({\mathbb Q})\ar@{^{(}->}[rrr]^{\delta} \ar[d] & & & H\ar[rrr]^{{{\rm val}}}\ar[d]^{\prod_v{\rm res}_v}& & &I(L)/I(L)^2 \ar[d] \\
\prod_v J({\mathbb Q}_v)/2J({\mathbb Q}_v)\ar@{^{(}->}[rrr]^{\prod_v \delta_v} & & &\prod_v H_v\ar[rrr]^{\prod_v {{\rm val}}_v} & & &\prod_v I_v(L)/I_v(L)^2 
}$$ The $2$-Selmer group of $J$ over ${\mathbb Q}$ is then given, as in Prop. \[prop-selmer\_recharacterization\], by $${{\rm Sel}}^{(2)}({\mathbb Q},J) = \{ \xi\in H : {{\rm val}}(\xi)\in I_S(L)/I_S(L)^2,\  {{\rm res}}_v(\xi)\in \delta_v(J({\mathbb Q}_v))\hbox{ for all } v\in S\}.$$ The Selmer group is thus contained in $H\subseteq L^\times/(L^\times)^2$, and more precisely, $${{\rm Sel}}^{(2)}({\mathbb Q},J)\subseteq \{ \xi \in H : {{\rm val}}(\xi) \in G,\ \text{res}_\infty(\xi) \in J_\infty \}$$ where $J_\infty=\delta_\infty(J({\mathbb R}))$, the group $ G$ is the product $ \prod_{v\in S\setminus\{ \infty \}} G_v \subseteq I(L)/I(L)^2,$ and 