in [@stoll]). The examples come from hyperelliptic curves $y^2=f(x)$ such that $f(x)$ defines the maximal real subfield of a cyclotomic extension of ${\mathbb Q}$, and the degree of $f(x)$ is $p$, a Sophie Germain prime. We obtain the following result.

\[thm-sophie-intro\] Let $q\geq 7$ be a prime such that $p=(q-1)/2$ is also prime, and let $L={\mathbb Q}(\zeta_q)^+$ be the maximal totally real subfield of ${\mathbb Q}(\zeta_q)$. Let $f(x)$ be the minimal polynomial of $\zeta_q+\zeta_q^{-1}$ or $-(\zeta_q+\zeta_q^{-1})$, let $C/{\mathbb Q}$ be the hyperelliptic curve $y^2=f(x)$, of genus $g=(p-1)/2$, and let $J/{\mathbb Q}$ be its jacobian. Then, there are constants $\rho_\infty$ and $j_\infty$, that depend on $q$, such that $$\begin{aligned}
{{\rm rank\,}}_{\mathbb Z}(J({\mathbb Q}))\leq \dim_{{\mathbb F}_2}{{\rm Sel}}^{(2)}({\mathbb Q},J)  \leq \rho_{\infty} + j_\infty + \dim_{{\mathbb F}_2} ({{\rm Cl}}^+(L)[2]),\end{aligned}$$ where $\rho_{\infty}+j_\infty\leq p-1$. Further, if one of the following conditions is satisfied,

1.  the Davis–Taussky conjecture holds (Conjecture \[conj-taussky\]), or

2.  the prime $2$ is inert in the extension ${\mathbb Q}(\zeta_p)^+/{\mathbb Q}$, or

3.  $q\leq 92459$,

then $\rho_{\infty}=0$ and $j_\infty=g=(p-1)/2$, and $\dim_{{\mathbb F}_2}{{\rm Sel}}^{(2)}({\mathbb Q},J) \leq  g + \dim_{{\mathbb F}_2}({{\rm Cl}}(L)[2]).$

In fact, if the Davis–Taussky conjecture holds (see Remark \[rem-davistausskyoddclassnumber\]), then the bound of Theorem \[thm-sophie-intro\] becomes $\dim_{{\mathbb F}_2}{{\rm Sel}}^{(2)}({\mathbb Q},J) \leq  g$.

The 