Class Groups and Selmer Groups in Special Families

This paper establishes that, for elliptic curves over $\mathbb{Q}$ with a rational 3-isogeny, the 3-isogeny Selmer group over the dihedral field $K=\mathbb{Q}(\zeta_3,m^{1/3})$ is sandwiched between two class-group 3-ranks. For Type I curves $E_a:y^2=x^3+a$ with $a\notin K^{*2}$, writing $L=K(\sqrt{a})$ and $S=S_a(K)\cup S_3(K)$, the $\mathbb{F}_3$-dimension of $\mathrm{Sel}_\phi(E_a/K)$ lies between $\dim_{\mathbb{F}_3}\mathrm{Cl}_S(L)[3]$ and $\dim_{\mathbb{F}_3}\mathrm{Cl}_S(L)[3]+|S(L)|+r_1+r_2$; when $a\in K^{*2}$, the same sandwich holds with $K$ in place of $L$ and an additive constant $2$. This matters because Selmer groups are hard to compute directly, whereas class-group 3-ranks are computable, so the bounds convert class-group data into constraints on Selmer ranks and hence on the arithmetic of the curve. Combining the lower bound with the unboundedness of the 3-part of the class groups of these fields $K$, the paper concludes that $\dim_{\mathbb{F}_3}\mathrm{Sel}_\phi(E_a/K)$, and therefore the full 3-Selmer dimension, becomes arbitrarily large as $m$ varies. The same sandwich structure is developed for Type II curves $E_{a,b}:y^2=x^3+a(x-b)^2$, where the prime sets and Tamagawa-number computations are more elaborate.