Bounding Selmer Groups of Superelliptic Jacobians via Class Groups

Abstract

Let $K$ be a number field containing a primitive $p$-th root of unity $ζ_p$. Let $f(x)\in K[x]$ be a monic integral polynomial, and let $f_0$ denote its radical. Let $C/K$ be the superelliptic curve defined by $y^p=f(x)$, and let $J$ be its Jacobian variety. The variety $J$ admits multiplication by $ζ_p$ over $K$; in particular, the endomorphism induced by $Π:=1-ζ_p$ gives an isogeny of $J$ over $K$. Let $\operatorname{Sel}_Π(J)$ denote the Selmer group associated to $Π$. Under suitable hypotheses, we obtain bounds for $\operatorname{Sel}_Π(J)$ in terms of the $p$-torsion subgroup of the class group of $L:=K[x]/(f_0)$. Several examples illustrating the results are also discussed.