$3$-Selmer group, ideal class groups and cube sum problem
Somnath Jha, Dipramit Majumdar, Pratiksha Shingavekar
TL;DR
Addresses bounds for the phi-Selmer group of the Mordell curve E_a: y^2 = x^3 + a over K (the cyclotomic field) by relating it to the 3-part of the class group of the quadratic field L_a = K[X]/(X^2 - a), and proves a general upper and lower bound (Theorem type1selmer) with refinements (Theorem type1bounds). The approach exploits explicit 3-isogenies, local Kummer maps, and algebraic number theory to translate Selmer questions into unit-group and norm computations, especially at primes dividing 3 and away from 3. These bounds feed into the 3-Selmer group bounds over Q, enabling unconditional results for certain primes in the cube-sum problem and positive-density families of E_a with Sel^3(E_a/Q)=0 or F_3-rank 1, along with conditional results for some twists. By connecting isogeny-induced Selmer groups to class groups, the paper provides concrete arithmetic criteria and families illustrating the cube-sum phenomenon.
Abstract
Consider a Mordell curve $E_a:y^2=x^3+a$ with $a \in \mathbb Z$. These curves have a rational $3$-isogeny, say $\varphi$. We give an upper and a lower bound on the rank of the $\varphi$-Selmer group of $E_a$ over $\mathbb Q(ζ_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $\mathbb Q(ζ_3)$. Using our bounds on the Selmer groups, we prove some cases of the rational cube sum problem. Further, using these bounds, we give explicit families of the Mordell curves to show that for a positive proportion of $E_a$, ${\rm Sel}^3(E_{a}/\mathbb Q)=0$ (respectively ${\rm Sel}^3(E_{a}/\mathbb Q)$ has $\mathbb F_3$-rank $1$).