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$3$-Selmer group, ideal class groups and cube sum problem

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arxiv 2207.12487 v2 pith:Q4YGXPTM submitted 2022-07-25 math.NT

Somnath Jha , Dipramit Majumdar , Pratiksha Shingavekar This is my paper · ORCID
classification math.NT
keywords mathbbgroupselmerboundsclasscubecurvesgive
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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 $\mathbb Q(\zeta_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $\mathbb Q(\zeta_3)$. 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 $1$).

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. $\sqrt{-3}$-Selmer groups, ideal class groups and large $3$-Selmer ranks

    math.NT 2025-02 conditional novelty 6.0 of 10

    The 3-Selmer ranks of the elliptic curves y^2=x^3+a(x-b)^2 over Q(zeta_3) are controlled by ideal class group 3-ranks, yielding large-rank families and a positive proportion of rank-one curves.

  2. Relative $p$-class groups and $p$-Selmer groups

    math.NT 2024-12 conditional novelty 5.0 of 10

    For CM elliptic curves with j-invariant 0 or 1728, the p-Selmer rank of a twist is determined by the root number and a χ-component of a relative p-class group.

  3. Hilbert's 10th Problem via Mordell curves

    math.NT 2024-12 conditional novelty 4.0 of 10

    For five-sixths of primes p, Hilbert's 10th problem is shown unsolvable over Q(ζ3, ∛p), with a companion result for degree-12 extensions Q(ζ3, √D, ∛p).

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