arXiv subjects

Omer Avci

Publications and source records attributed to Omer Avci.

3 recordsLinked to original sources

arXiv2026

Existence and non-existence of rational elliptic curves with prescribed torsion subgroups over quadratic fields

Let $K=\mathbb{Q}(\sqrt{-p})$ be a quadratic field for an odd prime $p$. We show that there exist infinitely many primes $p$ for which no elliptic curve $E/\mathbb{Q}$ has torsion subgroup $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2N\mathbb{Z}$ over $K$ for $N=5,6$. We also prove that there exist infinitely many primes $p$ for which there are infinitely many elliptic curves $E/\mathbb{Q}$ with this torsion structure, conditional on the parity conjecture. Using these results, we obtain new torsion classification results over Kummer extensions of cyclotomic fields and over composites of $\mathbb{Z}_p$-extensions of number fields, refining and extending our previous work.

math.NT↗

arXiv2025

Torsion of Rational Elliptic Curves over the $\mathbb{Z}_p$-Extensions of Quadratic Fields

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. For a quadratic number field $K$ and an odd prime number $p$, let $L$ be a $\mathbb{Z}_p$-extension of $K$. We prove that $E(L)_{\text{tors}}=E(K)_{\text{tors}}$ when $p>5$. It enables us to classify the groups that can be realized as the torsion subgroup $E(L)_{\text{tors}}$, by using the classification of torsion subgroups over the quadratic fields.

math.NT↗

arXiv2024

Torsion of Rational Elliptic Curves over the Cyclotomic Extensions of $\mathbb{Q}$

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. In this article, we classify all groups that can arise as $E(\mathbb{Q}(\zeta_p))_{\text{tors}}$ up to isomorphism for any prime $p$. When $p - 1$ is not divisible by small integers such as $3, 4, 5, 7$, or $11$, we obtain a sharper classification. For any abelian number field $K$, the torsion subgroup $E(K)_{\text{tors}}$ is a subgroup of $E(\mathbb{Q}^{\text{ab}})_{\text{tors}}$. Our methods provide tools to eliminate non-realized torsion structures from the list of possibilities for $E(K)_{\text{tors}}$.

math.NT↗