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Tomislav Gužvić

Publications and source records attributed to Tomislav Gužvić.

6 recordsLinked to original sources
arXiv2026

Completing the classification of torsion subgroups for rational elliptic curves over sextic fields

We complete the classification of torsion subgroups $E(K)_{\text{tors}}$ that can occur for an elliptic curve $E/\mathbb{Q}$ over a sextic number field $K$. Previous work determined the complete set of these groups, leaving the existence of only one group in question: $C_3 \oplus C_{18}$. We prove that this group does not occur. Our proof relies on the theory of Galois representations attached to elliptic curves. The assumed existence of a $C_3 \oplus C_{18}$ torsion subgroup would impose strong, simultaneous constraints on the mod-$2$ and $3$-adic Galois representations of the curve. By applying the recent classification of $\ell$-adic Galois images for elliptic curves over $\mathbb{Q}$, we translate these arithmetic constraints into a problem of Diophantine geometry: the $j$-invariant of such a curve must correspond to a rational point on one of the finitely many modular curves. We then analyze these curves using classical methods and show that none have the necessary rational points corresponding to elliptic curves without complex multiplication, thereby proving our main result.

math.NT↗
arXiv2022

Torsion groups of elliptic curves over $\mathbb{Q}(μ_{p^{\infty}})$

Let $E/\mathbb{Q}$ be an elliptic curve and $p \in \{5,7,11 \}$ be a prime. We determine the possibilities for $E(\mathbb{Q}(ζ_{p}))_{tors}$. Additionally, we determine all the possibilities for $E(\mathbb{Q}(ζ_{16}))_{tors}$ and $E(\mathbb{Q}(ζ_{27}))_{tors}$. Using these results we are able to determine the possibilities for $E(\mathbb{Q}(μ_{p^{\infty}}))_{tors}$.

math.NT↗
arXiv2021

Torsion groups of Mordell curves over number fields of higher degree

Mordell curves over a number field $K$ are elliptic curves of the form $ y^2 = x^3 + c$, where $c \in K \setminus \{ 0 \}$. Let $p \geq 5$ be a prime number, $K$ a number field such that $[K:\mathbb{Q}] \in \{ 2p, 3p \}$ and let $E$ be a Mordell curve defined over $K$. We classify all the possible torsion subgroups $E(K)_{\text{tors}}$ for all Mordell curves $E$ defined over $\mathbb{Q}$ when $[K: \mathbb{Q}] \in \{2p, 3p \}$.

math.NT↗
arXiv2020

Torsion groups of elliptic curves over some infinite abelian extensions of $\mathbb{Q}$

We determine, for an elliptic curve $E/\mathbb{Q}$, all the possible torsion groups $E(K)_{tors}$, where $K$ is the compositum of all $\mathbb{Z}_{p}$-extensions of $\mathbb{Q}$. Furthermore, we prove that for an elliptic curve $E/\mathbb{Q}$ it holds that $E(\mathbb{Q}(μ_{p^{\infty}}))_{tors} = E(\mathbb{Q}(μ_{p}))_{tors}$, for all primes $p \geq 5$ and $E(\mathbb{Q}(μ_{3^{\infty}}))_{tors} = E(\mathbb{Q}(μ_{3^3}))_{tors}$, $E(\mathbb{Q}(μ_{2^{\infty}}))_{tors} = E(\mathbb{Q}(μ_{2^4}))_{tors}$.

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