arXiv subjects

Garen Chiloyan

Publications and source records attributed to Garen Chiloyan.

5 recordsLinked to original sources

arXiv2023

A classification of curious Galois groups as direct products

Let $N$ be a positive integer. Let $\operatorname{H}$ be a group of level $N$ and let $E$ be an elliptic curve defined over the rationals with $\textit{j}_{E} \neq 0, 1728$. Then the image $\overlineρ_{E,N}\left(\operatorname{Gal}\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)\right)$, of the mod-$N$ Galois representation attached to $E$, is conjugate to a subgroup of $\operatorname{H}$ if and only if $E$ corresponds to a non-cuspidal rational point on the modular curve $\operatorname{X}_{\operatorname{H}}$ generated by $\operatorname{H}$. In this article, we are interested when $\overlineρ_{E,N}(G_{\mathbb{Q}})$ is precisely $\operatorname{H}$. More precisely, we classify all groups $\operatorname{H}$ that are direct products of subgroups $\operatorname{H}$ for which $\operatorname{X}_{\operatorname{H}}$ contains infinitely many non-cuspidal rational points but there is no elliptic curve $E/\mathbb{Q}$ such that $\overlineρ_{E,N}\left(\operatorname{Gal}\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)\right)$ is conjugate to $\operatorname{H}$.

math.NT↗

arXiv2023

2-adic images of CM isogeny-torsion graphs

Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$. The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each element of $\mathcal{E}$ and an edge for each $\mathbb{Q}$-isogeny of prime degree that maps one elliptic curve in $\mathcal{E}$ to another elliptic curve in $\mathcal{E}$, with the degree of the isogeny recorded as a label of the edge. The isogeny-torsion graph associated to $\mathcal{E}$ is the isogeny graph associated to $\mathcal{E}$ where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over $\mathbb{Q}$ of the corresponding elliptic curve. The main result of the article is a classification of the 2-adic Galois image at each vertex of the isogeny-torsion graphs whose associated $\mathbb{Q}$-isogeny class consists of elliptic curves over $\mathbb{Q}$ with complex multiplication.

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arXiv2023

2-adic Galois images of non-CM isogeny-torsion graphs

Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$ without CM. The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each elliptic curve in $\mathcal{E}$ and an edge for each $\mathbb{Q}$-isogeny of prime degree that maps one elliptic curve in $\mathcal{E}$ to another elliptic curve in $\mathcal{E}$, with the degree recorded as a label of the edge. An isogeny-torsion graph is an isogeny graph where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over $\mathbb{Q}$ of the corresponding elliptic curve. Then, the main statement of the article is a classification of the $2$-adic image of Galois that occurs at each vertex of all isogeny-torsion graphs consisting of elliptic curves defined over $\mathbb{Q}$ without CM.

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arXiv2022

Infinite Families Of Isogeny-Torsion Graphs

Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$. The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each element of $\mathcal{E}$ and an edge for each $\mathbb{Q}$-isogeny of prime degree that maps one element of $\mathcal{E}$ to another element of $\mathcal{E}$, with the degree recorded as a label of the edge. The isogeny-torsion graph associated to $\mathcal{E}$ is the isogeny graph associated to $\mathcal{E}$ where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over $\mathbb{Q}$ of the corresponding elliptic curve. The main result of the article is a determination of which isogeny-torsion graphs associated to $\mathbb{Q}$-isogeny classes of elliptic curves defined over $\mathbb{Q}$ correspond to infinitely many $\textit{j}$-invariants.

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arXiv2020

A classification of isogeny-torsion graphs of $\mathbb{Q}$-isogeny classes of elliptic curves

Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$. The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each elliptic curve in the $\mathbb{Q}$-isogeny class $\mathcal{E}$, and an edge for each cyclic $\mathbb{Q}$-isogeny of prime degree between elliptic curves in the isogeny class, with the degree recorded as a label of the edge. In this paper, we define an isogeny-torsion graph to be an isogeny graph where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over $\mathbb{Q}$ of the corresponding elliptic curve. Then, the main result of the article is a classification of all the possible isogeny-torsion graphs that occur for $\mathbb{Q}$-isogeny classes of elliptic curves defined over the rationals.

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