arXiv subjects

Asimina S. Hamakiotes

Publications and source records attributed to Asimina S. Hamakiotes.

8 recordsLinked to original sources

arXiv2026

Towards a classification of $p^2$-discriminant ideal twins over number fields

Isogenous elliptic curves have the same conductor but not necessarily the same minimal discriminant ideal. In this article, we explicitly classify all $p^2$-isogenous elliptic curves defined over a number field with the same minimal discriminant ideal for odd prime $p$ where $X_0(p^2)$ has genus $0$, i.e., $p = 3$ or $5$. As a consequence, we give a list of all $p^2$-isogenous discriminant (ideal) twins over $\mathbb{Q}$ for such $p$.

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arXiv2026

Average twin prime conjecture for elliptic curves in arithmetic progressions

In 1988, Koblitz conjectured an asymptotic formula for the number of primes $p \le x$ for which the reduction of an elliptic curve over $\mathbb{Q}$ has prime order. Building on the work of Balog, Cojocaru, and David, who proved this conjecture on average in 2011, we extend the result to primes $p$ lying in arithmetic progressions. Our asymptotic formula holds uniformly for moduli up to a fixed power of $\log x$. We also show that the resulting average constant matches the theoretical constant predicted by Lee, Mayle, and Wang in 2025 using Galois representations.

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arXiv2026

On the Density of Coprime Reductions of Elliptic Curves

Given non-CM elliptic curves $E_1$ and $E_2$ over $\mathbb{Q}$, we study the natural density of primes $p$ of good reduction for which the orders of the groups $E_1(\mathbb{F}_p)$ and $E_2(\mathbb{F}_p)$ are coprime. This problem may be viewed as an elliptic curve analogue of the classical question concerning the density of coprime integer pairs. Motivated by Zywina's refinement of the Koblitz conjecture, we formulate a conjecture for the density of such primes. We prove that the series defining this constant converges and admits an almost Euler product expansion. In the case of Serre pairs, we give a closed formula for the constant and use it to prove a moments result describing the distribution of these constants as $(E_1, E_2)$ varies.

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arXiv2025

Bounds for the relative class number problem for function fields

We establish bounds on a finite separable extension of function fields in terms of the relative class number, thus reducing the problem of classifying extensions with a fixed relative class number to a finite computation. We also solve the relative class number two problem in all cases where the base field has constant field not equal to $\mathbb{F}_2$.

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arXiv2025

Genus formulas for families of modular curves

For each open subgroup $H\leq \operatorname{GL}_2(\widehat{\mathbb{Z}})$, there is a modular curve $X_H$, defined as a quotient of the full modular curve $X(N)$, where $N$ is the level of $H$. The genus formula of a modular curve is well known for $X_0(N)$, $X_1(N)$, $X(N)$, $X_{\mathrm{sp}}(N)$, $X_{\mathrm{ns}}(N)$, and $X_{S_4}(p)$ for $p$ prime. We explicitly work out the invariants of the genus formulas for $X_{\mathrm{sp}}^+(N)$, $X_{\mathrm{ns}}^+(N)$, and $X_{\text{arith},1}(M,MN)$. In Table $1$, we provide the invariants of the genus formulas for all of the modular curves listed.

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arXiv2024

The maximal abelian extension contained in a division field of an elliptic curve over $\mathbb{Q}$ with complex multiplication

Let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f\geq 1$. Let $E$ be an elliptic curve with CM by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j_{K,f})$, where $j_{K,f}=j(E)$. It has been shown by the author and Lozano-Robledo that $\operatorname{Gal}(\mathbb{Q}(j_{K,f},E[N])/\mathbb{Q}(j_{K,f}))$ is only abelian for $N=2,3$, and $4$. Let $p$ be a prime and let $n\geq 1$ be an integer. In this article, we bound the commutator subgroups of $\operatorname{Gal}(\mathbb{Q}(E[p^n])/\mathbb{Q})$ and classify the maximal abelian extensions contained in $\mathbb{Q}(E[p^n])/\mathbb{Q}$.

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arXiv2024

Sums of Powers by L'Hopital's Rule

For a positive integer $d$, let $p_d(n) := 0^d + 1^d + 2^d + \cdots + n^d$; i.e., $p_d(n)$ is the sum of the first $d^{\mathrm{th}}$-powers up to $n$. It's well known that $p_d(n)$ is a polynomial of degree $d+1$ in $n$. While this is usually proved by induction, once $d$ is not small it's a challenge as one needs to know the polynomial for the inductive step. We show how this difficulty can be bypassed by giving a simple proof that $p_d(n)$ is a polynomial of degree $d+1$ in $n$ by using L'Hopital's rule, and show how we can then determine the coefficients by Cramer's rule. This illustrates a general principle and the point of our paper: there's more than one path to a goal, different approaches have their advantages and disadvantages, and the more techniques one knows, the more likely one can successfully attack a problem.

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arXiv2023

Elliptic curves with complex multiplication and abelian division fields

Let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f\geq 1$. Let $E$ be an elliptic curve with CM by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j_{K,f})$, where $j_{K,f}=j(E)$. In this article, we classify the values of $N\geq 2$ and the elliptic curves $E$ such that (i) the division field $\mathbb{Q}(j_{K,f},E[N])$ is an abelian extension of $\mathbb{Q}(j_{K,f})$, and (ii) the $N$-division field coincides with the $N$-th cyclotomic extension of the base field.

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