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Nikola Adžaga

Publications and source records attributed to Nikola Adžaga.

14 recordsLinked to original sources
arXiv2026

Semi-stable models, local heights and quadratic Chabauty for $X_0(N)^*$

Quadratic Chabauty computations for $X_0(N)^*$ are complicated by local height contributions at primes of bad reduction. For squarefree $N$, we explain a strategy for constructing a global $p$-adic height for which all the contributions at the primes of bad reduction vanish. As our first main result, we show that such a $p$-adic height exists for all squarefree levels $N>714$. In order to do this, we first give an explicit description of the minimal regular model of $X_0(N)^*$ at primes of bad reduction, and show this model is semi-stable. Each component of this model gives rise to a linear condition which ensures that the contribution at that component vanishes. Using embeddings of quadratic orders into quaternion algebras, we obtain an upper bound for the number of these conditions; when the genus of $X_0(N)^*$ is greater than one more than this bound, there is enough freedom to choose a suitable correspondence, simplifying quadratic Chabauty computations. As an application, we determine the rational points on several curves $X_0(N)^*$ for which this was not done before.

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arXiv2026

Arithmetic of elliptic curves induced by regular Diophantine triples

We study elliptic curves induced by regular Diophantine triples, with emphasis on their torsion subgroups. We show that an elliptic curve $E$ induced by a regular Diophantine triple in integers necessarily has torsion subgroup $E(\mathbb{Q})_{\mathrm{tors}} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. Moreover, we develop a criterion for when such an elliptic curve acquires a point of order $3$ over a quadratic field. For a particular family $\{ k-1, k+1, 4k\}$, we use it to show that this does not happen. Finally, we study both the torsion and the generic rank of a family of elliptic curves induced by the $D(-k^2)$-triple $\{1, 2k^2, 2k^2+2k+1\}$.

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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.

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arXiv2026

The extensibility of the Diophantine triple $\{2, b, c\}$

The aim of this paper is to consider the extensibility of the Diophantine triple $\{2,b,c\}$, where $2<b<c$, and to prove that such a set cannot be extended to an irregular Diophantine quadruple. We succeed in that for some families of $c$'s (depending on $b$). As corollary, for example, we prove that for $b/2-1$ prime, all Diophantine quadruples $\{2,b,c,d\}$ with $2<b<c<d$ are regular.

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arXiv2024

Asymptotics of $D(q)$-pairs and triples via $L$-functions of Dirichlet charaters

Let $q$ be an integer. A $D(q)$-$m$-tuple is a set of $m$ distinct positive integers ${a_1, a_2, . . . , a_m}$ such that $a_ia_j + q$ is a perfect square for all $1 \leq i < j \leq m$. By counting integer solutions $x \in [1, b]$ of congruences $x^2 \equiv q (\mod b)$ with $b \leq N$, we count $D(q)$-pairs with both elements up to $N$, and give estimates on asymptotic behaviour. We show that for prime $q$, the number of such $D(q)$-pairs and $D(q)$-triples grows linearly with $N$. Up to a factor of $2$, the slope of this linear function is the quotient of the value of the $L$-function of an appropriate Dirichlet character (usually a Kronecker symbol) and of $ζ(2)$.

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arXiv2023

Computing quadratic points on modular curves $X_0(N)$

In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves $X_0(N)$ of genus up to $8$, and genus up to $10$ with $N$ prime, for which they were previously unknown. The values of $N$ we consider are contained in the set \[ \mathcal{L}=\{58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 \}.\] We obtain that all the non-cuspidal quadratic points on $X_0(N)$ for $N\in \mathcal{L}$ are CM points, except for one pair of Galois conjugate points on $X_0(103)$ defined over $\mathbb{Q}(\sqrt{2885})$. We also compute the $j$-invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings.

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arXiv2022

Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings

We complete the computation of all $\mathbb{Q}$-rational points on all the $64$ maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty--Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method combined with the Mordell-Weil sieve. Additionally, for square-free levels $N$, we classify all $\mathbb{Q}$-rational points as cusps, CM points (including their CM field and $j$-invariants) and exceptional ones. We further indicate how to use this to compute the $\mathbb{Q}$-rational points on all of their modular coverings.

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arXiv2021

Quadratic Chabauty for Atkin-Lehner Quotients of Modular Curves of Prime Level and Genus 4, 5, 6

We use the method of quadratic Chabauty on the quotients $X_0^+(N)$ of modular curves $X_0(N)$ by their Fricke involutions to provably compute all the rational points of these curves for prime levels $N$ of genus four, five, and six. We find that the only such curves with exceptional rational points are of levels $137$ and $311$. In particular there are no exceptional rational points on those curves of genus five and six. More precisely, we determine the rational points on the curves $X_0^+(N)$ for $N=137,173,199,251,311,157,181,227,263,163,197,211,223,269,271,359$.

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arXiv2019

On the extensions of the Diophantine triples in Gaussian integers

A Diophantine $m$-tuple is a set of $m$ distinct integers such that the product of any two distinct elements plus one is a perfect square. In this paper we study the extensibility of a Diophantine triple $\{k-1, k+1, 16k^3-4k\}$ in Gaussian integers $\mathbb{Z}[i]$ to a Diophantine quadruple. Similar one-parameter family, $\{k-1, k+1, 4k\}$, was studied in Franušić's previous paper, where it was shown that the extension to a Diophantine quadruple is unique (with an element $16k^3-4k$). The family of the triples of the same form $\{k-1, k+1, 16k^3-4k\}$ was already studied in rational integers. It appeared as a special case while solving the extensibility problem of Diophantine pair $\{k-1, k+1\}$, in which it was not possible to use the same method as in the other cases. As authors (Bugeaud, Dujella and Mignotte) point out, the difficulty appears because the gap between $k+1$ and $16k^3-4k$ is not sufficiently large. We find the same difficulty here while trying to use Diophantine approximations. Then we partially solve this problem by using linear forms in logarithms.

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arXiv2018

On the size of Diophantine $m$-tuples in imaginary quadratic number rings

A Diophantine $m$-tuple is a set of $m$ distinct integers such that the product of any two distinct elements plus one is a perfect square. It was recently proven that there is no Diophantine quintuple in positive integers. We study the same problem in the rings of integers of imaginary quadratic fields. By using a gap principle proven by Diophantine approximations, we show that $m\leqslant 42$. Our proof is relatively simple compared to the proofs of the similar results in positive integers.

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arXiv2017

Triples which are $D(n)$-sets for several $n$'s

For a nonzero integer $n$, a set of distinct nonzero integers $\{a_1,a_2,\ldots,a_m\}$ such that $a_ia_j+n$ is a perfect square for all $1\leq i<j\leq m$, is called a Diophantine $m$-tuple with the property $D(n)$ or simply $D(n)$-set. $D(1)$-sets are known as simply Diophantine $m$-tuples. Such sets were first studied by Diophantus of Alexandria, and since then by many authors. It is natural to ask if there exists a Diophantine $m$-tuple ($D(1)$-set) which is also a $D(n)$-set for some $n\neq 1$. This question was raised by Kihel and Kihel in 2001. They conjectured that there are no Diophantine triples which are also $D(n)$-sets for some $n\neq 1$. However, the conjecture does not hold, since, for example, $\{8, 21, 55\}$ is a $D(1)$ and $D(4321)$-triple, while $\{1, 8, 120\}$ is a $D(1)$ and $D(721)$-triple. We present several infinite families of Diophantine triples $\{a, b, c\}$ which are also $D(n)$-sets for two distinct $n$'s with $n\neq 1$, as well as some Diophantine triples which are also $D(n)$-sets for three distinct $n$'s with $n\neq 1$. We further consider some related questions.

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arXiv2016

On the extension of $D(-8k^2)$-pair $\{8k^2, 8k^2+1\}$

Let $n$ be a nonzero integer. A set of $m$ positive integers is called a $D(n)$-$m$-tuple if the product of any two of its distinct elements increased by $n$ is a perfect square. Let $k$ be a positive integer. By elementary means, we show that the $D(-8k^2)$-pair $\{8k^2, 8k^2+1\}$ can be extended to at most a quadruple (the third and fourth element can only be $1$ and $32k^2+1$). At the end, we suggest considering a $D(-k^2)$-triple $\{ 1, 2k^2, 2k^2+2k+1\}$ as possible future research direction.

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arXiv2015

Automated conjecturing of Frobenius numbers via grammatical evolution

Conjecturing formulas and other symbolic relations occurs frequently in number theory and combinatorics. If we could automate conjecturing, we could benefit not only from speeding up, but also from finding conjectures previously out of our grasp. Grammatical evolution, a genetic programming technique, can be used for automated conjecturing in mathematics. Concretely, this work describes how one can interpret the Frobenius problem as a symbolic regression problem, and then apply grammatical evolution to it. In this manner, a few formulas for Frobenius numbers of specific quadruples were found automatically. The sketch of the proof for one conjectured formula, using lattice point enumeration method, is provided as well. Same method can easily be used on other problems to speed up and enhance the research process.

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