Related papers: The quartic Fermat equation in Hilbert class field…

In 1888, Hilbert proved that every non-negative quartic form f=f(x,y,z) with real coefficients is a sum of three squares of quadratic forms. His proof was ahead of its time and used advanced methods from topology and algebraic geometry. Up…

Let $D$ be a square-free integer other than 1. Let $K$ be the quadratic field ${\mathbb Q}(\sqrt D)$. Let $\delta \in \{1,2\}$ with $\delta=2$ if $D\equiv 1 \pmod 4$. To each prime ideal $\mathcal P$ in $K$ that splits in $K/\mathbb Q$ we…

Let p be an odd prime. Let F_p^* be the no-null part of the finite field of p elements. Let K=\Q(zeta) be a p-cyclotomic field and O_K be its ring of integers. Let pi be the prime ideal of K lying over p. Let sigma : zeta --> zeta^v be the…

Recent results of Freitas, Kraus, Sengun and Siksek give sufficient criteria for the asymptotic Fermat's Last Theorem to hold over various number fields. In this paper, we prove asymptotic results about the solutions of the Diophantine…

We consider the Diophantine equation $x^4 + y^4 - w^2 = n$ for $n \in \mathbb{Z}$, which is related to near misses for the quartic case of Fermat's Last Theorem. For certain $n$ we show that the set of solutions is infinite, or more…

We solve unconditionally the class number one problem for the $2$-parameter family of real quadratic fields $\mathbb{Q}(\sqrt{d})$ with square-free discriminant $d=(an)^2+4a$ for positive odd integers $a$ and $n$.

In this paper we are interested in solving the Fermat-type equations x^5+y^5=dz^p where d is a positive integer and p a prime number $\ge 7$. We describe a new method based on modularity theorems which allows us to improve all the results…

Fix a rational prime $r \geq 5$. In this article, we study the integer solutions of the generalized Fermat equation of signature $(2p,2q,r)$, namely $x^{2p}+y^{2q}=z^r$, where the primes $p,q \geq 5$ are varying. For each rational prime $r…

For a finite subgroup $\Gamma\subset \mathrm{SL}(2,\mathbb{C})$ and $n\geq 1$, we construct the (reduced scheme underlying the) Hilbert scheme of $n$ points on the Kleinian singularity $\mathbb{C}^2/\Gamma$ as a Nakajima quiver variety for…

We prove a version of Hilbert's Irreducibility Theorem in the quadratic case, giving a quantitative improvement to a result of Bilu-Gillibert in this restricted setting. As an application, we give improvements to several quantitative…

We prove that the Fermat-type equation $x^3 + y^3 = z^p$ has no solutions $(a,b,c)$ satisfying $abc \ne 0$ and $\gcd(a,b,c)=1$ when $-3$ is not a square mod~$p$. This improves to approximately $0.844$ the Dirichlet density of the set of…

In this article we give an analogue of Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields. Let $K$ be a real quadratic field and $\Om_K$ its ring of integers. Let $\Gamma$ be a congruence subgroup of $\SL_2(\Om_K)$…

Within the scope of elementary number theory, we prove that, as the main result, if $1 \leq x < y < z$ are integers such that at least one of $y, z, x+y$ is prime then $x^{n}+y^{n} \neq z^{n}$ for every odd integer $n \geq 3$. This result…

Lifting problem for universal quadratic forms asks for totally real number fields $K$ that admit a positive definite quadratic form with coefficients in $\mathbb{Z}$ that is universal over the ring of integers of $K$. In this paper, we show…

Via a novel application of Iwasawa theory, we study Hilbert's tenth problem for number fields occurring in $\mathbb{Z}_p$-towers of imaginary quadratic fields $K$. For a odd prime $p$, the lines $(a,b) \in \mathbb{P}^1(\mathbb{Z}_p)$ are…

We determine the minimal number of variables $\Gamma^*(d, K)$ which guarantees a nontrivial solution for every additive form of degree $d=4$ over the four ramified quadratic extensions $\mathbb{Q}_2(\sqrt{2}), \mathbb{Q}_2(\sqrt{10}),…

Let $\ell>3$ be a prime such that $\ell \equiv 3 \pmod{4}$ and $\mathbb{Q}(\sqrt{\ell})$ has class number 1. Then Hirzebruch and Zagier noticed that the class number of $\mathbb{Q}(\sqrt{-\ell})$ can be expressed as $h(-\ell) =…

We prove an asymptotic formula for class numbers of totlally imaginary quartic number fields, ie for number fields of degree 4 over Q with only complex embeddings. After previous work for real quadratic fields (Sarnak) and complex cubic…

Let $n \in \mathbb{Z}_{\geq 2}$. We study the generalized Fermat equation \[x^{13}+y^{13}=z^n, \quad x,y,z \in \mathbb{Z}, \quad \gcd(x,y,z)=1.\] Using a combination of techniques, including the modular method, classical descent, unit…

Let $p$ be a prime number such that $p=2$ or $p\equiv 1\pmod 4$. Let $\varepsilon_p$ denote the fundamental unit of $\mathbb{Q}(\sqrt{p})$ and let $a$ be a positive square-free integer. The main aim of this paper is to determine explicitly…