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

N.Garc\'ia-Fritz and H.Pasten showed that Hilbert's 10th problem is unsolvable in the ring of integers of number fields of the form $\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$ for positive proportions of primes $p$ and $q$. We improve their…

Let $K = \mathbb Q(\sqrt{-q})$, where $q$ is a prime congruent to $3$ modulo $4$. Let $A = A(q)$ denote the Gross curve over the Hilbert class field $H$ of $K$. In this note we use Magma to calculate the values $L(E/H, 1)$ for all such…

Let p denote an odd prime. For all p-admissible conductors c over a quadratic number field \(K=\mathbb{Q}(\sqrt{d})\), p-ring spaces \(V_p(c)\) modulo c are introduced by defining a morphism \(\psi:\,f\mapsto V_p(f)\) from the divisor…

In a recent paper, Freitas and Siksek proved an asypmtotic version of Fermat's Last Theorem for many totally real fields. We prove an extension of their result to generalized Fermat equations of the form $A x^p+B y^p+ C z^p=0$, where $A$,…

Let C : y^2=f(x) be a hyperelliptic curve defined over the rationals. Let K be a number field and suppose f factors over K as a product of irreducible polynomials f=f_1 f_2...f_r. We shall define a "Selmer set" corresponding to this…

The aim of this paper is to give some properties of Hilbert genus fields and construct the Hilbert genus fields of the fields $L_{m,d}:=\mathbb{Q}(\zeta_{2^m},\sqrt{d})$, where $m\geq 3$ is a positive integer and $d$ is a square-free…

The purpose of the paper is to study some problems raised by Hayman and Gundersen about the existence of non-trivial entire and meromorphic solutions for the Fermat type functional equation $f^n+g^n+h^n=1$. Hayman showed that no non-trivial…

We show that Hilbert's Tenth Problem is undecidable for complementary subrings of number fields and that the p-adic and archimedean ring versions of Mazur's conjectures do not hold in these rings. More specifically, given a number field K,…

Let $C_k$ be a smooth projective curve over a global field $k$, which is neither rational nor elliptic. Harris-Silverman, when $p=0$, and Schweizer, when $p>0$ together with an extra condition on the Jacobian variety…

We give a cyclotomic proof of the fact that the equation $\frac{x^p + y^p}{x+y} = p^e z^p$ has no solutions in coprime integers $x,y,z$ and $p > 3$, a prime. This implies in particular Fermat's Last Theorem.

We show that for $5/6$-th of all primes $p$, Hilbert's 10-th Problem is unsolvable for $\mathbb{Q}(\zeta_3, \sqrt[3]{p})$. We also show that there is an infinite set $S$ of square free integers such tha Hilbert's 10-th Problem is unsolvable…

The paper studies the supersingular locus of the characteristic p moduli space of principally polarized abelian 8-folds that are equipped with an action of a maximal order in a quaternion algebra, that is non-split at the infinite place,…

We show that the generalized Fermat equations with signatures (5,5,7), (5,5,19), and (7,7,5) (and unit coefficients) have no non-trivial primitive integer solutions. Assuming GRH, we also prove the nonexistence of non-trivial primitive…

We provide explicit ranges for $\sigma$ for which the asymptotic formula \begin{equation*} \int_0^T|\zeta(1/2+it)|^4|\zeta(\sigma+it)|^{2j}dt \;\sim\; T\sum_{k=0}^4a_{k,j}(\sigma)\log^k T \quad(j\in\mathbb N) \end{equation*} holds as…

Let $p$ be an odd prime, and $m,r \in \mathbb{Z}^+$ with $m$ coprime to $p$. In this paper we investigate the real quadratic fields $K = \mathbb{Q}(\sqrt{m^2p^{2r} + 1})$. We first show that for $m < C$, where constant $C$ depends on $p$,…

In this work we will investigate a certain generalization of the so called S-lemma in higher degrees. The importance of this generalization is, that it is closely related to Hilbert's 1888 theorem about tenary quartics. In fact, if such a…

For a square-free integer $t$, Byeon \cite{byeon} proved the existence of infinitely many pairs of quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{tD})$ with $D > 0$ such that the class numbers of all of them are indivisible…

From some works of P. Furtw\"angler and H.S. Vandiver, we put the basis of a new cyclotomic approach to Fermat's last theorem for p>3 and to a stronger version called SFLT, by introducing governing fields of the form Q(exp(2 i pi/q-1)) for…

We confirm several conjectures of Sun involving quadratic residues modulo odd primes. For any prime $p\equiv 1\pmod 4$ and integer $a\not\equiv0\pmod p$, we prove that \begin{align*}&(-1)^{|\{1\le k<\frac p4:\ (\frac kp)=-1\}|}\prod_{1\le…

Let Q be a non-singular quadratic form with integer coefficients. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q=0. When Q is positive definite we provide improved upper bounds…