Related papers: On the asymptotic Fermat's Last Theorem over numbe…

We study Fermat's Last Theorem and Catalan's conjecture in the context of weak arithmetics with exponentiation. We deal with expansions (B,e) of models of arithmetical theories (in the language L=(0,1,+,x,<)) by a binary (partial or total)…

It is shown that the quartic Fermat equation $x^4 +y^4=1$ has nontrivial integral solutions in the Hilbert class field $\Sigma$ of any quadratic field $K=\mathbb{Q}(\sqrt{-d})$ whose discriminant satisfies $-d \equiv 1$ (mod 8). A corollary…

We prove an asymptotic formula for the number of multi-quadratic number fields of bounded discriminant with a power-saving error term. Furthermore, we explicitly calculate the leading coefficient and extend our result to totally real…

We derive a bound on the precision of state estimation for finite dimensional quantum systems and prove its attainability in the generic case where the spectrum is non-degenerate. Our results hold under an assumption called local asymptotic…

In this paper we find a new lower bound on the number of imaginary quadratic extensions of the function field $\mathbb{F}_{q}(x)$ whose class groups have elements of a fixed odd order. More precisely, for $q$, a power of an odd prime, and…

Let $\mathcal{F}(h)$ be the number of imaginary quadratic fields with class number $h$. In this note, we improve the error term in Soundararajan's asymptotic formula for the average of $\mathcal{F}(h)$. Our argument leads to a similar…

Given a number field $k$ and a quadratic extension $K_2$, we give an explicit asymptotic formula for the number of isomorphism classes of cubic extensions of $k$ whose Galois closure contains $K_2$ as quadratic subextension, ordered by the…

Can any element in a sufficiently large finite field be represented as a sum of two $d$th powers in the field? In this article, we recount some of the history of this problem, touching on cyclotomy, Fermat's last theorem, and diagonal…

It is shown that the sum of class numbers of orders in totally complex quartic fields with no real quadratic subfield obeys an asymptotic law similar to the prime numbers, as the bound on the regulators tends to infinity. Here only orders…

We consider a random field, defined on an integer-valued d-dimensional lattice, with covariance function satisfying a condition more general than summability. Such condition appeared in the well-known Newman's conjecture concerning the…

Let $n>1$ be an odd integer. We prove that there are infinitely many imaginary quadratic fields of the form $\mathbb{Q}(\sqrt{x^2-2y^n})$ whose ideal class group has an element of order $n$. This family gives a counter example to a…

We consider saddle point integrals in d variables whose phase function is neither real nor purely imaginary. Results analogous to those for Laplace (real phase) and Fourier (imaginary phase) integrals hold whenever the phase function is…

Let $q$ be a quadratic form over a field $F$ and let $L$ be a field extension of $F$ of odd degree. It is a classical result that if $q_L$ is isotropic (resp. hyperbolic) then $q$ is isotropic (resp. hyperbolic). In turn, given two…

Let $F_0=\mathbf Q(\sqrt{-d})$ be an imaginary quadratic field with $3\nmid d$ and let $K_0=\mathbf Q(\sqrt{3d})$. Let $\varepsilon_0$ be the fundamental unit of $K_0$ and let $\lambda$ be the Iwasawa $\lambda$-invariant for the cyclotomic…

We prove some new instances of a conjecture of Bachoc, Couvreur and Z\'emor that generalizes Freiman's $3k-4$ Theorem to a multiplicative version in a function field setting. As a consequence we find that if $F$ is a rational function field…

We give necessary and sufficient conditions on a squarefree integer $d$ for there to be non-trivial solutions to $x^{3} + y^{3} = z^{3}$ in $\Q(\sqrt{d})$, conditional on the Birch and Swinnerton-Dyer conjecture. These conditions are…

The ordinary continued fractions expansion of a real number is based on the Euclidean division. Variants of the latter yield variants of the former, all encompassed by a more general Dynamical Systems framework. For all these variants the…

In this note, we prove the existence of a secondary term in the asymptotic formula of the cubic moment of quadratic Dirichlet L-functions $$\sum_{\substack{d - \mathrm{monic \, \& \, sq. \, free} \mathrm{deg}\, d \, = \, D}}…

Let $K$ be a number field, and $S$ a finite set of places in $K$ containing all infinite places. We present an implementation for solving the $S$-unit equation $x + y = 1$, $x,y \in\mathscr{O}_{K,S}^\times$ in the computer algebra package…

Gauge theories and perturbative gravity in four dimensions are governed by a tower of infinite-dimensional symmetries which arise from tree-level soft theorems. However, aside from the leading soft theorems which are all-loop exact,…