Related papers: Lifting problem for universal quadratic forms

We study anisotropic universal quadratic forms over semi-global fields; i.e., over one-variable function fields over complete discretely valued fields. In particular, given a semi-global field $F$, we compute both the $m$-invariant of $F$…

A positive-definite integral quadratic form is called regular if it represents every positive integer which is locally represented. In this article, we classify all regular diagonal quadratic forms of rank greater than 3.

Let $l$ be a rational prime greater than or equal to $3$ and $k$ be a given positive integer. Under a conjecture due to Langland and an assumption on upper bound for the regulator of fields of the form $\mathbb{Q}\left(\sqrt[l]a\right)$, we…

Let $f$ be a positive definite (non-classic) integral quaternary quadratic form. We say $f$ is strongly $s$-regular if it satisfies a regularity property on the number of representations of squares of integers. In this article, we prove…

We consider the problem of classifying all positive-definite integer-valued quadratic forms that represent all positive odd integers. Kaplansky considered this problem for ternary forms, giving a list of 23 candidates, and proving that 19…

Caraiani and Newton have proven that if $F$ is an imaginary quadratic number field such that $X_0(15)$ has rank $0$ over $F$, then every elliptic curve over $F$ is modular. This paper is concerned with the quadratic fields…

An integral basis of the simplest number fields of degree 3,4 and 6 over $\mathbb{Q}$ are well-known, and widely investigated. We generalize the simplest number fields to any degree, and show that an integral basis of these fields is…

In this article we show that the form $x^2 + iy^2 + z^2 + iw^2$ represents all gaussian integers. The main tools used in this proof are Fermat's little theorem (over finite field extensions), the Mordell-Niven theorem (representation of…

A (positive definite integral) quadratic form is called almost 2-universal if it represents all (positive definite integral) binary quadratic forms except those in only finitely many equivalence classes. Oh [7] determined all almost…

It has been shown by Madden that there are only finitely many quadratic extensions of k(x), k a finite field, in which the ideal class group has exponent two and the infinity place of k(x) ramifies. We give a characterization of such fields…

In this paper, we begin the study of the Fermat equation $x^n+y^n=z^n$ over real biquadratic fields. In particular, we prove that there are no non-trivial solutions to the Fermat equation over $\mathbb{Q}(\sqrt{2},\sqrt{3})$ for $n\geq 4$.

We enumerate all totally real number fields F with root discriminant delta_F <= 14. There are 1229 such fields, each with degree [F:QQ] <= 9.

We give a Euclidean division algorithm for the real quadratic fields $\mathbb{Q}(\sqrt{m})$ for $m \in \{2, 3, 6, 7, 11, 19\}$, with the property that the norm of the remainder depends on the first Euclidean minimum of the field. In each…

We prove that Z in definable in Q by a formula with 2 universal quantifiers followed by 7 existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Q-morphisms, whether there exists one that…

Let $K/\mathbb{Q}$ be a finite extension. We prove that the minimal height of polynomials of degree $n$ of which all roots are in $K^\times$ increases exponentially in $n$. We determine the implied constant exactly for totally real $K$ and…

We study the class number one problem for real quadratic fields $\mathbb{Q}(\sqrt{9m^2+ 4m})$, where $m$ is an odd integer. We show that for $m \equiv 1 \pmod 3$ there is only one such field with class number one and only one such field…

The main purpose of this paper is to find all the prime numbers p for which whenever we add to p an odd square less than p we obtain a number which has at most two different prime factors. We solve completely the cases $p\equiv 1,3,5 \pmod…

In 2016, in the work related to Galois representations, Greenberg conjectured the existence of multi-quadratic $p$-rational number fields of degree $2^{t}$ for any odd prime number $p$ and any integer $t \geq 1$. Using the criteria provided…

We introduce a new class of generalised quadratic forms over totally real number fields, which is rich enough to capture the arithmetic of arbitrary systems of quadrics over the rational numbers. We explore this connection through a version…

For any integer $l\geq 1$, let $p_1, p_2, \ldots, p_{l+2}$ be distinct prime numbers $\geq 5.$ For all real numbers $X>1,$ we let $N_{3,l}(X)$ denote the number of real quadratic fields $K$ whose absolute discriminant $d_K\leq X$ and $d_K$…