Number Field Of Degree Research Articles

An upper bound on the number of classes of perfect unary forms in totally real number fields

Let K be a totally real number field of degree n over [Formula: see text], with discriminant and regulator [Formula: see text], respectively. In this paper, using a similar method to van Woerden, we prove that the number of classes of perfect unary forms, up to equivalence and scaling, can be bounded above by [Formula: see text], where [Formula: see text] is the discriminant of the field K, [Formula: see text] is the additive Hermite–Humbert constant over positive-definite unary forms for K and [Formula: see text] is the covering radius of the log-unit lattice. In particular, when K is Galois over [Formula: see text] and n is a prime number, the number of homothety classes of unary forms is upper bounded by [Formula: see text], where [Formula: see text] is the regulator of K. Moreover, if K is a maximal totally real subfield of a cyclotomic field, the number of homothety classes of perfect unary forms is upper bounded by [Formula: see text].

Log-free zero density estimates for automorphic L-functions

We prove a log-free zero density estimate for automorphic L-functions defined over a number field k. This work generalizes and sharpens the method of pseudo-characters and the large sieve used earlier by Kowalski and Michel. As applications, we demonstrate for a particular family of number fields of degree n over k (for any n) that an effective Chebotarev density theorem and a bound on ℓ-torsion in class groups hold for almost all fields in the family.

Moments of ideal class counting functions

We consider the counting function of ideals in a given ideal class of a number field of degree d. This describes, at least conjecturally, the Fourier coefficients of an automorphic form on GL(d), typically not a Hecke eigenform and not cuspidal. We compute its moments, and also investigate the moments of the corresponding cuspidal projection.

Brauer-Manin obstructions on hyperelliptic curves

We describe a practical algorithm for computing Brauer-Manin obstructions to the existence of rational points on hyperelliptic curves defined over number fields. This offers advantages over descent based methods in that its correctness does not rely on rigorous class and unit group computations of large degree number fields. We report on experiments showing it to be a very effective tool for deciding existence of rational points: Among a random samples of curves over Q of genus at least 5 we were able to decide existence of rational points for over 99% of curves. We also demonstrate its effectiveness for high genus curves, giving an example of a genus 50 hyperelliptic curve with a Brauer-Manin obstruction to the Hasse Principle. The main theoretical development allowing for this algorithm is an extension of the descent theory for abelian torsors to a framework of torsors with restricted ramification.

Torsion points on elliptic curves over number fields of small degree

We determine the set $S(d)$ of possible prime orders of $K$-rational points on elliptic curves over number fields $K$ of degree $d$, for $d = 4$, $5$, $6$, and $7$.

On fixed divisors of the values of the minimal polynomials over Z of algebraic numbers

Let $K$ be a number field of degree $n$, $A$ be its ring of integers, and $A_n$ (resp. $K_n$) be the set of elements of $A$ ( resp. $K$) which are primitive over $\mathbb Q$. For any $\gamma \in {K_n}$, let $F_{\gamma} (x)$ be the unique irreducible polynomial in $\mathbb Z[x]$, such that its leading coefficient is positive and $F_{\gamma} ({\gamma}) = 0$. Let $i(\gamma)=\gcd_{x\in\mathbb Z}F_{\gamma}(x)$, $i(K)=\lcm_{\theta\in{A_n}}i(\theta)$ and $\hat{\imath}(K) = \lcm_{\gamma\in{K_n}}i(\gamma)$. For any $\gamma \in {K_n}$, there exists a unique pair $(\theta,d)$, where $\theta\in A_n$ and $d$ is a positive integer such that $\gamma=\theta/d$ and $\theta\not\equiv 0\pmod{p}$ for any prime divisor $p$ of $d$. In this paper, we study the possible values of $\nu_{p}(d)$ when $p | i(\gamma)$. We introduce and study a new invariant of $K$ defined using $\nu_{p}(d)$, when $\gamma$ describes $K_n$. In the last theorem of this paper, we establish a generalisation of a theorem of MacCluer.

On unit signatures and narrow class groups of odd degree abelian number fields

For an abelian number field of odd degree, we study the structure of its 2 2 -Selmer group as a bilinear space and as a Galois module. We prove structural results and make predictions for the distribution of unit signature ranks and narrow class groups in families where the degree and Galois group are fixed.

Fourier non-uniqueness sets from totally real number fields

Let K be a totally real number field of degree n \geq 2 . The inverse different of K gives rise to a lattice in \mathbb{R}^n . We prove that the space of Schwartz Fourier eigenfunctions on \mathbb{R}^n which vanish on the “component-wise square root” of this lattice, is infinite dimensional. The Fourier non-uniqueness set thus obtained is a discrete subset of the union of all spheres \sqrt{m}S^{n-1} for integers m \geq 0 and, as m \rightarrow \infty , there are \sim c_{K} m^{n-1} many points on the m -th sphere for some explicit constant c_{K} , proportional to the square root of the discriminant of K . This contrasts a recent Fourier uniqueness result by Stoller (2021) Using a different construction involving the codifferent of K , we prove an analogue for discrete subsets of ellipsoids. In special cases, these sets also lie on spheres with more densely spaced radii, but with fewer points on each. We also study a related question about existence of Fourier interpolation formulas with nodes “ \sqrt{\Lambda} ” for general lattices \Lambda \subset \mathbb{R}^n . Using results about lattices in Lie groups of higher rank we prove that if n \geq 2 and a certain group \Gamma_{\Lambda} \leq \operatorname{PSL}_2(\mathbb{R})^n is discrete, then such interpolation formulas cannot exist. Motivated by these more general considerations, we revisit the case of one radial variable and prove, for all n \geq 5 and all real \lambda > 2 , Fourier interpolation results for sequences of spheres \sqrt{2 m/ \lambda}S^{n-1} , where m ranges over any fixed cofinite set of non-negative integers. The proof relies on a series of Poincaré type for Hecke groups of infinite covolume and is similar to the one in Stoller (2021).

Euclidean ideal classes in Galois number fields of odd prime degree

Weinberger in 1972, proved that the ring of integers of a number field with unit rank at least $1$ is a principal ideal domain if and only if it is a Euclidean domain, provided the generalised Riemann hypothesis holds. Lenstra extended the notion of Euclidean domains in order to capture Dedekind domains with finite cyclic class group and proved an analogous theorem in this setup. More precisely, he showed that the class group of the ring of integers of a number field with unit rank at least $1$ is cyclic if and only if it has a Euclidean ideal class, provided the generalised Riemann hypothesis holds. The aim of this paper is to show the following. Suppose that $\mathbf{K}_1$ and $\mathbf{K}_2$ are two Galois number fields of odd prime degree with cyclic class groups and Hilbert class fields that are abelian over $\mathbb{Q}$. If $\mathbf{K}_1\mathbf{K}_2$ is ramified over $\mathbf{K}_i$, then at least one $\mathbf{K}_i$ ($i \in \{1,2\}$) must have a Euclidean ideal class.

Unconditional Chebyshev biases in number fields

Chebyshev’s bias is the phenomenon according to which for most x, the interval [2,x] contains more primes congruent to 3 modulo 4 than primes congruent to 1 modulo 4. We present new families of examples of analogous phenomena when counting prime ideals in number fields of higher degree where the bias takes place for all large enough x. Our proofs are unconditional.

Elliptic curves over totally real quartic fields not containing √5 are modular

We prove that every elliptic curve defined over a totally real number field of degree 4 not containing 5 \sqrt {5} is modular. To this end, we study the quartic points on four modular curves.

Squarefree values of polynomial discriminants I

We determine the density of monic integer polynomials of given degree $$n>1$$ that have squarefree discriminant; in particular, we prove for the first time that the lower density of such polynomials is positive. Similarly, we prove that the density of monic integer polynomials f(x), such that f(x) is irreducible and $${\mathbb Z}[x]/(f(x))$$ is the ring of integers in its fraction field, is positive, and is in fact given by $$\zeta (2)^{-1}$$ . It also follows from our methods that there are $$\gg X^{1/2+1/n}$$ monogenic number fields of degree n having associated Galois group $$S_n$$ and absolute discriminant less than X, and we conjecture that the exponent in this lower bound is optimal.

Jordan constants of quaternion algebras over number fields and simple abelian surfaces over fields of positive characteristic

AbstractWe compute and provide a detailed description on the Jordan constants of the multiplicative subgroup of quaternion algebras over number fields of small degree. As an application, we determine the Jordan constants of the multiplicative subgroup of the endomorphism algebras of simple abelian surfaces over fields of positive characteristic.

An improvement on Schmidt’s bound on the number of number fields of bounded discriminant and small degree

Abstract We prove an improvement on Schmidt’s upper bound on the number of number fields of degree n and absolute discriminant less than X for $6\leq n\leq 94$ . We carry this out by improving and applying a uniform bound on the number of monic integer polynomials, having bounded height and discriminant divisible by a large square, that we proved in a previous work [7].

The unit equation over cyclic number fields of prime degree

Let $\ell \ne 3$ be a prime. We show that there are only finitely many cyclic number fields $F$ of degree $\ell$ for which the unit equation $$\lambda + \mu = 1, \qquad \lambda,~\mu \in \mathcal{O}_F^\times$$ has solutions. Our result is effective. For example, we deduce that the only cyclic quintic number field for which the unit equation has solutions is $\mathbb{Q}(\zeta_{11})^+$.

A Density of Ramified Primes

Let K be a cyclic number field of odd degree over {mathbb {Q}} with odd narrow class number, such that 2 is inert in K/{mathbb {Q}}. We define a family of number fields {K(p)}_p, depending on K and indexed by the rational primes p that split completely in K/{mathbb {Q}}, in which p is always ramified of degree 2. Conditional on a standard conjecture on short character sums, the density of such rational primes p that exhibit one of two possible ramified factorizations in K(p)/{mathbb {Q}} is strictly between 0 and 1 and is given explicitly as a formula in terms of the degree of the extension K/{mathbb {Q}}. Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals.

A cubic ring of integers with the smallest Pythagoras number

We prove that the ring of integers in the totally real cubic subfield $K^{(49)}$ of the cyclotomic field $\mathbb{Q}(\zeta_7)$ has Pythagoras number equal to $4$. This is the smallest possible value for a totally real number field of odd degree. Moreover, we determine which numbers are sums of integral squares in this field, and use this knowledge to construct a diagonal universal quadratic form in five variables.

$${{\mathbb {Q}}}$$-curves over odd degree number fields

By reformulating and extending results of Elkies, we prove some results on \({{\mathbb {Q}}}\)-curves over number fields of odd degree. We show that, over such fields, the only prime isogeny degrees \(\ell \) that a \({{\mathbb {Q}}}\)-curve without CM may have are those degrees that are already possible over \({{\mathbb {Q}}}\) itself (in particular, \(\ell \le 37\)), and we show the existence of a bound on the degrees of cyclic isogenies between \({{\mathbb {Q}}}\)-curves depending only on the degree of the field. We also prove that the only possible torsion groups of \({{\mathbb {Q}}}\)-curves over number fields of degree not divisible by a prime \(\ell \le 7\) are the 15 groups that appear as torsion groups of elliptic curves over \({{\mathbb {Q}}}\). Complementing these theoretical results, we give an algorithm for establishing whether any given elliptic curve E is a \({{\mathbb {Q}}}\)-curve, that involves working only over \({{\mathbb {Q}}}(j(E))\).

Uniformity in Mordell–Lang for curves

Consider a smooth, geometrically irreducible, projective curve of genus $g\ge 2$ defined over a number field of degree $d \ge 1$. It has at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of $g$, $d$ and the Mordell–Weil rank of the curve's Jacobian, thereby answering in the affirmative a question of Mazur. In addition we obtain uniform bounds, in $g$ and $d$, for the number of geometric torsion points of the Jacobian which lie in the image of an Abel–Jacobi map. Both estimates generalize our previous work for one-parameter families. Our proof uses Vojta's approach to the Mordell Conjecture, and the key new ingredient is the generalization of a height inequality due to the second- and third-named authors.

Residual Galois representations of elliptic curves with image contained in the normaliser of a nonsplit Cartan

It is known that if $p>37$ is a prime number and $E/\mathbb{Q}$ is an elliptic curve without complex multiplication, then the image of the mod $p$ Galois representation $$ \bar{\rho}_{E,p}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}(E[p]) $$ of $E$ is either the whole of $\operatorname{GL}(E[p])$, or is \emph{contained} in the normaliser of a non-split Cartan subgroup of $\operatorname{GL}(E[p])$. In this paper, we show that when $p>1.4\times 10^7$, the image of $\bar{\rho}_{E,p}$ is either $\operatorname{GL}(E[p])$, or the \emph{full} normaliser of a non-split Cartan subgroup. We use this to show the following result, partially settling a question of Najman. For $d\geq 1$, let $I(d)$ denote the set of primes $p$ for which there exists an elliptic curve defined over $\mathbb{Q}$ and without complex multiplication admitting a degree $p$ isogeny defined over a number field of degree $\leq d$. We show that, for $d\geq 1.4\times 10^7$, we have $$ I(d)=\{p\text{ prime}:p\leq d-1\}. $$