Juergen Klueners
We prove that the number of quartic $S_4$--extensions of the rationals of given discriminant $d$ is $O_\eps(d^{1/2+\eps})$ for all $\eps>0$. For a prime number $p$ we derive that the dimension of the space of octahedral modular forms of weight 1 and conductor $p$ or $p^2$ is bounded above by $O(p^{1/2}\log(p)^2)$.
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We obtain upper bounds for the number of monic irreducible polynomials over $\mathbb Z$ of a fixed degree $n$ and a growing height $H$ for which the field generated by one of its roots has a given discriminant. We approach it via counting square-free parts of polynomial discriminants via two complementing approaches. In turn, this leads to a lower bound on the number of distinct discriminants of fields generated by roots of polynomials of degree $n$ and height at most $H$. We also give an upper bound for the number of trinomials of bounded height with given square-free part of the discriminant…
Let $N_n(X)$ denote the number of degree $n$ number fields with discriminant bounded by $X$. In this note, we improve the best known upper bounds on $N_n(X)$, finding that $N_n(X) = O(X^{ c (\log n)^2})$ for an explicit constant $c$.
Given a finite transitive permutation group $G$, we investigate number fields $F/\mathbb{Q}$ of Galois group $G$ whose discriminant is only divisible by small prime powers. This generalizes previous investigations of number fields with squarefree discriminant. In particular, we obtain a comprehensive result on number fields with cubefree discriminant. Our main tools are arithmetic-geometric, involving in particular an effective criterion on ramification in specializations of Galois covers.
We give an explicit upper bound for the number of equivalence classes of binary forms with rational integral coefficients of given degree and given discriminant, and with given splitting field. Further, we give an explicit upper bound for the number of irreducible binary forms with rational integral coefficients with given invariant order. Our bounds depend on as few parameters as possible. For instance, we show that the number of equivalence classes of irreducible binary forms with rational integral coefficients of degree r with given invariant order has an upper bound depending only on r. We…
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 multi-quadratic number fields.
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.
In the paper we obtain the asymtotic number of integral quadratic polynomials with bounded heights and discriminants as the upper bound of heights tends to infinity.
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.
In the paper we partially solved the problem of the distribution of the discriminants of integral polynomials in the cubic case. We proved the asymptotic formula for the number of integral cubic polynomials having bounded height and bounded discriminant. The main term of the obtained formula is the best possible for the considered range of polynomial discriminant values.
Let n be an odd number and F an imaginary quadratic field with odd discriminant. We show that there exists infinitely many cubic fields K such that the class number of K is divisible by n and the Galois closure of K contains F.
We give an upper bound on the number of extensions of a fixed number field of prescribed degree and discriminant less than X; these bounds improve on work of Schmidt. We also prove various related results, such as lower bounds for the number of extensions and upper bounds for Galois extensions.
There are consecutive cubic fields with large class numbers, when ordered by discriminant
研究显示，按判别式排序的三次域中可存在任意长的序列，其中所有域的类数均超过给定界限。
For every prime number p, we show the existence of a solvable number field L ramified only at {p and infinity whose p-Hilbert Class field tower is infinite.
In this paper, we obtain an asymptotic formula for the number of imaginary quadratic fields with prime discriminant and class number up to $H$, as $H\to \infty$. Previously, such an asymptotic was only known under the assumption of the Generalized Riemann Hypothesis, by the recent work of Holmin, Jones, Kurlberg, McLeman and Petersen.
Let $K$ be a number field and $d_K$ the absolute value of the discrimant of $K/\mathbb{Q}$. We consider the root discriminant $d_L^{\frac{1}{[L:\mathbb{Q}]}}$ of extensions $L/K$. We show that for any $N>0$ and any positive integer n, the set of length n solvable extensions of $K$ with root discriminant less than $N$ is finite. The result is motivated by the study of class field towers.
Counting wildly ramified quartic extensions with prescribed discriminant and Galois closure group
本文针对2进数域上具有指定判别式赋值与Galois闭包群的完全分歧四次扩张建立计数公式，并据此改进Serre质量公式，为数域算术统计提供工具。
Geometry-of-numbers over number fields and the density of ADE families of curves having squarefree discriminant
本文研究ADE型Dynkin图所产生曲线族中判别式无平方因子的密度，借助Thorne–Laga框架并将Bhargava的数的几何方法推广到数域，证明其全局密度等于各局部密度之积。
Class number of real quadratic fields of explicit discriminant
本文针对判别式平方根有已知连分数展开的一类实二次域展开研究，证明了除有限个可能例外外，其类数$h(D)$均大于1，推进了实二次域类数相关问题的研究。
We present an algorithm for computing discriminants and prime ideal decomposition in number fields. The algorithm is a refinement of a p-adic factorization method based on Newton polygons of higher order. The running-time and memory requirements of the algorithm appear to be very good: for a given prime number p, it computes the p-valuation of the discriminant and the factorization of p in a number field of degree 1000 in a few seconds, in a personal computer.
For a binary quadratic form $Q$, we consider the action of $\mathrm{SO}_Q$ on a two-dimensional vector space. This representation yields perhaps the simplest nontrivial example of a prehomogeneous vector space that is not irreducible, and of a coregular space whose underlying group is not semisimple. We show that the nondegenerate integer orbits of this representation are in natural bijection with orders in cubic fields having a fixed "lattice shape". Moreover, this correspondence is discriminant-preserving: the value of the invariant polynomial of an element in this representation agrees with…
Given an elliptic curve over a field of characteristic different from 2,3, its discriminant defines a $μ_{12}$-torsor over the field. In this paper, we give an explicit description of this $μ_{12}$-torsor in terms of the 3-torsion points and of the 4-torsion points on the given elliptic curve. %In addition, we show that such a description involves the Weil pairing in a certain way. As an application, we generalize a result of Coates on the 12-th root of the discriminant of an elliptic curve.
We study some divisibility properties related to the factors of the discriminant of the characteristic polynomial of generalized Fibonacci sequences $(G_n)_{n\ge0}$ defined by $G_0=0$, $G_1=1$ and $G_n=pG_{n-1}+qG_{n-2}$ for $n\ge2$, where $p,q$ are given integers. As corollaries, we give some divisibility properties on some well known sequences.
In this paper we describe how to use the algorithmic methods provided by Hunter and Pohst in order to give a complete classification of number fields of degree 8 and signature (2,3) with absolute discriminant less than a certain bound. The choice of this bound comes from the local corrections given by prime ideals to the lower estimates for discriminants obtained with the Odlyzko-Poitou-Serre method.
The problem of analyzing the number of number field extensions $L/K$ with bounded (relative) discriminant has been the subject of renewed interest in recent years, with significant advances made by Schmidt, Ellenberg-Venkatesh, Bhargava, Bhargava-Shankar-Wang, and others. In this paper, we use the geometry of numbers and invariant theory of finite groups, in a manner similar to Ellenberg and Venkatesh, to give an upper bound on the number of extensions $L/K$ with fixed degree, bounded relative discriminant, and specified Galois closure.
We classify all the number fields with signature (4,2), (6,1), (1,4) and (3,3) having discriminant lower than a specific upper bound. This completes the search for minimum discriminants for fields of degree 8 and continues it in the degree 9 case. We recall the theoretical tools and the algorithmic steps upon which our procedure is based, then we focus on the novelties due to a new implementation of this process on the computer algebra system PARI/GP; finally, we make some remarks about the final results, among which the existence of a number field with signature $(3,3)$ and small discriminant…
Let $K=\mathbb{Q}(\sqrt[n]{a})$ be an extension of degree $n$ of the field $\Q$ of rational numbers, where the integer $a$ is such that for each prime $p$ dividing $n$ either $p\nmid a$ or the highest power of $p$ dividing $a$ is coprime to $p$; this condition is clearly satisfied when $a, n$ are coprime or $a$ is squarefree. The paper contains an explicit formula for the discriminant of $K$ involving only the prime powers dividing $a,n$.
On the number of binary quadratic forms having discriminant $1-4p$, $p$ prime
本文研究了具有判别式1-4p的二元二次型数量，并提出了一种随机欧拉乘积模型来解释Hurwitz类数的分布。
We prove inequalities that compare the regulator of a number field with its absolute discriminant. We refine some ideas in Silverman's work in 1984 where such general inequalities are first proven. In order to prove our main theorems, we combine these refinements with the authors' recent results on bounding the product of heights of relative units in a number field extension.
We study some counting questions concerning products of positive integers $u_1, \ldots, u_n$ which form a non-zero perfect square, or more generally, a perfect $k$-th power. We obtain an asymptotic formula for the number of such integers of bounded size and in particular improve and generalize a result of D. I. Tolev (2011). We also use similar ideas to count the discriminants of number fields which are multiquadratic extensions of $\mathbb{Q}$ and improve and generalize a result of N. Rome (2017).
We define the notion of a weighted discriminant and corresponding counting function for number fields, and what it means for these counting functions to have a mass formula for a set of primes. We extend a result of Kedlaya to show that any proper counting function for a finite group $Γ$ has a mass formula for the set of primes not dividing $|Γ|$. We also prove that if $Γ$ is an $\ell$-group for some prime $\ell$, then there are only finitely many weighted discriminant counting functions for $Γ$-extensions of $\Q$ that have a mass formula for all primes. Finally, we enumerate all such counting…
We study the asymptotic count of dihedral quartic extensions over a fixed number field with bounded norm of the relative discriminant. The main term of this count (including a summation formula for the constant) can be found in the literature (see Cohen--Diaz y Diaz--Olivier for the statement without proof and see Klüners for a proof), but a power-saving for the error term has not been explicitly determined except in the case that the base field is $\mathbb{Q}$. In this article, we describe the argument for obtaining both the explicit main term and a power-saving error term for the number of $…
The Thue-Siegel method is applied to derive an upper bound for the number of solutions to Thue's equation $F(x,y) = 1$ where $F$ is a quartic diagonalizable form with negative discriminant. Computation is used in this argument to handle forms whose discriminant is small in absolute value. We then apply our results to bound the number of integral points on a certain family of elliptic curves.
In this note we give a brief survey of the most elementary criteria used to determine the surjectivity of the trace operator on the ring of integers of a number field $K$. Furthermore, we introduce an easy to state yet unknown surjectivity criterion depending only on the prime factorization of the degree $n$ of $K$ and on the squarefree part of the discriminant $d_K$.
In this paper we construct number fields ramified at 2 and 3 only, with various moderate-sized non-solvable Galois groups. We construct these fields by specializing three point covers, some from the literature and some new here. The specialization points come from solutions to a generalized Fermat equation. We conclude with a specific suggestion for further research and two general conjectures.
On the sparsity of non-diagonalisable integer matrices and matrices with a given discriminant
研究整数矩阵集合中特征多项式具有固定判别式d的矩阵数量，对d=0的情况改进了之前的结果，对d≠0的情况提供了上界估计。
In this paper we consider a tower of number fields $\cdots \supseteq K(1) \supseteq K(0) \supseteq K$ arising naturally from a continuous $p$-adic representation of $\mathrm{Gal}(\bar{\mathbb{Q}}/K)$, referred to as a $p$-adic Lie tower over $K$. A recent conjecture of Daqing Wan hypothesizes, for certain $p$-adic Lie towers of curves over $\mathbb{F}_p$, a stable (polynomial) growth formula for the genus. Here we prove the analogous result in characteristic zero, namely: the $p$-adic valuation of the discriminant of the extension $K(i)/K$ is given by a polynomial in $i,p^i$ for $i$ sufficient…
The Brezing-Weng method is a general framework to generate families of pairing-friendly elliptic curves. Here, we introduce an improvement which can be used to generate more curves with larger discriminants. Apart from the number of curves this yields, it provides an easy way to avoid endomorphism rings with small class number.
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 $ζ(2)^{-1}$. 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 discrim…
Consider tuples of separable algebras over a common local or global number field, related to each other by specified resolvent constructions. Under the assumption that all ramification is tame, simple group-theoretic calculations give best possible divisibility relations among the discriminants. We show that for many resolvent constructions, these divisibility relations continue to hold even in the presence of wild ramification.
We determine, asymptotically, the number of quintic fields having bounded discriminant. Specifically, we prove that the asymptotic number of quintic fields having absolute discriminant at most X is a constant times X. In contrast with the quartic case, we also show that a density of 100% of quintic fields, when ordered by absolute discriminant, have Galois closure with full Galois group $S_5$. The analogues of these results are also proven for orders in quintic fields. Finally, we give an interpretation of the various constants appearing in these theorems in terms of local masses of quintic ri…
Let $k$ be a number field and $O$ the ring of integers. In the previous paper [T06] we study the Dirichlet series counting discriminants of cubic algebras of $O$ and derive some density theorems on distributions of the discriminants by using the theory of zeta functions of prehomogeneous vector spaces. In this paper we consider these objects under imposing finite number of splitting conditions at non-archimedean places. Especially the explicit formulae of residues at $s=1$ and 5/6 under the conditions are given.
We consider the simplest quartic number fields $\mathbb{K}_m$ defined by the irreducible quartic polynomials $$x^4-mx^3-6x^2+mx+1,$$ where $m$ runs over the positive rational integers such that the odd part of $m^2+16$ is squarefree. In this paper, we study the common index divisor $I(\mathbb K_m)$ and determine explicitly the prime ideal decomposition for any prime number in any simplest quartic number fields $\mathbb{K}_m$. On the other hand, we establish an asymptotic formula for the number of simplest quartic fields with discriminant $\leq x$ and given index.
Let $n$ be a squarefree positive odd integer. We will show that there exist infinitely many imaginary quadratic number fields with discriminant divisible by $n$ and-at the same time-having an element of order $n$ in the class group. We then apply our result to prove that for a given squarefree positive odd integer $n$ there exist infinitely many $N$ such that $n$ divides both $N$ and $r(N)$, where $r(N)$ is the representation number of $N$ as sums of three squares.
For any integer n >= 2 and any nonnegative integers r,s with r+2s = n, we give an unconditional construction of infinitely many monic irreducible polynomials of degree n with integer coefficients having squarefree discriminant and exactly r real roots. These give rise to number fields of degree n, signature (r,s), Galois group S_n, and squarefree discriminant; we may also force the discriminant to be coprime to any given integer. The number of fields produced with discriminant in the range [-N, N] is at least c N^(1/(n-1)). A corollary is that for each n \geq 3, infinitely many quadratic numbe…
In this article, we establish an average behaviour of the normalised Fourier coefficients of the Hecke eigenforms supported at the integers represented by any primitive integral positive definite binary quadratic form of fixed discriminant $D < 0$ when the class number $h(D) = 1$. We also obtain a quantitative result for the number of sign changes of the sequence of the normalised Fourier coefficients $λ_{f}(n)$ of the Hecke eigenforms $f$ where $n$ is represented by any primitive integral positive definite binary quadratic form of fixed discriminant $D < 0$ when the class number $h(D) = 1$ in…
For a given form $F\in \mathbb Z[x_1,\dots,x_s]$ we apply the circle method in order to give an asymptotic estimate of the number of $m$-tuples $\mathbf x_1, \dots, \mathbf x_m$ spanning a linear space on the hypersurface $F(\mathbf x) = 0$ with the property that $\det ( (\mathbf x_1, \dots, \mathbf x_m)^t \, (\mathbf x_1, \dots, \mathbf x_m)) = b$. This allows us in some measure to count rational linear spaces on hypersurfaces whose underlying integer lattice is primitive.
We give an improvement of a result of J. Martinet on Stickelberger's congruences for the absolute norms of relative discriminants of number fields, by using classical arguments of class field theory.
Towards a classification of $p^2$-discriminant ideal twins over number fields
本文显式分类了数域上对于亏格为0的奇素数 $p$（即3或5）具有相同最小判别式理想的 $p^2$-同源椭圆曲线，并列出了 $\mathbb{Q}$ 上所有此类 $p^2$-同源判别式理想孪生。
We will give an explicit upper bound for the number of solutions to cubic inequality |F(x, y)| \leq h, where F(x, y) is a cubic binary form with integer coefficients and positive discriminant D. Our upper bound is independent of h, provided that h is smaller than D^{1/4}.
Let $K=\mathbb Q(θ)$ be an algebraic number field with $θ$ a root of an irreducible trinomial $f(x)=x^6+ax+b$ belonging to $\mathbb{Z}[x]$. In this paper, for each prime number $p$ we compute the highest power of $p$ dividing the discriminant of $K$ in terms of the prime powers dividing $a,~b$ and discriminant of $f(x)$. An explicit $p$-integral basis of $K$ is also given for each prime $p$ and a method is described to obtain an integral basis of $K$ from these $p$-integral bases which is illustrated with examples.
This paper was motivated by a recent paper by Krumm and Pollack investigating modulo-$p$ behaviour of quadratic twists with rational points of a given hyperelliptic curve, conditional on the abc-conjecture. We extend those results to twisted Galois covers with arbitrary Galois groups. The main point of this generalization is to interpret those results as statements about the sets of specializations of a given Galois cover under restrictions on the discriminant. In particular, we make a connection with existing heuristics about the distribution of discriminants of Galois extensions such as the …
We study the space of binary cubic and quadratic forms over the ring of integers $O$ of an algebraic number field $k$. By applying the theory of prehomogeneous vector spaces founded by M. Sato and T. Shintani, we can associate the zeta functions for these spaces. Applying these zeta functions, we derive some density theorems on the distributions of discriminants of cubic algebras of $O$. In the case $k$ is a quadratic field, we give a correction term as well as the main term. These are generalizations of Shintani's asymptotic formulae of the mean values of class numbers of binary cubic forms o…
Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants
本文结合 Fourier 分析与算术方法，研究 Galois 群含于 A_n 的 n 次多项式的定量 Hilbert 不可约性，并给出判别式为殆素数的首一多项式及数域数量的下界。
Squarefree values of polynomial discriminants II
本文确定具有无平方因子判别式及截出极大序的整系数二元型密度，证明算术 Bertini 猜想，并改进 $S_n$ 数域计数下界、修复 Nakagawa 的相关结果。
The minimal integral Mahler measure of a number field $K$, $M(\mathcal{O}_K)$, is the minimal Mahler measure of a non-torsion primitive element of $\mathcal{O}_K$. Upper and lower bounds, which depend on the discriminant, are known. We show that for cubics, the lower bounds are sharp with respect to its growth as a function of discriminant. We construct an algorithm to compute $M(\mathcal{O}_K)$ for all cubics with absolute value of the discriminant bounded by $N$.
Prime isogenous discriminant ideal twins
本文研究了素数等价判别理想双生的分类，证明在数域中，当p∈{2,3,5,7,13}时，存在有限个这样的双生。
Using Kummer theory for a finite extension K of \Qp(ζ)(where p is a prime number and ζa primitive p-th root of~1), we compute the ramification filtration and the discriminant of an arbitrary elementary abelian p-extension of K. We also develop the analogous Artin-Schreier theory for finite extensions of \Fp((π)) and derive similar results for their elementary abelian p-extensions.
For any distinct two primes $p_1\equiv p_2\equiv 3$ $(\text{mod }4)$, let $h(-p_1)$, $h(-p_2)$ and $h(p_1p_2)$ be the class numbers of the quadratic fields $\mathbb{Q}(\sqrt{-p_1})$, $\mathbb{Q}(\sqrt{-p_2})$ and $\mathbb{Q}(\sqrt{p_1p_2})$, respectively. Let $ω_{p_1p_2}:=(1+\sqrt{p_1p_2})/2$ and let $Ψ(ω_{p_1p_2})$ be the Hirzebruch sum of $ω_{p_1p_2}$. We show that $h(-p_1)h(-p_2)\equiv h(p_1p_2)Ψ(ω_{p_1p_2})/n$ $(\text{mod }8)$, where $n=6$ (respectively, $n=2$) if $\min\{p_1,p_2\}>3$ (respectively, otherwise). We also consider the real quadratic order with conductor $2$ in $\mathbb{Q}(\sqr…
Let $n\in\mathbb{N}$ be fixed, $Q>1$ be a real parameter and $\mathcal{P}_n(Q)$ denote the set of polynomials over $\mathbb{Z}$ of degree $n$ and height at most $Q$. In this paper we investigate the following counting problems regarding polynomials with small discriminant $D(P)$ and pairs of polynomials with small resultant $R(P_1,P_2)$: (i) given $0\le v\le n-1$ and a sufficiently large $Q$, estimate the number of polynomials $P\in\mathcal{P}_n(Q)$ such that $$0<|D(P)|\le Q^{2n-2-2v};$$ (ii) given $0\le w\le n$ and a sufficiently large $Q$, estimate the number of pairs of polynomials $P_1,P_2…
An infinite family of pairs of distinct quartic Galois CM-fields with the same discriminant and regulator
本文构造了无限多个具有相同判别式和调节数的虚二元四次域和虚循环四次域的配对，并给出了具有相同判别式、调节数和类数的示例。
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