The modern theory of class field towers has its origins in the study of the p-class field tower over a quadratic imaginary number field, so it is fitting that this problem be the first in the discipline to be nearing a solution. We survey the state of the subject and present a new cohomological condition for a quadratic imaginary number field to have an infinite p-class field tower (for p odd). Under an additional hypothesis, we refine this to a necessary and sufficient condition and describe an algorithm for evaluating this condition for a given quadratic imaginary number field.
Let $d$ be a positive square-free integer. In this paper we shall investigate the structure of the $2$-class group of the cyclotomic $\mathbb{Z}_2$-extension of the imaginary biquadratic number field $\mathbb{Q}(\sqrt{d},\sqrt{-1})$. Furthermore, we deduce the structure of the $2$-class group of cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}(\sqrt{-d})$.
The $p$-group generation algorithm is used to verify that the Hilbert $3$-class field tower has length $3$ for certain imaginary quadratic fields $K$ with $3$-class group $\mathrm{Cl}_3(K) \cong [3,3]$. Our results provide the first examples of finite $p$-class towers of length $> 2$ for an odd prime $p$.
We formulate and for the most part prove a conjecture in the style of Mazur-Greenberg for the nonvanishing of central values of Rankin-Selberg $L$-functions attached to elliptic curves in abelian extensions of imaginary quadratic fields. This in particular generalizes the theorem of Rohrlich on $L$-functions of elliptic curves in cyclotomic towers to the setting of abelian extensions of imaginary quadratic fields, corresponding to families of degree-four $L$-functions given by $\operatorname{GL}(2)\times\operatorname{GL}(2)$ Rankin-Selberg $L$-functions. It also generalizes the theorems of Roh…
Cohen and Lenstra have given a heuristic which, for a fixed odd prime $p$, leads to many interesting predictions about the distribution of $p$-class groups of imaginary quadratic fields. We extend the Cohen-Lenstra heuristic to a non-abelian setting by considering, for each imaginary quadratic field $K$, the Galois group of the $p$-class tower of $K$, i.e. $G_K:=\mathrm{Gal}(K_\infty/K)$ where $K_\infty$ is the maximal unramified $p$-extension of $K$. By class field theory, the maximal abelian quotient of $G_K$ is isomorphic to the $p$-class group of $K$. For integers $c\geq 1$, we give a heur…
Inspired by the Odlyzko root discriminant and Golod--Shafarevich $p$-group bounds, Martinet (1978) asked whether an imaginary quadratic number field $K/\mathbb{Q}$ must always have an infinite Hilbert $2$-class field tower when the class group of $K$ has $2$-rank $4$, or equivalently when the discriminant of $K$ has $5$ prime factors. No negative results are known. Benjamin (2001, 2002) and Sueyoshi (2004, 2009, 2010) systematically established infinite $2$-towers for many $K$ in question, by casework on the associated Rédei matrices. Others, notably Mouhib (2010), have also made progress, but…
We classify all complex quadratic number fields with 2-class group of type (2,2^m) whose Hilbert 2-class fields have class groups of 2-rank equal to 2. These fields all have 2-class field tower of length 2. We still don't know examples of fields with 2-class field tower of length 3, but the smallest candidate is the field with discriminant -1015.
For each odd prime p>=5, there exist finite p-groups G with derived quotient G/D(G)=C(p)xC(p) and nearly constant transfer kernel type k(G)=(1,2,...,2) having two fixed points. It is proved that, for p=7, this type k(G) with the simplest possible case of logarithmic abelian quotient invariants t(G)=(11111,111,21,21,21,21,21,21) of the eight maximal subgroups is realized by exactly 98 non-metabelian Schur sigma-groups S of order 7^11 with fixed derived length dl(S)=3 and metabelianizations S/D(D(S)) of order 7^7. For p=5, the type k(G) with t(G)=(2111,111,21,21,21,21) leads to infinitely many n…
Let $p$ be an odd prime. For a number field $K$, we let $K_\infty$ be the maximal unramified pro-$p$ extension of $K$; we call the group $\mathrm{Gal}(K_\infty/K)$ the $p$-class tower group of $K$. In a previous work, as a non-abelian generalization of the work of Cohen and Lenstra on ideal class groups, we studied how likely it is that a given finite $p$-group occurs as the $p$-class tower group of an imaginary quadratic field. Here we do the same for an arbitrary real quadratic field $K$ as base. As before, the action of $\mathrm{Gal}(K/\mathbb{Q})$ on the $p$-class tower group of $K$ plays …
The $p$-group generation algorithm from computational group theory is used to obtain information about large quotients of the pro-2 group $G = \text{Gal} (k^{nr,2}/k)$ for $k = \mathbb{Q}(\sqrt{d})$ with $d = -445, -1015, -1595, -2379$. In each case we are able to narrow the identity of $G$ down to one of a finite number of explicitly given finite groups. From this follow several results regarding the corresponding 2-class tower. This is a revised version of ANT-0302.
All current techniques for showing that a number field has an infinite p-class field tower depend on one of various forms of the Golod-Shafarevich inequality. Such techniques can also be used to restrict the types of p-groups which can occur as Galois groups of finite p-class field towers. In the case that the base field is a quadratic imaginary number field, the theory culminates in showing that a finite such group must be of one of three possible presentation types. By keeping track of the error terms arising in standard proofs of Golod-Shafarevich type inequalities, we prove a Golod-Shafare…
In this article we classify the complex quadratic number fields k with 2-class group of type (2,2,2) whose Hilbert 2-class fields have a 2-class group of rank 2, and then determine the length of their 2-class field towers.
The structure of the Galois group of the maximal unramified p-extension of an imaginary quadratic field is restricted in various ways. In this paper we construct a family of finite 3-groups satisfying these restrictions. We prove several results about this family and characterize them as finite extensions of certain quotients of a Sylow pro-3 subgroup of SL_2(Z_3). We verify that the first group in the family does indeed arise as such a Galois group and provide a small amount of evidence that this may hold for the other members. If this were the case then it would imply that there is no upper …
We establish large sieve inequalities for power moduli in imaginary quadratic number fields, extending earlier work of Baier and Bansal for the Gaussian field.
We show by adopting Schertz's argument with the Siegel-Ramachandra invariant that singular values of certain quotients of the $Δ$-function generate ring class fields over imaginary quadratic fields.
The families of simplest cubic, simplest quartic and simplest sextic fields and the related Thue equations are well known. The family of simplest cubic Thue equations was already studied in the relative case, over imaginary quadratic fields. In the present paper we give a similar extension of simplest quartic and simplest sextic Thue equations over imaginary quadratic fields. We explicitly give the solutions of these infinite parametric families of Thue equations over arbitrary imaginary quadratic fields.
A formula for the sum of quadratic residues modulus a prime $p=4n-1$ is studied. We relate some terms on this formula with roots of quadratics and provide an exhaustive analysis of new concepts based on these roots. A number of formulas for the sum of the quadratic residues are obtained. We finalize the paper by obtaining several identities involving $h(-p)$ the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p}).$
A positive definite Hermitian lattice is said to be 2-universal if it represents all positive definite binary Hermitian lattices. We find all 2-universal ternary and quaternary Hermitian lattices over imaginary quadratic number fields.
The goal is to obtain an asymptotic formula for the number of quadratic extensions with bounded discriminant of a some quadratic number field with odd class number. This extends an already known result for Q.
We extend formulae of Mertens and Mirsky on the asymptotic behaviour of the standard Euler function to the Euler functions of principal rings of integers of imaginary quadratic number fields, giving versions in angular sector and with congruences.
In this paper we extend methods of Rubin to prove the Gras conjecture for abelian extensions of a given imaginary quadratic field k and prime numbers p which divide the number of roots of unity in k.
We describe an efficient algorithm to calculate generators of power integral bases in composites of totally real fields with imaginary quadratic fields. We show that the calculation can be reduced to solving index form equations in the original totally real fields. We illustrate our method by investigating monogenity in the infinite parametric family of imaginary quadratic extensions of the simplest quartic fields.
We classify all quadratic imaginary number fields that have a Euclidean ideal class. There are seven of them, they are of class number at most two, and in each case the unique class that generates the class-group is moreover norm-Euclidean.
A formula for the class number $h$ of the imaginary quadratic field $Q(\sqrt{-p}$ is obtained by counting on a specific way the quadratic residues of a prime number of the form $p=4n-1.$ Formulas for the sum of the quadratic residues are also found.
We construct an Euler system attached to a weight 2 modular form twisted by a Groessencharacter of an imaginary quadratic field, and apply this to bounding Selmer groups.
We study an analog of Serre's conjecture over imaginary quadratic fields. In particular, we ask whether the weight recipe of Buzzard, Diamond and Jarvis will hold in this setting. Using a program written by the author, we provide computational evidence that this is in fact the case. In order to justify the method used in the program, we prove that a modular symbols method will work for arbitrary weights over imaginary quadratic fields.
In this paper, we study the moments of central values of Hecke $L$-functions associated with quadratic characters in $\mathbb{Q}(i)$ and $\mathbb{Q}(ω)$ with $ω= exp(2πi/3)$ and establish some quantitative non-vanishing result for these $L$-values.
Let p be an odd prime number. In this paper, we show existence of certain infinite families of imaginary quadratic fields in which p splits and whose Iwasawa λ-invariant of the cyclotomic Zp-extension is equal to 1.
We quantify a recent theorem of Wiles on class numbers of imaginary quadratic fields by proving an estimate for the number of negative fundamental discriminants down to -X whose class numbers are indivisible by a given prime and whose imaginary quadratic fields satisfy any given set of local conditions. This estimate matches the best results in the direction of the Cohen-Lenstra heuristics for the number of imaginary quadratic fields with class number indivisible by a given prime. This general result is applied to study rank 0 twists of certain elliptic curves.
Let $K$ be a non-cylotomic imaginary quadratic field of class number 1 and $E/K$ is an elliptic curve with $E(K)[2]\simeq \mathbb{Z}_1.$ We determine the odd-order torsion groups that can arise as $E(L)_{\text{tor}}$ where $L$ is a quadratic extension of $K.$
In this paper, we prove a one level density result for the low-lying zeros of quadratic Hecke $L$-functions of imaginary quadratic number fields of class number one. As a corollary, we deduce, essentially, that at least $(19-\cot (1/4))/16 = 94.27... \%$ of the $L$-functions under consideration do not vanish at $1/2$.
Let $k$ be an imaginary quadratic number field, and $F/k$ a finite abelian extension of Galois group $G$. We show that a Gross conjecture concerning the leading terms of Artin $L$-series holds for $F/k$ and all rational primes which are split in $k$ and which do not divide $6$.
We present an adaptation of Voronoi theory for imaginary quadratic number fields of class number greater than 1. This includes a characterisation of extreme Hermitian forms which is analogous to the classic characterisation of extreme quadratic forms as well as a version of Voronoi's famous algorithm which may be used to enumerate all perfect Hermitian forms for a given imaginary quadratic number field in dimensions 2 and 3. We also present an application of the algorithm which allows to determine generators of the general linear group of an $Ø_K$-lattice.
In this paper, we will talk about the titled elliptic curve defined over imaginary quadratic fields such as $\mathbb{Q}(\sqrt{-q})$, where $q$ is congruent to 3 modulo 8 and $(p,q)=1$.
The purpose of this article is to give an overview of the series of papers [BK1], [BK2] concerning the $p$-adic Beilinson conjecture of motives associated to Hecke characters of an imaginary quadratic field $K$, for a prime $p$ which splits in $K$.
We prove a non-minimal modularity lifting theorem for ordinary Galois representations over imaginary quadratic fields, conditional on a local-global compatibility conjecture for ordinary torsion classes.
In this work, we compute the perfect forms for all imaginary quadratic fields of absolute discriminant up to $5000$ and study the number and types of the polytopes that arise. We prove a bound on the combinatorial types of polytopes that can arise regardless of discriminant and give a volumetric argument for a lower bound on the number of perfect forms as well as a heuristic for a better lower bound for imaginary quadratic fields of sufficiently large absolute discriminant.
We extend the Cohen-Lenstra heuristics to the setting of ray class groups of imaginary quadratic number fields, viewed as exact sequences of Galois modules. By asymptotically estimating the mixed moments governing the distribution of a cohomology map, we prove these conjectures in the case of $4$-ranks.
For a given odd integer $n>1$, we provide some families of imaginary quadratic number fields of the form $\mathbb{Q}(\sqrt{x^2-t^n})$ whose ideal class group has a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z}$.
In this paper we continue the study of codes over imaginary quadratic fields and their weight enumerators and theta functions. We present new examples of non-equivalent codes over rings of characteristic $p=2$ and $p=5$ which have the same theta functions. We also look at a generalization of codes over imaginary quadratic fields, providing examples of non-equivalent pairs with the same theta function for $p=3$ and $p=5$.
We extend to $p=2$ and $p=3$ the result of Gillard and Schneps which says that the $μ$-invariant of Katz $p$-adic $L$ functions attached to imaginary quadratic fields is zero for $p>3$. The arithmetic interpretation of this fact is given in the introduction.
We give an explicit construct of a harmonic weak Maass form $F_Θ$ that is a "lift" of $Θ^3$, where $Θ$ is the classical Jacobi theta function. Just as the Fourier coefficients of $Θ^3$ are related to class numbers of imaginary quadratic fields, the Fourier coefficients of the "holomorphic part" of $F_Θ$ are associated to class numbers of real quadratic fields.
We give criteria of the solvability of the diophantine equation $p=x^2+ny^2$ over some imaginary quadratic fields where $p$ is a prime element. The criteria becomes quite simple in special cases.
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 conjecture by H. Wada \cite{WA70} on the structure of ideal class groups.
We relate non-critical special values $p$-adic $L$-functions associated to algebraic Hecke characters of an imaginary quadratic number field with class number one to $p$-adic Coleman function called the $p$-adic Eisenstein-Kronecker series, when the conductors of the algebraic Hecke characters are divisible by $p$.
The main purpose of this paper is to extend results on isomorphism types of the abelianized absolute Galois group $\mathcal G_K^{ab}$, where $K$ denotes imaginary quadratic field. In particular, we will show that if the class number $h_K$ of an imaginary quadratic field $K$ different from $\mathbb Q(i)$, $\mathbb Q(\sqrt{-2})$ is a fixed prime number $p$ then there are only two isomorphism types of $\mathcal G_K^{ab}$ which could occur. For instance, this result implies that imaginary quadratic fields of the discriminant $D_K$ belonging to the set $\{-35, -51, -91, -115, -123, -187, -235,$ $ -…
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.
We develop a criterion for a normal basis, and prove that the singular values of certain Siegel functions form normal bases of ray class fields over imaginary quadratic fields other than $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. This result would be an answer for the Lang-Schertz conjecture on a ray class field with modulus generated by an integer ($\geq2$).
