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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
$\text{Gal}(K_{\infty }/K)$
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
$\text{Gal}(K/\mathbb{Q})$
on the
$p$
-class tower group of
$K$
plays a crucial role; however, the presence of units of infinite order in the ground field significantly complicates the possibilities for the groups that can occur. We also sharpen our results in the imaginary quadratic field case by removing a certain hypothesis, using ideas of Boston and Wood. In the appendix, we show how the probabilities introduced for finite
$p$
-groups can be extended in a consistent way to the infinite pro-
$p$
groups which can arise in both the real and imaginary quadratic settings.
For
$\unicode[STIX]{x1D6FC}$
an algebraic integer of any degree
$n\geqslant 2$
, it is known that the discriminants of the orders
$\mathbb{Z}[\unicode[STIX]{x1D6FC}^{k}]$
go to infinity as
$k$
goes to infinity. We give a short proof of this result.
We give an algebraic proof of a class number formula for dihedral extensions of number fields of degree 2q, where q is any odd integer. Our formula expresses the ratio of class numbers as a ratio of orders of cohomology groups of units and allows one to recover similar formulas which have appeared in the literature. As a corollary of our main result, we obtain explicit bounds on the (finitely many) possible values which can occur as ratio of class numbers in dihedral extensions. Such bounds are obtained by arithmetic means, without resorting to deep integral representation theory.
Let K be an imaginary quadratic field different from
$\open{Q}(\sqrt {-1})$
and
$\open{Q}(\sqrt {-3})$
. For a positive integer N, let KN be the ray class field of K modulo
$N {\cal O}_K$
. By using the congruence subgroup ± Γ1(N) of SL2(ℤ), we construct an extended form class group whose operation is basically the Dirichlet composition, and explicitly show that this group is isomorphic to the Galois group Gal(KN/K). We also present an algorithm to find all distinct form classes and show how to multiply two form classes. As an application, we describe Gal(KNab/K) in terms of these extended form class groups for which KNab is the maximal abelian extension of K unramified outside prime ideals dividing
$N{\cal O}_K$
.
The aim of this paper is to study the group of elliptic units of a cyclic extension
$L$
of an imaginary quadratic field
$K$
such that the degree
$[L:K]$
is a power of an odd prime
$p$
. We construct an explicit root of the usual top generator of this group, and we use it to obtain an annihilation result of the
$p$
-Sylow subgroup of the ideal class group of
$L$
.
We collect some statements regarding equivalence of the parities of various class numbers and signature ranks of units in prime power cyclotomic fields. We correct some misstatements in the literature regarding these parities by providing an example of a prime cyclotomic field where the signature rank of the units and the signature rank of the circular units are not equal.
A number field K with a ring of integers 𝒪K is called a Pólya field, if the 𝒪K-module of integer-valued polynomials on 𝒪K has a regular basis, or equivalently all its Bhargava factorial ideals are principal [1]. We generalize Leriche's criterion [8] for Pólya-ness of Galois closures of pure cubic fields, to general S3-extensions of ℚ. Also, we prove for a real (resp. imaginary) Pólya S3-extension L of ℚ, at most four (resp. three) primes can be ramified. Moreover, depending on the solvability of unit norm equation over the quadratic subfield of L, we determine when these sharp upper bounds can occur.
In this paper we investigate the moments and the distribution of
$L(1,\unicode[STIX]{x1D712}_{D})$
, where
$\unicode[STIX]{x1D712}_{D}$
varies over quadratic characters associated to square-free polynomials
$D$
of degree
$n$
over
$\mathbb{F}_{q}$
, as
$n\rightarrow \infty$
. Our first result gives asymptotic formulas for the complex moments of
$L(1,\unicode[STIX]{x1D712}_{D})$
in a large uniform range. Previously, only the first moment has been computed due to the work of Andrade and Jung. Using our asymptotic formulas together with the saddle-point method, we show that the distribution function of
$L(1,\unicode[STIX]{x1D712}_{D})$
is very close to that of a corresponding probabilistic model. In particular, we uncover an interesting feature in the distribution of large (and small) values of
$L(1,\unicode[STIX]{x1D712}_{D})$
, which is not present in the number field setting. We also obtain
$\unicode[STIX]{x1D6FA}$
-results for the extreme values of
$L(1,\unicode[STIX]{x1D712}_{D})$
, which we conjecture to be the best possible. Specializing
$n=2g+1$
and making use of one case of Artin’s class number formula, we obtain similar results for the class number
$h_{D}$
associated to
$\mathbb{F}_{q}(T)[\sqrt{D}]$
. Similarly, specializing to
$n=2g+2$
we can appeal to the second case of Artin’s class number formula and deduce analogous results for
$h_{D}R_{D}$
, where
$R_{D}$
is the regulator of
$\mathbb{F}_{q}(T)[\sqrt{D}]$
.
Let
$p\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}4$
be a prime number. We use a number field variant of Vinogradov’s method to prove density results about the following four arithmetic invariants: (i)
$16$
-rank of the class group
$\text{Cl}(-4p)$
of the imaginary quadratic number field
$\mathbb{Q}(\sqrt{-4p})$
; (ii)
$8$
-rank of the ordinary class group
$\text{Cl}(8p)$
of the real quadratic field
$\mathbb{Q}(\sqrt{8p})$
; (iii) the solvability of the negative Pell equation
$x^{2}-2py^{2}=-1$
over the integers; (iv)
$2$
-part of the Tate–Šafarevič group
$\unicode[STIX]{x0428}(E_{p})$
of the congruent number elliptic curve
$E_{p}:y^{2}=x^{3}-p^{2}x$
. Our results are conditional on a standard conjecture about short character sums.
Let K be a totally real number field of degree r. Let K∞ denote the cyclotomic -extension of K, and let L∞ be a finite extension of K∞, abelian over K. The goal of this paper is to compare the characteristic ideal of the χ-quotient of the projective limit of the narrow class groups to the χ-quotient of the projective limit of the rth exterior power of totally positive units modulo a subgroup of Rubin–Stark units, for some
$\overline{\mathbb{Q}_{2}}$
-irreducible characters χ of Gal(L∞/K∞).
We construct a random model to study the distribution of class numbers in special families of real quadratic fields
${\open Q}(\sqrt d )$
arising from continued fractions. These families are obtained by considering continued fraction expansions of the form
$\sqrt {D(n)} = [f(n),\overline {u_1,u_2, \ldots ,u_{s-1} ,2f(n)]} $
with fixed coefficients u1, …, us−1 and generalize well-known families such as Chowla's 4n2 + 1, for which analogous results were recently proved by Dahl and Lamzouri [‘The distribution of class numbers in a special family of real quadratic fields’, Trans. Amer. Math. Soc. (2018), 6331–6356].
Let
$k$
be an imaginary quadratic field with
$\operatorname{Cl}_{2}(k)\simeq V_{4}$
. It is known that the length of the Hilbert
$2$
-class field tower is at least
$2$
. Gerth (On 2-class field towers for quadratic number fields with
$2$
-class group of type
$(2,2)$
, Glasgow Math. J. 40(1) (1998), 63–69) calculated the density of
$k$
where the length of the tower is
$1$
; that is, the maximal unramified
$2$
-extension is a
$V_{4}$
-extension. In this paper, we shall extend this result for generalized quaternion, dihedral, and semidihedral extensions of small degrees.
For any odd prime
$\ell$
, let
$h_{\ell }(-d)$
denote the
$\ell$
-part of the class number of the imaginary quadratic field
$\mathbb{Q}(\sqrt{-d})$
. Nontrivial pointwise upper bounds are known only for
$\ell =3$
; nontrivial upper bounds for averages of
$h_{\ell }(-d)$
have previously been known only for
$\ell =3,5$
. In this paper we prove nontrivial upper bounds for the average of
$h_{\ell }(-d)$
for all primes
$\ell \geqslant 7$
, as well as nontrivial upper bounds for certain higher moments for all primes
$\ell \geqslant 3$
.
Boston, Bush and Hajir have developed heuristics, extending the Cohen–Lenstra heuristics, that conjecture the distribution of the Galois groups of the maximal unramified pro-
$p$
extensions of imaginary quadratic number fields for
$p$
an odd prime. In this paper, we find the moments of their proposed distribution, and further prove there is a unique distribution with those moments. Further, we show that in the function field analog, for imaginary quadratic extensions of
$\mathbb{F}_{q}(t)$
, the Galois groups of the maximal unramified pro-
$p$
extensions, as
$q\rightarrow \infty$
, have the moments predicted by the Boston, Bush and Hajir heuristics. In fact, we determine the moments of the Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic function fields, leading to a conjecture on Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic number fields.
We develop a theory of commensurability of groups, of rings, and of modules. It allows us, in certain cases, to compare sizes of automorphism groups of modules, even when those are infinite. This work is motivated by the Cohen–Lenstra heuristics on class groups.
In this paper we describe how to compute smallest monic polynomials that define a given number field
$\mathbb{K}$
. We make use of the one-to-one correspondence between monic defining polynomials of
$\mathbb{K}$
and algebraic integers that generate
$\mathbb{K}$
. Thus, a smallest polynomial corresponds to a vector in the lattice of integers of
$\mathbb{K}$
and this vector is short in some sense. The main idea is to consider weighted coordinates for the vectors of the lattice of integers of
$\mathbb{K}$
. This allows us to find the desired polynomial by enumerating short vectors in these weighted lattices. In the context of the subexponential algorithm of Biasse and Fieker for computing class groups, this algorithm can be used as a precomputation step that speeds up the rest of the computation. It also widens the applicability of their faster conditional method, which requires a defining polynomial of small height, to a much larger set of number field descriptions.
In this paper, we present novel algorithms for finding small relations and ideal factorizations in the ideal class group of an order in an imaginary quadratic field, where both the norms of the prime ideals and the size of the coefficients involved are bounded. We show how our methods can be used to improve the computation of large-degree isogenies and endomorphism rings of elliptic curves defined over finite fields. For these problems, we obtain improved heuristic complexity results in almost all cases and significantly improved performance in practice. The speed-up is especially high in situations where the ideal class group can be computed in advance.
Using the elements of order four in the narrow ideal class group, we construct generators of the maximal elementary
$2$
-class group of real quadratic number fields with even discriminant which is a sum of two squares and with fundamental unit of positive norm. We then give a characterization of when two of these generators are equal in the narrow sense in terms of norms of Gaussian integers.
In previous work, Ohno conjectured, and Nakagawa proved, relations between the counting functions of certain cubic fields. These relations may be viewed as complements to the Scholz reflection principle, and Ohno and Nakagawa deduced them as consequences of ‘extra functional equations’ involving the Shintani zeta functions associated to the prehomogeneous vector space of binary cubic forms. In the present paper, we generalize their result by proving a similar identity relating certain degree-
$\ell$
fields to Galois groups
$D_{\ell }$
and
$F_{\ell }$
, respectively, for any odd prime
$\ell$
; in particular, we give another proof of the Ohno–Nakagawa relation without appealing to binary cubic forms.
The discriminant of a trinomial of the form
$x^{n}\pm \,x^{m}\pm \,1$
has the form
$\pm n^{n}\pm (n-m)^{n-m}m^{m}$
if
$n$
and
$m$
are relatively prime. We investigate when these discriminants have nontrivial square factors. We explain various unlikely-seeming parametric families of square factors of these discriminant values: for example, when
$n$
is congruent to 2 (mod 6) we have that
$((n^{2}-n+1)/3)^{2}$
always divides
$n^{n}-(n-1)^{n-1}$
. In addition, we discover many other square factors of these discriminants that do not fit into these parametric families. The set of primes whose squares can divide these sporadic values as
$n$
varies seems to be independent of
$m$
, and this set can be seen as a generalization of the Wieferich primes, those primes
$p$
such that
$2^{p}$
is congruent to 2 (mod
$p^{2}$
). We provide heuristics for the density of these sporadic primes and the density of squarefree values of these trinomial discriminants.