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An explicit formula for the mean value of
$\vert L(1,\chi )\vert ^2$
is known, where
$\chi $
runs over all odd primitive Dirichlet characters of prime conductors p. Bounds on the relative class number of the cyclotomic field
${\mathbb Q}(\zeta _p)$
follow. Lately, the authors obtained that the mean value of
$\vert L(1,\chi )\vert ^2$
is asymptotic to
$\pi ^2/6$
, where
$\chi $
runs over all odd primitive Dirichlet characters of prime conductors
$p\equiv 1\ \ \pmod {2d}$
which are trivial on a subgroup H of odd order d of the multiplicative group
$({\mathbb Z}/p{\mathbb Z})^*$
, provided that
$d\ll \frac {\log p}{\log \log p}$
. Bounds on the relative class number of the subfield of degree
$\frac {p-1}{2d}$
of the cyclotomic field
${\mathbb Q}(\zeta _p)$
follow. Here, for a given integer
$d_0>1$
, we consider the same questions for the nonprimitive odd Dirichlet characters
$\chi '$
modulo
$d_0p$
induced by the odd primitive characters
$\chi $
modulo p. We obtain new estimates for Dedekind sums and deduce that the mean value of
$\vert L(1,\chi ')\vert ^2$
is asymptotic to
$\frac {\pi ^2}{6}\prod _{q\mid d_0}\left (1-\frac {1}{q^2}\right )$
, where
$\chi $
runs over all odd primitive Dirichlet characters of prime conductors p which are trivial on a subgroup H of odd order
$d\ll \frac {\log p}{\log \log p}$
. As a consequence, we improve the previous bounds on the relative class number of the subfield of degree
$\frac {p-1}{2d}$
of the cyclotomic field
${\mathbb Q}(\zeta _p)$
. Moreover, we give a method to obtain explicit formulas and use Mersenne primes to show that our restriction on d is essentially sharp.
Given a profinite group G of finite p-cohomological dimension and a pro-p quotient H of G by a closed normal subgroup N, we study the filtration on the Iwasawa cohomology of N by powers of the augmentation ideal in the group algebra of H. We show that the graded pieces are related to the cohomology of G via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups H, we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on H. We apply our study to prove lower bounds on the p-ranks of class groups of certain nonabelian extensions of
$\mathbb {Q}$
and to give a new proof of the vanishing of Massey triple products in Galois cohomology.
In this paper, we prove one divisibility of the Iwasawa–Greenberg main conjecture for the Rankin–Selberg product of a weight two cusp form and an ordinary complex multiplication form of higher weight, using congruences between Klingen Eisenstein series and cusp forms on $\mathrm {GU}(3,1)$, generalizing an earlier result of the third-named author to allow nonordinary cusp forms. The main result is a key input in the third-named author’s proof of Kobayashi’s $\pm $-main conjecture for supersingular elliptic curves. The new ingredient here is developing a semiordinary Hida theory along an appropriate smaller weight space and a study of the semiordinary Eisenstein family.
We apply a method inspired by Popa's intertwining-by-bimodules technique to investigate inner conjugacy of MASAs in graph $C^*$-algebras. First, we give a new proof of non-inner conjugacy of the diagonal MASA ${\mathcal {D}}_E$ to its non-trivial image under a quasi-free automorphism, where $E$ is a finite transitive graph. Changing graphs representing the algebras, this result applies to some non quasi-free automorphisms as well. Then, we exhibit a large class of MASAs in the Cuntz algebra ${\mathcal {O}}_n$ that are not inner conjugate to the diagonal ${\mathcal {D}}_n$.
Let $K={\mathbf {Q}}(\theta )$ be an algebraic number field with $\theta$ a root of an irreducible polynomial $x^5+ax+b\in {\mathbf {Z}}[x]$. In this paper, for every rational prime $p$, we provide necessary and sufficient conditions on $a,\,~b$ so that $p$ is a common index divisor of $K$. In particular, we give sufficient conditions on $a,\,~b$ for which $K$ is non-monogenic. We illustrate our results through examples.
We investigate a novel geometric Iwasawa theory for
${\mathbf Z}_p$
-extensions of function fields over a perfect field k of characteristic
$p>0$
by replacing the usual study of p-torsion in class groups with the study of p-torsion class group schemes. That is, if
$\cdots \to X_2 \to X_1 \to X_0$
is the tower of curves over k associated with a
${\mathbf Z}_p$
-extension of function fields totally ramified over a finite nonempty set of places, we investigate the growth of the p-torsion group scheme in the Jacobian of
$X_n$
as
$n\rightarrow \infty $
. By Dieudonné theory, this amounts to studying the first de Rham cohomology groups of
$X_n$
equipped with natural actions of Frobenius and of the Cartier operator V. We formulate and test a number of conjectures which predict striking regularity in the
$k[V]$
-module structure of the space
$M_n:=H^0(X_n, \Omega ^1_{X_n/k})$
of global regular differential forms as
$n\rightarrow \infty .$
For example, for each tower in a basic class of
${\mathbf Z}_p$
-towers, we conjecture that the dimension of the kernel of
$V^r$
on
$M_n$
is given by
$a_r p^{2n} + \lambda _r n + c_r(n)$
for all n sufficiently large, where
$a_r, \lambda _r$
are rational constants and
$c_r : {\mathbf Z}/m_r {\mathbf Z} \to {\mathbf Q}$
is a periodic function, depending on r and the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on
${\mathbf Z}_p$
-towers of curves, and we prove our conjectures in the case
$p=2$
and
$r=1$
.
Let K be a number field, let A be a finite-dimensional K-algebra, let
$\operatorname {\mathrm {J}}(A)$
denote the Jacobson radical of A and let
$\Lambda $
be an
$\mathcal {O}_{K}$
-order in A. Suppose that each simple component of the semisimple K-algebra
$A/{\operatorname {\mathrm {J}}(A)}$
is isomorphic to a matrix ring over a field. Under this hypothesis on A, we give an algorithm that, given two
$\Lambda $
-lattices X and Y, determines whether X and Y are isomorphic and, if so, computes an explicit isomorphism
$X \rightarrow Y$
. This algorithm reduces the problem to standard problems in computational algebra and algorithmic algebraic number theory in polynomial time. As an application, we give an algorithm for the following long-standing problem: Given a number field K, a positive integer n and two matrices
$A,B \in \mathrm {Mat}_{n}(\mathcal {O}_{K})$
, determine whether A and B are similar over
$\mathcal {O}_{K}$
, and if so, return a matrix
$C \in \mathrm {GL}_{n}(\mathcal {O}_{K})$
such that
$B= CAC^{-1}$
. We give explicit examples that show that the implementation of the latter algorithm for
$\mathcal {O}_{K}=\mathbb {Z}$
vastly outperforms implementations of all previous algorithms, as predicted by our complexity analysis.
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].
Let $p$ be a prime number. Kęstutis Česnavičius proved that for an abelian variety $A$ over a global field $K$, the $p$-Selmer group $\mathrm {Sel}_{p}(A/L)$ grows unboundedly when $L$ ranges over the $(\mathbb {Z}/p\mathbb {Z})$-extensions of $K$. Moreover, he raised a further problem: is $\dim _{\mathbb {F}_{p}} \text{III} (A/L) [p]$ also unbounded under the above conditions? In this paper, we give a positive answer to this problem in the case $p \neq \mathrm {char}\,K$. As an application, this result enables us to generalize the work of Clark, Sharif and Creutz on the growth of potential $\text{III}$ in cyclic extensions. We also answer a problem proposed by Lim and Murty concerning the growth of the fine Tate–Shafarevich groups.
Given a singular modulus
$j_0$
and a set of rational primes S, we study the problem of effectively determining the set of singular moduli j such that
$j-j_0$
is an S-unit. For every
$j_0 \neq 0$
, we provide an effective way of finding this set for infinitely many choices of S. The same is true if
$j_0=0$
and we assume the Generalised Riemann Hypothesis. Certain numerical experiments will also lead to the formulation of a “uniformity conjecture” for singular S-units.
In this paper, we obtain a precise formula for the one-level density of L-functions attached to non-Galois cubic Dedekind zeta functions. We find a secondary term which is unique to this context, in the sense that no lower-order term of this shape has appeared in previously studied families. The presence of this new term allows us to deduce an omega result for cubic field counting functions, under the assumption of the Generalised Riemann Hypothesis. We also investigate the associated L-functions Ratios Conjecture and find that it does not predict this new lower-order term. Taking into account the secondary term in Roberts’s conjecture, we refine the Ratios Conjecture to one which captures this new term. Finally, we show that any improvement in the exponent of the error term of the recent Bhargava–Taniguchi–Thorne cubic field counting estimate would imply that the best possible error term in the refined Ratios Conjecture is
$O_\varepsilon (X^{-\frac 13+\varepsilon })$
. This is in opposition with all previously studied families in which the expected error in the Ratios Conjecture prediction for the one-level density is
$O_\varepsilon (X^{-\frac 12+\varepsilon })$
.
Let
$m>1$
and
$\mathfrak {d} \neq 0$
be integers such that
$v_{p}(\mathfrak {d}) \neq m$
for any prime p. We construct a matrix
$A(\mathfrak {d})$
of size
$(m-1) \times (m-1)$
depending on only of
$\mathfrak {d}$
with the following property: For any tame
$ \mathbb {Z}/m \mathbb {Z}$
-number field K of discriminant
$\mathfrak {d}$
, the matrix
$A(\mathfrak {d})$
represents the Gram matrix of the integral trace-zero form of K. In particular, we have that the integral trace-zero form of tame cyclic number fields is determined by the degree and discriminant of the field. Furthermore, if in addition to the above hypotheses, we consider real number fields, then the shape is also determined by the degree and the discriminant.
A class of exotic
$_3F_2(1)$
-series is examined by integral representations, which enables the authors to present relatively easier proofs for a few remarkable formulae. By means of the linearization method, these
$_3F_2(1)$
-series are further extended with two integer parameters. A general summation theorem is explicitly established for these extended series, and several sample summation identities are highlighted as consequences.
By analogy with the trace of an algebraic integer
$\alpha $
with conjugates
$\alpha _1=\alpha , \ldots , \alpha _d$
, we define the G-measure
$ {\mathrm {G}} (\alpha )= \sum _{i=1}^d ( |\alpha _i| + 1/ | \alpha _i | )$
and the absolute
${\mathrm G}$
-measure
${\mathrm {g}}(\alpha )={\mathrm {G}}(\alpha )/d$
. We establish an analogue of the Schur–Siegel–Smyth trace problem for totally positive algebraic integers. Then we consider the case where
$\alpha $
has all its conjugates in a sector
$| \arg z | \leq \theta $
,
$0 < \theta < 90^{\circ }$
. We compute the greatest lower bound
$c(\theta )$
of the absolute G-measure of
$\alpha $
, for
$\alpha $
belonging to
$11$
consecutive subintervals of
$]0, 90 [$
. This phenomenon appears here for the first time, conforming to a conjecture of Rhin and Smyth on the nature of the function
$c(\theta )$
. All computations are done by the method of explicit auxiliary functions.
Let E be an elliptic curve with positive rank over a number field K and let p be an odd prime number. Let
$K_{\operatorname {cyc}}$
be the cyclotomic
$\mathbb {Z}_p$
-extension of K and
$K_n$
its nth layer. The Mordell–Weil rank of E is said to be constant in the cyclotomic tower of K if for all n, the rank of
$E(K_n)$
is equal to the rank of
$E(K)$
. We apply techniques in Iwasawa theory to obtain explicit conditions for the rank of an elliptic curve to be constant in this sense. We then indicate the potential applications to Hilbert’s tenth problem for number rings.
Let
$\alpha $
be a totally positive algebraic integer of degree d, with conjugates
$\alpha _1=\alpha , \alpha _2, \ldots , \alpha _d$
. The absolute
$S_k$
-measure of
$\alpha $
is defined by
$s_k(\alpha )= d^{-1} \sum _{i=1}^{d}\alpha _i^k$
. We compute the lower bounds
$\upsilon _k$
of
$s_k(\alpha )$
for each integer in the range
$2\leq k \leq 15$
and give a conjecture on the results for integers
$k>15$
. Then we derive the lower bounds of
$s_k(\alpha )$
for all real numbers
$k>2$
. Our computation is based on an improvement in the application of the LLL algorithm and analysis of the polynomials in the explicit auxiliary functions.
Using a recent breakthrough of Smith [18], we improve the results of Fouvry and Klüners [4, 5] on the solubility of the negative Pell equation. Let
$\mathcal {D}$
denote the set of positive squarefree integers having no prime factors congruent to
$3$
modulo
$4$
. Stevenhagen [19] conjectured that the density of d in
$\mathcal {D}$
such that the negative Pell equation
$x^2-dy^2=-1$
is solvable with
$x, y \in \mathbb {Z}$
is
$58.1\%$
, to the nearest tenth of a percent. By studying the distribution of the
$8$
-rank of narrow class groups
$\operatorname {\mathrm {Cl}}^+(d)$
of
$\mathbb {Q}(\sqrt {d})$
, we prove that the infimum of this density is at least
$53.8\%$
.
Let
$F_{2^n}$
be the Frobenius group of degree
$2^n$
and of order
$2^n ( 2^n-1)$
with
$n \ge 4$
. We show that if
$K/\mathbb {Q} $
is a Galois extension whose Galois group is isomorphic to
$F_{2^n}$
, then there are
$\dfrac {2^{n-1} +(-1)^n }{3}$
intermediate fields of
$K/\mathbb {Q} $
of degree
$4 (2^n-1)$
such that they are not conjugate over
$\mathbb {Q}$
but arithmetically equivalent over
$\mathbb {Q}$
. We also give an explicit method to construct these arithmetically equivalent fields.
The Euler–Mascheroni constant
$\gamma =0.5772\ldots \!$
is the
$K={\mathbb Q}$
example of an Euler–Kronecker constant
$\gamma _K$
of a number field
$K.$
In this note, we consider the size of the
$\gamma _q=\gamma _{K_q}$
for cyclotomic fields
$K_q:={\mathbb Q}(\zeta _q).$
Assuming the Elliott–Halberstam Conjecture (EH), we prove uniformly in Q that
In other words, under EH, the
$\gamma _q /\!\log q$
in these ranges converge to the one point distribution at
$1$
. This theorem refines and extends a previous result of Ford, Luca and Moree for prime
$q.$
The proof of this result is a straightforward modification of earlier work of Fouvry under the assumption of EH.
Iwasawa theory of elliptic curves over noncommutative
$GL(2)$
extension has been a fruitful area of research. Over such a noncommutative p-adic Lie extension, there exists a structure theorem providing the structure of the dual Selmer groups for elliptic curves in terms of reflexive ideals in the Iwasawa algebra. The central object of this article is to study Iwasawa theory over the
$PGL(2)$
extension and connect it with Iwasawa theory over the
$GL(2)$
extension, deriving consequences to the structure theorem when the reflexive ideal is the augmentation ideal of the center. We also show how the dual Selmer group over the
$GL(2)$
extension being torsion is related with that of the
$PGL(2)$
extension.