Let $\mathbf{Was}(\mathbb{K})$ denote the group of Washington's cyclotomic units of any abelian number field $\mathbb{K}$. If $\mathbb{K}$ coincides with its genus field in the narrow sense, we give a $Λ$-basis of $\lim\limits\_{\xleftarrow{}} \mathbf{Was}(\mathbb{K}\_k^{+})$ where $(\mathbb{K}\_k)\_{k \geqslant 0}$ denotes the cyclotomic $\mathbb{Z}\_p$-tower of $\mathbb{K}$ and $Λ$ denotes the Iwasawa's algebra. This results from a $\mathbb{Z} [1/2]$-basis of $\mathbf{Was}(\mathbb{K}) \otimes\_{\mathbb{Z}} \mathbb{Z} [1/2]$ that we give under the same hypothesis.
The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials. We formulate some basic properties of unitary cyclotomic polynomials and study how they are connected with cyclotomic, inclusion-exclusion and Kronecker polynomials. Further, we derive some related arithmetic function identities involving the unitary analog of the Dirichlet convolution.
We present a method to deal with the values of polynomials of type $P(z)=\prod_{d\in D}(1-z^d)^{j_d}$ on the unit circle. We use it to improve the known bounds on various measures of coefficients of cyclotomic and similar polynomials.
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.
We investigate the interrelationships of three notions of primary units in the local cyclotomic field of $p$-th roots of~1($p$ being an odd prime number), especially with reference to global units.
Denote by $ε$ a primitive root of $N^{th}$-unity. In this paper, we show that the unit cyclotomic multiple zeta values for $μ_N$ generate all the cyclotomic multiple zeta values for $μ_N$ in cases $N=2,3,4$. Moreover, the unit cyclotomic multiple zeta values for $μ_N$ can be written as $\mathbb{Q}$-linear combinations of $\left(ζ\binom{1}ε\right)^n, \left(ζ\binom{1}{ε^{-1}}\right)^n$ and lower depth terms in each weight $n$ in case of $N=2,3$ and $4$. By detailed analysis of the motivic Galois action, we compute the coefficients of $\left(ζ\binom{1}ε\right)^n, \left(ζ\binom{1}{ε^{-1}}\right)^n…
For a real abelian field and for an odd prime p splitting in the field, we study a map between the p-parts of the class group and of the quotient of units modulo Cyclotomic Units, respectively, along the cyclotomic Z_p-extension of the field. We determine the kernel and the cokernel of this map assuming Greenberg's conjecture on the vanishing of the lambda-invariant of the extension.
In this paper, we obtain a formula for the special value of Euler-Dirichlet $L$-function $L_E(s,χ)$ at $s=1$. This leads to another class number formula of $\mathbb{Q}(μ_{m})^{+}$, the maximal real subfield of $m$th cyclotomic field. From this formula, we construct a new type of cyclotomic units in $\mathbb{Q}(μ_{p^{n}})$, which implies a similar Kummer's relation between the ideal class number of $\mathbb{Q}(μ_{p^{n}})^{+}$ and the unit index.
Kurihara established a refinement of the minus-part of the Iwasawa main conjecture for totally real number fields using the higher Fitting ideals. In this paper, we study the higher Fitting ideals of the plus-part of the Iwasawa module associated to the cyclotomic Z_p-extension of Q(μ_p) for an odd prime number p by similar methods as in Kurihara's work. We define the higher cyclotomic ideals C_i, which are ideals of the Iwasawa algebra defined by the Kolyvagin derivatives of cyclotomic units, and prove that they give upper bounds of the higher Fitting ideals. Our result can be regarded as a r…
In this essay, we see how prime cyclotomic fields (cyclotomic fields obtained by adjoining a primitive p-th root of unity to Q, where p is an odd prime) can lead to elegant proofs of number theoretical concepts. We namely develop the notion of primary units in a cyclotomic field, demonstrate their equivalence to real units in this case, and show how this leads to a proof of a special case of Fermat's Last Theorem. We finally modernize Dirichlet's solution to Pell's Equation.
We prove the rank of the group of signatures of the circular units (hence also the full group of units) of ${\mathbb Q}( ζ_m)^+$ tends to infinity with $m$. We also show the signature rank of the units differs from its maximum possible value by a bounded amount for all the real subfields of the composite of an abelian field with finitely many odd prime-power cyclotomic towers. In particular, for any prime $p$ the signature rank of the units of ${\mathbb Q}( ζ_{p^n})^+$ differs from $φ(p^n)/2$ by an amount that is bounded independent of $n$. Finally, we show conditionally that for general cyclo…
We define a "period ring-valued beta function" and give a reciprocity law on its special values. The proof is based on some results of Rohrlich and Coleman concerning Fermat curves. We also have the following application. Stark's conjecture implies that the exponential of the derivatives at $s=0$ of partial zeta functions are algebraic numbers which satisfy a reciprocity law under certain conditions. It follows from Euler's formulas and properties of cyclotomic units when the base field is the rational number field. In this paper, we provide an alternative (and partial) proof by using the reci…
Let p > 2 be a prime. Let Q(zeta) be the p-cyclotomic field. Let pi be the prime ideal of Q(zeta) lying over p. This article aims to describe some pi-adic congruences characterizing the structure of the p-class group and of the unit group of the field Q(zeta). For the unit group, this article supplements the 1954 and 1956 papers of Denes on this topic. A complete summarizing of the results obtained follows in the Introduction section of the paper (pages 3 to 6). This new version of the article with the same title, submitted with reference math.NT/0407430 25 Jul 2004: - corrects several typing …
Let $\mathcal{A}_n$ be the $2$-part of the ideal class group of the $n$-th layer of the cyclotomic $\mathbb{Z}_2$-extension of a real quadratic number field $F$. The cardinality of $\mathcal{A}_n$ is related to the index of cyclotomic units in the full group of units. We present a method to study the latter index. As an application we show that the sequence of the $\mathcal{A}_n$'s stabilizes for the real fields $F=\mathbb{Q}(\sqrt{f})$ for any integer $0<f<10000$. Equivalently Greenberg's conjecture holds for those fields.
We describe Greenberg's pseudo-null conjecture, and prove a result describing conditions under which the pseudo-null conjecture for a number field $K$ implies the conjecture for finite extensions of $K$. We then apply the result to the cyclotomic $\mathbb{Z}_p$-tower above a cyclotomic field of prime roots of unity, verifying the conjecture for a large class of cyclotomic fields.
Let $n$ be an even positive integer with at most three distinct prime factors and let $\ze_n$ be a primitive $n$-th root of unity. In this study, we made an attempt to find the lowest-degree $0,1$-polynomial $f(x) \in \Q[x]$ having at least three terms such that $f(\ze_n)$ is a minimal vanishing sum of the $n$-th roots of unity.
Hilbert's Satz 90 tells us that for a given cyclic extension $K/k$, a unit of norm $1$ in $K$ can be written as a quotient of conjugate elements in $K$. For the extensions $\mathbb{Q}(ζ_p)/\mathbb{Q}$ with $p$ prime $> 3$, Newman proved a refinement of Hilbert's Satz 90 that gives a sufficient and necessary condition for which a unit of norm $1$ in $\mathbb{Q}(ζ_p)$ can be written as a quotient of conjugate units. In order to obtain this result, Newman proved a stronger result that gives a unique representation of units of norm $1$ as a product of a power of $\dfrac{1 - ζ_p^e}{1 - ζ_p}$ with a…
In this note, we investigate the p-adic behavior of Weil numbers in the cyclotomic $\mathbb Z\_p$-extension of $\mathbb Q(ζ\_p).$ We determlne the characteristic ideal of the quotient of semi-local units by Weil numbers in terms of the characteristic ideals of some classical modules that appear in Iwasawa Theory.
In the context of cyclotomic fields, it is still unknown whether there exist Euler systems other than the ones derived from cyclotomic units. Nevertheless, we first give an exposition on how norm-compatible units are generated by any Euler system, following work of Coates. Then we prove that the units obtained from Euler systems and the cyclotomic units generate the same $\mathbb{Z}_{p}$-module for any odd prime $p$. The techniques adopted for the Iwasawa theoretic proof in latter part of this article originated in Rubin's work on main conjectures of Iwasawa theory.
The Laplacian functional determinants for conformal scalars and coexact one-forms are evaluated in closed form on inhomogeneous lens spaces of certain orders, including all odd primes when the essential part of the expression is given, formally as a cyclotomic unit
The $k$-higher Mahler measure of a nonzero polynomial $P$ is the integral of $\log^k|P|$ on the unit circle. In this note, we consider Lehmer's question (which is a long-standing open problem for $k=1$) for $k>1$ and find some interesting formulae for 2- and 3-higher Mahler measure of cyclotomic polynomials.
We give a new proof of a conjecture of Darmon, Lauder and Rotger regarding the computation of the $\mathcal L$-invariant of the adjoint of a weight one modular form in terms of units and $p$-units. While in our previous work with Rotger the essential ingredient was the use of Galois deformations techniques following the computations of Bellaïche and Dimitrov, we propose a new approach exclusively using the properties of Beilinson--Flach classes. One of the key ingredients is the computation of a cyclotomic derivative of a cohomology class in the framework of Perrin-Riou theory, which can be se…
We prove that all elliptic curves defined over the cyclotomic $\mathbb{Z}_p$-extension of a real quadratic field are modular under the assumption that the algebraic part of the central value of a twisted $L$-function is a $p$-adic unit. Our result is a generalization of a result of X. Zhang, which is a real quadratic analogue of a result of Thorne.
Let z be a primitive fifth root of unity and let F be the cyclotomic field F=Q(z). Let O be the ring of integers. We compute the Voronoi polyhedron of binary Hermitian forms over F and classify GL_2(O)-conjugacy classes of perfect forms. The combinatorial data of this polyhedron can be used to compute the cohomology of the arithmetic group GL_2(O) and Hecke eigen forms.
We construct certain elements in the integral motivic cohomology group $H^3_{\cal M}(E \times E',\Q(2))_{\ZZ}$, where $E$ and $E'$ are elliptic curves over $\Q$. When $E$ is not isogenous to $E'$ these elements are analogous to `cyclotomic units' in real quadratic fields as they come from modular parametrisations of the elliptic curves. We then find an analogue of the class number formula for real quadratic fields. Finally we use the Beilinson conjectures for $E \times E'$ to deduce them for products of $n$ elliptic curves. A certain amount of this paper is expository in nature.
Following Hasse's example, various authors have been deriving divisibility properties of minus class numbers of cyclotomic fields by carefully examining the analytic class number formula. In this paper we will show how to generalize these results to CM-fields by using class field theory. Although we will only need some special cases, we have also decided to include a few results on Hasse's unit index for CM-fields as well, because it seems that our proofs are more direct than those given by Hasse.
Unit group computations are a cryptographic primitive for which one has a fast quantum algorithm, but the required number of qubits is $\tilde O(m^5)$. In this work we propose a modification of the algorithm for which the number of qubits is $\tilde O(m^2)$ in the case of cyclotomic fields. Moreover, under a recent conjecture on the size of the class group of $\mathbb{Q}(ζ_m + ζ_m^{-1})$, the quantum algorithms is much simpler because it is a hidden subgroup problem (HSP) algorithm rather than its error estimation counterpart: continuous hidden subgroup problem (CHSP). We also discuss the (min…
Let $ζ_k$ be a $k$-th primitive root of unity, $m\geqϕ(k)+1$ an integer and $Φ_k(X)\in\mathbb Z [X]$ the $k$-th cyclotomic polynomial. In this paper we show that the pair $(-m+ζ_k,\mathcal N)$ is a canonical number system, with $\mathcal N=\{0,1,\dots,|Φ_k(m)|\}$. Moreover we also discuss whether the two bases $-m+ζ_k$ and $-n+ζ_k$ are multiplicatively independent for positive integers $m$ and $n$ and $k$ fixed.
For each odd prime $p$, let $ζ_p$ denote a primitive $p$-th root of unity. In this paper, we study the determinants of some matrices with cyclotomic unit entries. For instance, we show that when $p\equiv 3\pmod4$ and $p>3$ the determinant of the matrix $\(\frac{1-ζ_p^{j^2k^2}}{1-ζ_p^{j^2}}\)_{1\le j,k\le (p-1)/2}$ can be written as $(-1)^{\frac{h(-p)+1}{2}}(a_p+b_pi\sqrt{p})$ with $a_p,b_p\in\frac12\Z$ and $$\begin{cases}ν_p(a_p)=ν_p(b_p)=\frac{p-3}{8}&\mbox{if}\ p\equiv 3\pmod8, \\ν_p(a_p)=ν_p(b_p)+1=\frac{p+1}{8}&\mbox{if}\ p\equiv 7\pmod8,\end{cases}$$ where $ν_p(x)$ denotes the $p$-adic or…
Let $\mathbb B_n:=\mathbb Q(\cos(π/2^{n+1}))$. For the relative norm map $\mathrm{N}_{n/n-1} \colon \mathcal O_{\mathbb B_n}^\times \rightarrow \mathcal O_{\mathbb B_{n-1}}^\times$ on the units group, we define $RE_n:=\mathrm{N}_{n/n-1}^{-1}(\{\pm 1\})$, $RE_n^+:=\mathrm{N}_{n/n-1}^{-1}(\{1\})$. Komatsu conjectured that $\mathrm{Tr} ε^2 \geq 2^n(2^{n+1}-1)$ for $ε\in RE_n -\{\pm 1\}$. Morisawa and Okazaki showed that it holds for $ε\in RE_n -RE_n^+$. In this paper we study the case $ε\in RE_n^+$. We conjecture that $\min \{\mathrm{Tr} ε^2 \mid ε\in RE_n^+-\{\pm 1\}\}= 2^n(1+8c_n)$, where $c_1:…
Let $φ_p(z)=(z-1)^p+2-ζ_p$, where $ζ_p\in\bar{\mathbb{Q}}$ is a primitive $p$-th root of unity for some odd prime $p$. Building on previous work, we show that the $n$-th iterate $φ_p^n(z)$ has Galois group $[C_p]^n$, an iterated wreath product of cyclic groups, whenever $p$ is not a Wieferich prime.
Let $G$ be a finite group and let $N/E$ be a tamely ramified $G$-Galois extension of number fields. We show how Stickelberger's factorization of Gauss sums can be used to determine the stable isomorphism class of various arithmetic $\mathbb{Z}[G]$-modules attached to $N/E$. If $\mathcal{O}_N$ and $\mathcal{O}_E$ denote the rings of integers of $N$ and $E$ respectively, we get in particular that $\mathcal{O}_N\otimes_{\mathcal{O}_E}\mathcal{O}_N$ defines the trivial class in the class group $\mathrm{Cl}(\mathbb{Z}[G])$ and, if $N/E$ is also assumed to be locally abelian, that the square root of…
Let $Ψ_n(x)$ be the monic polynomial having precisely all non-primitive $n$th roots of unity as its simple zeros. One has $Ψ_n(x)=(x^n-1)/Φ_n(x)$, with $Φ_n(x)$ the $n$th cyclotomic polynomial. The coefficients of $Ψ_n(x)$ are integers that like the coefficients of $Φ_n(x)$ tend to be surprisingly small in absolute value, e.g. for $n<561$ all coefficients of $Ψ_n(x)$ are $\le 1$ in absolute value. We establish various properties of the coefficients of $Ψ_n(x)$.
Let $N$ be a power of $2$ or $3$, and $μ_{N}$ the set of $N$-th roots of unity. We show that the ring of motivic periods of Mixed Tate motives over $\mathbb{Z}[μ_{N},\frac{1}{N}]$ is spanned by the motivic cyclotomic multiple zeta values of level $N$. This implies that the action of the motivic Galois group of mixed Tate motives over $\mathbb{Z}[μ_{N},\frac{1}{N}]$ on the motivic fundamental group of $\mathbb{G}_{m}-μ_{N}$ is faithful. This is a generalization of the known results for $N\in\{1,2,3,4,8\}$ by Deligne and Brown. We also discuss cyclotomic multiple zeta values of weight $2$ of oth…
Suppose $4|n$, $n\geq 8$, $F=F_n=\mathbb{Q}(ζ_n+\barζ_n)$, and there is one prime $\mathfrak{p}=\mathfrak{p}_n$ above $2$ in $F_n$. We study amalgam presentations for $\operatorname{PU_{2}}(\mathbb{Z}[ζ_n, 1/2])$ and $\operatorname{PSU_{2}}(\mathbb{Z}[ζ_n, 1/2])$ with the Clifford-cyclotomic group in quantum computing as a subgroup. These amalgams arise from an action of these groups on the Bruhat-Tits tree $Δ=Δ_{\mathfrak{p}}$ for $\operatorname{SL_{2}}(F_\mathfrak{p})$ constructed via the Hamilton quaternions. We explicitly compute the finite quotient graphs and the resulting amalgams for $8…
In this article we prove that (1-zeta+zeta^2) is a unit in the ring of integers of the cyclotomic field where zeta is a primitive n-th root of unity and n is coprime to 2 and 3. We also prove that for prime n, N_{Q(zeta)/Q}(1-zeta-zeta^2)=L(p) the p-th Lucas number thus completing the study of norms of quadratic polynomials in zeta that only have coefficients equal to 1 or -1 and both numbers appear.
Recently the authors showed that the algebraic integers of the form $-m+ζ_k$ are bases of a canonical number system of $\mathbb{Z}[ζ_k]$ provided $m\geq ϕ(k)+1$, where $ζ_k$ denotes a $k$-th primitive root of unity and $ϕ$ is Euler's totient function. In this paper we are interested in the questions whether two bases $-m+ζ_k$ and $-n+ζ_k$ are multiplicatively independent. We show the multiplicative independence in case that $0<|m-n|<10^6$ and $|m|,|n|> 1$.
For primes p greater than 3, we propose a conjecture that relates the values of cup products in the Galois cohomology of the maximal unramified outside p extension of a cyclotomic field on cyclotomic p-units to the values of p-adic L-functions of cuspidal eigenforms that satisfy mod p congruences with Eisenstein series. Passing up the cyclotomic and Hida towers, we construct an isomorphism of certain spaces that allows us to compare the value of a reciprocity map on a particular norm compatible system of p-units to what is essentially the two-variable p-adic L-function of Mazur and Kitagawa.
Concentration and equi-distribution, near the unit circle, in Solomyak's set, of the union of the Galois conjugates and the beta-conjugates of a Parry number $β$ are characterized by means of the Erdős-Turán approach, and its improvements by Mignotte and Amoroso, applied to the analytical function $f_β(z) = -1 + \sum_{i \geq 1} t_i z^i$ associated with the Rényi $β$-expansion $d_β(1)= 0.t_1 t_2 ...$ of unity. Mignotte's discrepancy function requires the knowledge of the factorization of the Parry polynomial of $β$. This one is investigated using theorems of Cassels, Dobrowolski, Pinner and Vaa…
Let $K_{m}=\Bbb{Q}(ζ_{m})$ where $ζ_{m}$ is a primitive $m$th root of unity. Let $p>2$ be prime and let $C_{p}$ denote the group of order $p.$ The ring of algebraic integers of $K_{m}$ is $\Cal{O}_{m}=\Bbb{Z}[ζ_{m}].$ Let $Λ_{m,p}$ denote the order $\Cal{O}_{m}[C_{p}]$ in the algebra $K_{m}[C_{p}].$ Consider the kernel group $D(Λ_{m,p})$ and the Swan subgroup $T(Λ_{m,p}).$ If $(p,m)=1$ these two subgroups of the class group coincide. Restricting to when there is a rational prime $p$ that is prime in $\Cal{O}_{m}$ requires $m=4$ or $q^{n}$ where $q>2$ is prime. For each such $m$, $3 \leq m \leq…
Given a numerical semigroup $S$, we let $\mathrm P_S(x)=(1-x)\sum_{s\in S}x^s$ be its semigroup polynomial. We study cyclotomic numerical semigroups; these are numerical semigroups $S$ such that $\mathrm P_S(x)$ has all its roots in the unit disc. We conjecture that $S$ is a cyclotomic numerical semigroup if and only if $S$ is a complete intersection numerical semigroup and present some evidence for it. Aside from the notion of cyclotomic numerical semigroup we introduce the notion of cyclotomic exponents and polynomially related numerical semigroups. We derive some properties and give some ap…
The construction of the $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbf{Q}_p)$ uses in an essential way Fontaine's theory of cyclotomic $(φ,Γ)$-modules. Here \emph{cyclotomic} means that $Γ= \mathrm{Gal}(\mathbf{Q}_p(μ_{p^\infty})/\mathbf{Q}_p)$ is the Galois group of the cyclotomic extension of $\mathbf{Q}_p$. In order to generalize the $p$-adic local Langlands correspondence to $\mathrm{GL}_2(L)$, where $L$ is a finite extension of $\mathbf{Q}_p$, it seems necessary to have at our disposal a theory of Lubin-Tate $(φ,Γ)$-modules. Such a generalization has been carried out t…
We examine when units in a field are the norms of elements in an extension field, given certain conditions. We apply these results to the study of the $\ell$-class groups in lifts of the anti-cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}(i)$.
Nonabelian Chabauty for the Thrice-punctured Line over Cyclotomic Fields
International Centre for Mathematical Sciences（国际数学科学中心）；Korea Institute for Advanced Study（韩国高等研究院）；School of Mathematics, The University of Edinburgh（爱丁堡大学数学学院）；Institut für Mathematik, Carl von Ossietzky Universität Oldenburg（奥尔登堡卡尔·冯·奥西茨基大学数学研究所）
Let $X_{0}=\mathbb{P}^{1} - (\{0,\infty\} \cup μ_{N})\text{ }/\text{ }\mathbb{F}_{q}$, with $N \in \mathbb{N}^{\ast}$ and $\mathbb{F}_{q}$ of characteristic $p>0$ and containing a primitive $N$-th root of unity. We establish an explicit theory of the crystalline pro-unipotent fundamental groupoid of $X_{0}$. In part I, we have computed explicitly the Frobenius, and in particular cyclotomic $p$-adic multiple zeta values. In part II, we use part I to understand the algebraic relations of cyclotomic $p$-adic multiple zeta values via explicit formulas ; this is in particular a study of the harmoni…
The $n^{th}$ cyclotomic polynomial $Φ_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Its coefficients are the subject of intensive study and some formulas are known for them. Here we are interested in formulas which are valid for all natural numbers $n$. In these a host of famous number theoretical objects such as Bernoulli numbers, Stirling numbers of both kinds and Ramanujan sums make their appearance, sometimes even at the same time! In this paper we present a survey of these formulas which until now were scattered in the literature and introduce an unified approach…
