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.
Let $F$ be a number field unramified at an odd prime $p$ and $F_\infty$ be the $\mathbf{Z}_p$-cyclotomic extension of $F$. Let $A$ be an abelian variety defined over $F$ with good supersingular reduction at all primes of $F$ above $p$. Büyükboduk and the first named author have defined modified Selmer groups associated to $A$ over $F_\infty$. Assuming that the Pontryagin dual of these Selmer groups are torsion $\mathbf{Z}_p[[\mathrm{Gal}(F_\infty/F)]]$-modules, we give an explicit sufficient condition for the rank of the Mordell-Weil group $A(F_n)$ to be bounded as $n$ varies.
Let p be an odd prime, and k_\infty the cyclotomic Z_p-extension of an abelian field k. For a finite set S of rational primes which does not include p, we will consider the maximal S-ramified abelian pro-p extension M_S(k_\infty) over k_\infty. We shall give a formula of the Z_p-rank of Gal(M_S(k_\infty)/k_\infty). In the proof of this formula, we also show that M_{q}(k_\infty)/L(k_\infty) is a finite extension for every real abelian field k and every rational prime q distinct from p, where L(k_\infty) is the maximal unramified abelian pro-p extension over k_\infty.
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 prove an analogue of the Tate conjecture on homomorphisms of abelian varieties over infinite cyclotomic extensions of finitely generated fields of characteristic zero.
Let $K$ be an imaginary quadratic field, $N$ be a positive integer, $f(z)$ be a newform of level $Γ_1(N)$, and $A_f$ be the abelian variety associated to $f$. For each $τ\in K$ ($\operatorname{Im} τ>0$), we construct a certain point $P_τ$ on $A_f$ defined over an extended ring class field of $K$ of level $N$. Our construction generalizes Birch's construction of the Heegner points to the abelian varieties associated to modular forms of level $Γ_1(N)$ and nontrivial character. Then, we show that $P_τ$'s satisfy the distribution and congruence relations of an Euler system, which implies that it s…
We mainly study a polynomial $f_{1,n}(x)=x^{n-1} + 2x^{n-2} + 3x^{n-3} + \cdots + kx^{n-k} + \cdots + (n-1)x + n$ over $\mathbb{Z}$ and the Galois group of the minimal splitting field. First, we show that an arbitrary root $α_{n}$ of $f_{1,n}(x)$ satisfies $|α_{n}|\to 1$ ($n\to \infty$), and discuss the irreducibility of $f_{1,n}(x)$ over $\mathbb{Z}$ for several type $n$. After that, we show that the Galois group of $f_{1,n}(x)$ is Symmetric group $S_{n-1}$ for several type $n$. Although those roots of $f_{1,n}(x)=0$ don't draw an exact circle, it looks like a circle on complex plane. Moreove…
Let K denote the quadratic field $\mathbb{Q}(\sqrt{d})$ where d=$-1$ or $-3$. Let E be an elliptic curve defined over K. In this paper, we analyze the torsion subgroups of E in the maximal elementary abelian $2$-extension of $K$.
In this article, we prove that every finite abelian group $G$ of odd order occurs as a subgroup of the class group of infinitely many real cyclotomic fields.
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.
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…
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…
We call a (q-1)-th Kummer extension of a cyclotomic function field a quasi-cyclotomic function field if it is Galois, but non-abelian, over the rational function field with the constant field of q elements. In this paper, we determine the structure of the Galois groups of a kind of quasi-cyclotomic function fields over the base field. We also give the genus formulae of them.
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 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…
This book deals with the elementary theory of cyclotomic fields, cyclotomic function fields, abelian extensions, genus fields, $p$--extensions, Drinfeld modules and class field theory. The text is in Spanish.
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.
Let p be an odd prime satisfying Vandiver's conjecture. We consider two objects, the Galois group X of the maximal unramified abelian pro-p extension of the compositum of all Z_p-extensions of the pth cyclotomic field and the Galois group G of the unramified pro-p extension of the cyclotomic field of all p-power roots of unity. We give a lower bound for the height of the annihilator of X as an Iwasawa module. Under some mild assumptions on Bernoulli numbers, we provide a necessary and sufficient condition for G to be abelian. The bound and the condition in the two results are given in terms of…
Following the prequel work \cite{VO3}, we prove a generalization of "Mazur's conjecture" for $L$-functions of elliptic curves in abelian extensions of imaginary quadratic fields, including the assertion that the Mordell-Weil rank of an elliptic curve in the ${\bf{Z}}_p^2$-extension is finitely generated modulo Heegner points. The novelty of the approach here is to use the existence of a suitable $p$-adic $L$-function to reduce the problem to a minimal nonvanishing criterion, which should be applicable to a broader class of problems than considered here.
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.
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…
Let A be an abelian variety defined over a number field K and let Kab be the maximal abelian extension of K. We show that there only finitely many torsion points of A which are defined over Kab iff A has no abelian subvariety with complex multiplication over K. We use this to give another proof of Ribet's result that A has only finitely many torsion points which are defined over the cyclotomic extension of K.
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, by using Kurihara's methods and Mazur-Rubin theory, we study the higher Fitting ideals of the plus-part of Iwasawa modules associated the cyclotomic Z_p-extension of abelian fields for an odd prime number p. We define the higher cyclotomic ideals C_i for any non-negative integer i, which are ideals of the Iwasawa algebra defined by the Kolyvagin derivative classes of circular units, and prove that they give upper and lower bounds of t…
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 $k$-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…
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
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.
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.
Let $F$ be a number field. Let $p$ be a prime number. Washington proved the $\ell$-part of the class numbers in cyclotomic $\mathbb{Z}_p$ extension of $F$ is bounded when $F$ is an abelian number field and $\ell\neq p$ is a prime. By class number formula, this is essentially a mod $\ell$ nonvanishing property of Drichlet L-functions at $s=0$. In \cite{Sinnot}, Sinnot gave a different proof by algebraic methods. In this article, we show that Sinnot's method can prove nonvanishing properties of Drichlet L-functions at $s=-k$, where $k\geq 0$ is an integer. Since the Lichtenbaurn conjecture which…
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.
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…
We give a completely normal element in the maximal real subfield of a cyclotomic field over the field of rational numbers, which is different from that of Okada. This result is a consequence of the criterion for a normal element developed in [Normal bases of ray class fields over imaginary quadratic fields, Math. Zeit.]. Furthermore, we find a completely normal element in certain extension of modular function fields in terms of a quotient of the modular discriminant function.
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.
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 $k_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of an algebraic number field $k$. We denote by $S$ a finite set of prime numbers which does not contain $p$, and $S(k_\infty)$ the set of primes of $k_\infty$ lying above $S$. In the present paper, we will study the structure of the Galois group $\mathcal{X}_S (k_\infty)$ of the maximal pro-$p$ extension unramified outside $S (k_\infty)$ over $k_\infty$. We mainly consider the question whether $\mathcal{X}_S (k_\infty)$ is a non-abelian free pro-$p$ group or not. In the former part, we treat the case when $k$ is an imaginary quadratic f…
