Cyclotomic Extension Research Articles

Iwasawa main conjecture for the Carlitz cyclotomic extension and applications

We prove an Iwasawa Main Conjecture for the class group of the $$\mathfrak {p}$$-cyclotomic extension $$\mathcal {F}$$ of the function field $$\mathbb {F}_q(\theta )$$ ($$\mathfrak {p}$$ is a prime of $$\mathbb {F}_q[\theta ]$$), showing that its Fitting ideal is generated by a Stickelberger element. We use this and a link between the Stickelberger element and a $$\mathfrak {p}$$-adic L-function to prove a close analog of the Ferrero–Washington Theorem for $$\mathcal {F}$$ and to provide information on the $$\mathfrak {p}$$-adic valuations of the Bernoulli-Goss numbers $$\beta (j)$$ (i.e., on the values of the Carlitz-Goss $$\zeta $$-function at negative integers).

Signature ranks of units in cyclotomic extensions of abelian number fields

We prove the rank of the group of signatures of the circular units (hence also the full group of units) of ${\mathbb Q}( \zeta_m)^+$ 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 ${\mathbb Q}( \zeta_{p^n})^+$ differs from $\varphi(p^n)/2$ by an amount that is bounded independent of $n$. Finally, we show conditionally that for general cyclotomic fields the unit signature rank can differ from its maximum possible value by an arbitrarily large amount.

Fitting ideals of class groups in Carlitz–Hayes cyclotomic extensions

We generalize some results of Greither and Popescu to a geometric Galois cover X→Y which appears naturally for example in extensions generated by pn-torsion points of a rank 1 normalized Drinfeld module (i.e. in subextensions of Carlitz–Hayes cyclotomic extensions of global fields of positive characteristic). We obtain a description of the Fitting ideal of class groups (or of their dual) via a formula involving Stickelberger elements and providing a link (similar to the one in [1]) with Goss ζ-function.

Arithmetic properties of signed Selmer groups at non-ordinary primes

We extend many results on Selmer groups for elliptic curves and modular forms to the non-ordinary setting. More precisely, we study the signed Selmer groups defined using the machinery of Wach modules over $\mathbf{Z}_p$-cyclotomic extensions. First, we provide a definition of residual and non-primitive Selmer groups at non-ordinary primes. This allows us to extend techniques developed by Greenberg (for $p$-ordinary elliptic curves) and Kim ($p$-supersingular elliptic curves) to show that if two $p$-non-ordinary modular forms are congruent to each other, then the Iwasawa invariants of their signed Selmer groups are related in an explicit manner. Our results have several applications. First of all, this allows us to relate the parity of the analytic ranks of such modular forms generalizing a recent result of the first-named author for $p$-supersingular elliptic curves. Second, we can prove a Kida-type formula for the signed Selmer groups generalizing results of Pollack and Weston.

Ranks of rational points of the Jacobian varieties of hyperelliptic curves

In this paper, we obtain bounds for the Mordell–Weil ranks over certain Zp-extensions (including cyclotomic Zp-extensions) of a wide range of abelian varieties defined over a number field F whose primes above p are totally ramified over F/Q. We assume that the abelian varieties may have good non-ordinary reduction at primes above p. Our work is a generalization of [5], in which the second author generalized Perrin-Riou's Iwasawa theory for elliptic curves over Q with supersingular reduction ([14]) to elliptic curves defined over the above-mentioned number field F. As a result, we obtain bounds of the Mordell–Weil ranks over cyclotomic extensions of the Jacobian varieties of y2=x3pN+axpN+b and y2pM=x3pN+axpN+b.

On the Birch-Swinnerton-Dyer conjecture and Schur indices

For every odd prime $p$, we exhibit families of irreducible Artin representations $\tau$ with the property that for every elliptic curve $E$ the order of the zero of the twisted $L$-function $L(E,\tau,s)$ at $s\!=\!1$ must be a multiple~of~$p$. Analogously, the multiplicity of $\tau$ in the Selmer group of $E$ must also be divisible by $p$. We give further examples where $\tau$ can moreover be twisted by any character that factors through the $p$-cyclotomic extension, and examples where the $L$-functions are those of twists of certain Hilbert modular forms by Dirichlet charaters. These results are conjectural, and rely on a standard generalisation of the Birch--Swinnerton-Dyer conjecture. Our main tool is the theory of Schur indices from representation theory.

First-order decidability and definability of integers in infinite algebraic extensions of the rational numbers

We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show the first-order undecidabilitv. We also obtain a structural sufficient condition for definability of the ring of integers over its field of fractions. In particular, we show that the following propositions hold: (1) For any rational prime q and any positive rational integer m. algebraic integers are definable in any Galois extension of Q where the degree of any finite subextension is not divisible by qm. (2) Given a prime q, and an integer m > 0, algebraic integers are definable in a cyclotomic extension (and any of its subfields) generated by any set $$\{ {\zeta _{{p^l}}}|l \in {Z_{ > 0,}}P \ne q$$ is any prime such that qm +1 ∧(p — 1)}. (3) The first-order theory of Any Abelina Extension of Q With Finitely Many Rational Primes is undecidable and rational integers are definable in these extensions. We also show that under a condition on the splitting of one rational Q generated elliptic curve over the field in question is enough to have a definition of Z and to show that the field is undecidable.

Gaussian periods in cyclotomic fields and relative traces as generators of intermediate subfields

We study relative traces to provide primitive elements for all the subfields of any cyclotomic field. We also build a primitive element for a cyclotomic extension such that every intermediate field is generated by its relative trace.

Analogue of the Hyodo Inequality for the Ramification Depth in Degree p2 Extensions

Ramification in complete discrete valuation fields is studied. For the case of a perfect residue field, there is a well-developed theory of ramification groups. Hyodo introduced the concept of ramification depth associated with the different of an extension and obtained an inequality that combines the concept of ramification depth in a degree p2 cyclotomic extension with the concept of ramification depth in a degree p subextension. The paper gives a detailed consideration of the structure of degree p2 extensions that can be obtained by a composite of two degree p extensions. In this case, it is not required that the residue field be perfect. Using the concepts of wild and ferocious extensions and the defect of the main unit, degree p2 extensions are classified and more accurate estimates for the ramification depth are obtained. In a number of cases, exact formulas for ramification depth are presented.

Formal Duality in Finite Cyclic Groups

The notion of formal duality in finite Abelian groups appeared recently in relation to spherical designs, tight sphere packings, and energy minimizing configurations in Euclidean spaces. For finite cyclic groups it is conjectured that there are no primitive formally dual pairs besides the trivial one and the TITO configuration. This conjecture has been verified for cyclic groups of prime power order, as well as of square-free order. In this paper, we will confirm the conjecture for other classes of cyclic groups, namely almost all cyclic groups of order a product of two prime powers, with finitely many exceptions for each pair of primes, or whose order $N$ satisfies $p\mid\!\mid N$, where $p$ a prime satisfying the so-called self-conjugacy property with respect to $N$. For the above proofs, various tools were needed: the field descent method, used chiefly for the circulant Hadamard conjecture, the techniques of Coven & Meyerowitz for sets that tile $\mathbb{Z}$ or $\mathbb{Z}_N$ by translations, dubbed herein as the polynomial method, as well as basic number theory of cyclotomic fields, especially the splitting of primes in a given cyclotomic extension.

The geometry of Hida families I: $$\Lambda $$-adic de Rham cohomology

We construct the $$\Lambda $$ -adic de Rham analogue of Hida’s ordinary $$\Lambda $$ -adic etale cohomology and of Ohta’s $$\Lambda $$ -adic Hodge cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of $$\mathbf {Q}_p$$ , we give a purely geometric proof of the expected finiteness, control, and $$\Lambda $$ -adic duality theorems. Following Ohta, we then prove that our $$\Lambda $$ -adic module of differentials is canonically isomorphic to the space of ordinary $$\Lambda $$ -adic cuspforms. In the sequel (Cais, Compos Math, to appear) to this paper, we construct the crystalline counterpart to Hida’s ordinary $$\Lambda $$ -adic etale cohomology, and employ integral p-adic Hodge theory to prove $$\Lambda $$ -adic comparison isomorphisms between all of these cohomologies. As applications of our work in this paper and (Cais, Compos Math, to appear), we will be able to provide a “cohomological” construction of the family of $$(\varphi ,\Gamma )$$ -modules attached to Hida’s ordinary $$\Lambda $$ -adic etale cohomology by Dee (J Algebra 235(2), 636–664, 2001), as well as a new and purely geometric proof of Hida’s finiteness and control theorems. We are also able to prove refinements of the main theorems in Mazur and Wiles (Compos Math 59(2):231–264, 1986) and Ohta (J Reine Angew Math 463:49–98, 1995).

Galois groups of some iterated polynomials over cyclotomic extensions

Let \(\varphi _p(z)=(z-1)^p+2-\zeta _p\), where \(\zeta _p\in \bar{\mathbb {Q}}\) is a primitive pth root of unity. Building on previous work, we show that the nth iterate \(\varphi _p^n(z)\) has Galois group \([C_p]^n\), an iterated wreath product of cyclic groups, whenever p is not a Wieferich prime.

Bounds on the Tamagawa numbers of a crystalline representation over towers of cyclotomic extensions

In this paper, we study the Tamagawa numbers of a crystalline representation over a tower of cyclotomic extensions under certain technical conditions on the representation. In particular, we show that we may improve the asymptotic bounds given in the thesis of Arthur Laurent in certain cases.

Wach Modules, Regulator Maps, and $\varepsilon$-Isomorphisms in Families

Abstract We prove the “local $\varepsilon$-isomorphism” conjecture of Fukaya and Kato [13] for certain crystalline families of $G_{{\mathbf{Q}_p}}$-representations. This conjecture can be regarded as a local analog of the Iwasawa main conjecture for families. Our work extends earlier work of Kato for rank-$1$ modules (cf. [33]), of Benois and Berger for crystalline $G_{{\mathbf{Q}_p}}$-representations with respect to the cyclotomic extension (cf. [1]), as well as of Loeffler et al. (cf. [21]) for crystalline $G_{{\mathbf{Q}_p}}$-representations with respect to abelian $p$-adic Lie extensions of ${\mathbf{Q}_p}$. Nakamura [24, 25] has also formulated a version of Kato’s $\varepsilon$-conjecture for affinoid families of $(\varphi,\Gamma)$-modules over the Robba ring, and proved his conjecture in the rank-$1$ case. He used this case to construct an $\varepsilon$-isomorphism for families of trianguline $(\varphi,\Gamma)$-modules, depending on a fixed triangulation. Our results imply that this $\varepsilon$-isomorphism is independent of the chosen triangulation for certain crystalline families. The main ingredient of our proof consists of the construction of families of Wach modules generalizing work of Wach and Berger [6] and following Kisin’s approach to the construction of potentially semi-stable deformation rings [18].

Trigonometric series and special values of L-functions

Inspired by representations of the class number of imaginary quadratic fields, in this paper, we give explicit evaluations of trigonometric series having generalized harmonic numbers as coefficients in terms of odd values of the Riemann zeta function and special values of L-functions subject to the parity obstruction. The coefficients that arise in these evaluations are shown to belong to certain cyclotomic extensions. Furthermore, using best polynomial approximation of smooth functions under uniform convergence due to Jackson and their log-sine integrals, we provide approximations of real numbers by combinations of special values of L-functions corresponding to the Legendre symbol. Our method for obtaining these results rests on a careful study of generating functions on the unit circle involving generalized harmonic numbers and the Legendre symbol, thereby relating them to values of polylogarithms and then finally extracting Fourier series of special functions that can be expressed in terms of Clausen functions.

Cohomology and torsion cycles over the maximal cyclotomic extension

Abstract A classical theorem by K. Ribet asserts that an abelian variety defined over the maximal cyclotomic extension K of a number field has only finitely many torsion points. We show that this statement can be viewed as a particular case of a much more general one, namely that the absolute Galois group of K acts with finitely many fixed points on the étale cohomology with 𝐐 / 𝐙 {\mathbf{Q}/\mathbf{Z}} -coefficients of a smooth proper K ¯ {\overline{K}} -variety defined over K. We also present a conjectural generalization of Ribet’s theorem to torsion cycles of higher codimension. We offer supporting evidence for the conjecture in codimension 2, as well as an analogue in positive characteristic.

Multivariable (φ,Γ)-modules and locally analytic vectors

Let $K$ be a finite extension of $\mathbf{Q}_p$ and let $G_K = \mathrm{Gal}(\bar{\mathbf{Q}}_p/K)$. There is a very useful classification of $p$-adic representations of $G_K$ in terms of cyclotomic $(\varphi,\Gamma)$-modules (cyclotomic means that $\Gamma={\rm Gal}(K_\infty/K)$ where $K_\infty$ is the cyclotomic extension of $K$). One particularly convenient feature of the cyclotomic theory is the fact that any $(\varphi,\Gamma)$-module is overconvergent. Questions pertaining to the $p$-adic local Langlands correspondence lead us to ask for a generalization of the theory of $(\varphi,\Gamma)$-modules, with the cyclotomic extension replaced by an infinitely ramified $p$-adic Lie extension $K_\infty / K$. It is not clear what shape such a generalization should have in general. Even in the case where we have such a generalization, namely the case of a Lubin-Tate extension, most $(\varphi,\Gamma)$-modules fail to be overconvergent. In this article, we develop an approach that gives a solution to both problems at the same time, by considering the locally analytic vectors for the action of $\Gamma$ inside some big modules defined using Fontaine's rings of periods. We show that, in the cyclotomic case, we recover the ususal overconvergent $(\varphi,\Gamma)$-modules. In the Lubin-Tate case, we can prove, as an application of our theory, a folklore conjecture in the field stating that $(\varphi,\Gamma)$-modules attached to $F$-analytic representations are overconvergent.

Roots of unity and torsion points of abelian varieties

We answer a question raised by Hindry and Ratazzi concerning the intersection between cyclotomic extensions of a number field K and extensions of K generated by torsion points of an abelian variety over K. We prove that the property called \((\mu )\) in Hindry and Ratazzi (J Ramanujan Math Soc 25(1):81–111, 2010) holds for any abelian variety, while the same is not true for the stronger version of the property introduced in Hindry and Ratazzi (J Inst Math Jussieu 11(1):27–65, 2012).

On the degree of irrationality in Noether’s problem

Noether’s problem asks whether, for a given field [Formula: see text] and finite group [Formula: see text], the fixed field [Formula: see text] is a purely transcendental extension of [Formula: see text], where [Formula: see text] acts on the [Formula: see text] by [Formula: see text]. The field [Formula: see text] is naturally the function field for a quotient variety [Formula: see text]. We study the degree of irrationality [Formula: see text] of [Formula: see text] for an abelian group [Formula: see text], which is defined to be the minimal degree of a dominant rational map from [Formula: see text] to projective space. In particular, we give bounds for [Formula: see text] in terms of the arithmetic of cyclotomic extensions [Formula: see text].

Number of prime ideals in short intervals

Assuming a weaker form of the Riemann hypothesis for Dedekind zeta functions by allowing Siegel zeros, we extend a classical result of Cramér on the number of primes in short intervals to prime ideals of the ring of integers in cyclotomic extensions with norms belonging to such intervals. The extension is uniform with respect to the degree of the cyclotomic extension. Our approach is based on the arithmetic of cyclotomic fields and analytic properties of their Dedekind zeta functions together with a lower bound for the number of primes over progressions in short intervals subject to similar assumptions. Uniformity with respect to the modulus of the progression is obtained and the lower bound turns out to be best possible, apart from constants, as shown by the Brun–Titchmarsh theorem.