Abelian Extension Of Number Fields Research Articles

The special value 𝑢=1 of Artin-Ihara 𝐿-functions

We study the special value u = 1 u=1 of Artin-Ihara L L -functions associated to characters of the automorphism group of abelian covers of multigraphs. In particular, we show an annihilation statement analogous to a classical conjecture of Brumer on annihilation of class groups for abelian extensions of number fields and we also calculate the index of an ideal analogous to the classical Stickelberger ideal in algebraic number theory. Along the way, we make some observations about the number of spanning trees in abelian multigraph coverings that may be of independent interest.

Distribution of genus numbers of abelian number fields

AbstractWe study the quantitative behaviour of genus numbers of abelian extensions of number fields with given Galois group. We prove an asymptotic formula for the average value of the genus number and show that any given genus number appears only of the time.

On the Brumer--Stark conjecture

Let $H/F$ be a finite abelian extension of number fields with $F$ totally real and $H$ a CM field. Let $S$ and $T$ be disjoint finite sets of places of $F$ satisfying the standard conditions. The Brumer–Stark conjecture states that the Stickelberger element $\Theta_{S,T}^{H/F}$ annihilates the $T$-smoothed class group $\mathrm{Cl}^T(H)$. We prove this conjecture away from $p=2$, that is, after tensoring with ${\bf Z}[1/2]$. We prove a stronger version of this result conjectured by Kurihara that gives a formula for the 0th Fitting ideal of the minus part of the Pontryagin dual of $\mathrm{Cl}^T(H) \otimes {\bf Z}[1/2]$ in terms of Stickelberger elements. We also show that this stronger result implies Rubin's higher rank version of the Brumer--Stark conjecture, again away from 2. Our technique is a generalization of Ribet's method, building upon on our earlier work on the Gross--Stark conjecture. Here we work with group ring valued Hilbert modular forms as introduced by Wiles. A key aspect of our approach is the construction of congruences between cusp forms and Eisenstein series that are stronger than usually expected, arising as shadows of the trivial zeroes of $p$-adic $L$-functions. These stronger congruences are essential to proving that the cohomology classes we construct are unramified at $p$.

Abelian lifts of polynomials

Let p be a prime, and let f∈Z[x] be a non-constant polynomial whose leading coefficient is prime to p and that f has no repeated root modulo p. Suppose the mod-p irreducible factors of f all have the same degree. Then there are infinitely many Abelian extensions of number fields L/K such that OL/pOL≃(Z/p)[x]/f, where OL denotes the ring of integers of L and p⊂OK is a maximal ideal with OK/pOK≃Z/p. Moreover, under the generalized Riemann hypothesis for the Dedekind zeta functions of number fields, for fixed deg⁡(f) we can construct Abelian lifts in polynomial time in log⁡p. This confirms in a stronger form a question raised by Mihăilescu and Vuletescu concerning polynomial cyclic algebras. The proofs make use of a weak form of Artin's conjecture on primitive roots and polynomial time polynomial factoring algorithms over number fields.

Numerical evidence for higher order Stark-type conjectures

We give a systematic method of providing numerical evidence for higher order Stark-type conjectures such as (in chronological order) Stark's conjecture over $\mathbb{Q}$, Rubin's conjecture, Popescu's conjecture, and a conjecture due to Burns that constitutes a generalization of Brumer's classical conjecture on annihilation of class groups. Our approach is general and could be used for any abelian extension of number fields, independent of the signature and type of places (finite or infinite) that split completely in the extension. We then employ our techniques in the situation where $K$ is a totally real, abelian, ramified cubic extension of a real quadratic field. We numerically verify the conjectures listed above for all fields $K$ of this type with absolute discriminant less than $10^{12}$, for a total of $19197$ examples. The places that split completely in these extensions are always taken to be the two real archimedean places of $k$ and we are in a situation where all the $S$-truncated $L$-functions have order of vanishing at least two.

Commutative Hopf–Galois module structure of tame extensions

We prove three theorems concerning the Hopf–Galois module structure of fractional ideals in a finite tamely ramified extension of p-adic fields or number fields which is H-Galois for a commutative Hopf algebra H. Firstly, we show that if L/K is a tame Galois extension of p-adic fields then each fractional ideal of L is free over its associated order in H. We also show that this conclusion remains valid if L/K is merely almost classically Galois. Finally, we show that if L/K is an abelian extension of number fields then every ambiguous fractional ideal of L is locally free over its associated order in H.

Capitulation in abelian extensions of number fields

This paper studies the kernel and cokernel of the map $j:\mathit{Cl}_K \to \mathit{Cl}_L$ where $L$ is an abelian extension of a number field $K$, $\mathit{Cl}_L$ and $\mathit{Cl}_K$ are their respective ideal class groups and $j$ is induced by the l

Iwasawa theory and p-adic L-functions over ${\mathbb Z}_{p}^{2}$-extensions

We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of ℚp, over a Galois extension whose Galois group is an abelian p-adic Lie group of dimension 2. We use this regulator map to study p-adic representations of global Galois groups over certain abelian extensions of number fields whose localization at the primes above p is an extension of the above type. In the example of the restriction to an imaginary quadratic field of the representation attached to a modular form, we formulate a conjecture on the existence of a "zeta element", whose image under the regulator map is a p-adic L-function. We show that this conjecture implies the known properties of the 2-variable p-adic L-functions constructed by Perrin-Riou and Kim.

A Brumer–Stark conjecture for non-abelian Galois extensions

Let K/k be an abelian extension of number fields. The Brumer–Stark conjecture predicts that a group ring element constructed from special values of L-functions associated to K/k annihilates the ideal class group of K. Moreover it specifies that the generators obtained have special properties. The aim of this article is to state and study a generalization of this conjecture to non-abelian Galois extensions that is, in spirit, very similar to the original conjecture.

Index formulae for Stark units and their solutions

Let K= k be an abelian extension of number fields with a distinguished place of k that splits totally in K . In that situation, the abelian rank-one Stark conjecture predicts the existence of a unit in K , called the Stark unit, constructed from the values of the L-functions attached to the extension. In this paper, assuming the Stark unit exists, we prove index formulae for it. In a second part, we study the solutions of the index formulae and prove that they admit solutions unconditionally for quadratic, quartic and sextic (with some additional conditions) cyclic extensions. As a result we deduce a weak version of the conjecture (“up to absolute values”) in these cases and precise results on when the Stark unit, if it exists, is a square.

The Rank of Jacobian Varieties over the Maximal Abelian Extensions of Number Fields: Towards the Frey–Jarden Conjecture

AbstractFrey and Jarden asked if any abelian variety over a number field K has the infinite Mordell–Weil rank over the maximal abelian extension Kab. In this paper, we give an affirmative answer to their conjecture for the Jacobian variety of any smooth projective curve C over K such that #C(Kab) = ∞ and for any abelian variety of GL2-type with trivial character.

Trace forms of abelian extensions of number fields of type (1, 0)

This article is concerned with describing certain bilinear forms associated with finite abelian extensions N|K of an algebraic number field K. These abelian trace forms are described up to Witt equivalence, that is, they are described as elements in the Witt ring W(K). When the base field K has exactly one dyadic prime and no real embeddings, it is shown that the Witt class of every abelian trace form over K is a product of Witt classes of five specified types.

THE FRACTIONAL GALOIS IDEAL FOR ARBITRARY ORDER OF VANISHING

We propose a candidate, which we call the fractional Galois ideal after Snaith's fractional ideal, for replacing the classical Stickelberger ideal associated to an abelian extension of number fields. The Stickelberger ideal can be seen as gathering information about those L-functions of the extension which are non-zero at the special point s = 0, and was conjectured by Brumer to give annihilators of class-groups viewed as Galois modules. An earlier version of the fractional Galois ideal extended the Stickelberger ideal to include L-functions with a simple zero at s = 0, and was shown by the present author to provide class-group annihilators not existing in the Stickelberger ideal. The version presented in this paper deals with L-functions of arbitrary order of vanishing at s = 0, and we give evidence using results of Popescu and Rubin that it is closely related to the Fitting ideal of the class-group, a canonical ideal of annihilators.Finally, we prove an equality involving Stark elements and class-groups originally due to Büyükboduk, but under a slightly different assumption, the advantage being that we need none of the Kolyvagin system machinery used in the original proof.

A uniform relative Dobrowolski's lower bound over abelian extensions

Let L/K be an abelian extension of number fields. We prove a uniform lower bound for the height in L* outside roots of unity. This lower bound depends only on the degree [L : K].

The canonical fractional Galois ideal at [formula omitted

The Stickelberger elements attached to an abelian extension of number fields conjecturally participate, under certain conditions, in annihilator relations involving higher algebraic K-groups. In [Victor P. Snaith, Stark's conjecture and new Stickelberger phenomena, Canad. J. Math. 58 (2) (2006) 419–448], Snaith introduces canonical Galois modules hoped to appear in annihilator relations generalising and improving those involving Stickelberger elements. In this paper we study the first of these modules, corresponding to the classical Stickelberger element, and prove a connection with the Stark units in a special case.

Stark’s Question on Special Values of $L$\textendash Functions

In 1980, Stark posed a far reaching question regarding the second derivatives at $s=0$ of $L$--functions of order of vanishing two associated to abelian extensions of number fields (global fields of characteristic $0$) (see \cite{St2}). Over the years, various mathematicians, most notably Grant \cite{Gra}, Sands \cite{Sa}, and Tangedal \cite{Tan} have provided evidence in favor of an affirmative answer to Stark's question. In \cite{P2}, we extrapolated Stark's question to the appropriate class of $L$--functions associated to abelian extensions of function fields (global fields of characteristic $p>0$) and showed that, in general, it has a negative answer in that context. Unfortunately, the methods developed in \cite{P2} are specific to the geometric, characteristic $p>0 $ situation and cannot be carried over to the characteristic $0$ case. In the present paper, we develop new methods which permit us to prove that, in general, (even a weak form of) Stark's question has a negative answer in its original, characteristic $0$ formulation as well.

Twisted Zeta-Functions and Abelian Stark Conjectures

We present a new version “at s=1” of Rubin's refined, higher order Stark conjecture at s=0 for an abelian extension of number fields (K. Rubin, 1996, Ann. Inst. Fourier46, No. 1, 33–62). The key idea is to introduce a formalism of “twisted zeta-functions” to replace the L-functions underlying Rubin's conjecture. This achieves certain simplifications, notably eliminating Gauss sums in a natural way from our version at s=1. It also facilitates some further developments, including an important motivation of the present paper: the formulation of an analogous p-adic conjecture to be presented in a sequel.

Explicit units and the equivariant Tamagawa number conjecture

Let L/K be a finite abelian extension of number fields of group G . We study the Tamagawa number T Ω( L/K ) of h 0 (Spec L ), considered as a motive defined over K and with coefficients [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /][ G ]. For a large class of extensions L/K we interpret the conjectural vanishing of T Ω( L/K ) in terms of the existence of S -units in L satisfying a variety of explicit conditions. These explicit conditions are in the same spirit as, but are in general much finer than, the conditions studied by Rubin and Popescu. By using this approach we are able to complete the proof that T Ω( L/K ) vanishes for all of the genus field extensions considered by Fröhlich. In the course of proving this result we find that for certain extensions the vanishing of T Ω( L/K ) is a refinement of the main result of Solomon concerning "wild Euler systems."

On the equivariant structure of ideals in abelian extensions of local fields (with an appendix by W. Bley)

Let p be an odd rational prime and K a finite extension of ${\Bbb Q}\_p$ . We give a complete classification of those finite abelian extensions $L/K$ in which any ideal of the valuation ring of L is free over its associated order in ${\Bbb Q}\_p\[Gal(L/K)]$ . In an appendix W. Bley describes an algorithm which can be used to determine the structure of Galois stable ideals in abelian extensions of number fields. The algorithm is applied to give several new and interesting examples.

The trace form for tame, abelian extensions of number fields

Let N/K be a tame, abelian extension of number fields, whose Galois group is denoted by Γ. The basic object of study in this paper is the ring of integers of N, endowed with the trace form TN/K; the pair is then a Hermitian module (where we abbreviate ), and it restricts to a -Hermitian module (The reader is referred to Section 2 for the basics on Hermitian modules.) Ideally one would like to determine completely the class of this Hermitian module in K0H(ℤΓ), the Grothendieck group of ℤΓ-Hermitian modules modulo orthogonal sums; however, in general when Γ is even, one knows that even the ℚΓ-Hermitian module given by restricting (N, TN/K) is difficult to classify. (See for instance [S] and [F2].) To circumvent this difficulty we proceed in the following fashion, as suggested by the recent work of P. Lawrence (see [L]): let D = D(Γ) denote the anti-diagonal of Γ in Γ × Γ, that is to sayand let N(2) = (N⊗KN)D so that N(2) is a Galois algebra over K with Galois group Γ × Γ/D(Γ) ≅ Γ. Write for the trace form of N(2)/K, and define the -order it is then easy to see that is isomorphic as an -module to (see (3·1·6)). To be precise we ought really to write etc.; however, the base field will always be clear from the context.