Abelian Galois Group Research Articles

An equivariant Tamagawa number formula for Drinfeld modules and applications

We fix data $(K/F, E)$ consisting of a Galois extension $K/F$ of characteristic $p$ global fields with arbitrary abelian Galois group $G$ and a Drinfeld module $E$ defined over a certain Dedekind subring of $F$. For this data, we define a $G$-equivariant $L$-function $\Theta_{K/F}^E$ and prove an equivariant Tamagawa number formula for certain Euler-completed versions of its special value $\Theta_{K/F}^E(0)$. This generalizes Taelman's class number formula for the value $\zeta_F^E(0)$ of the Goss zeta function $\zeta_F^E$ associated to the pair $(F, E)$. Taelman's result is obtained from our result by setting $K=F$. As a consequence, we prove a perfect Drinfeld module analogue of the classical (number field) refined Brumer--Stark conjecture, relating a certain $G$-Fitting ideal of Taelman's class group $H(E/K)$ to the special value $\Theta_{K/F}^E(0)$ in question.

Number fields with prescribed norms (with an appendix by Yonatan Harpaz and Olivier Wittenberg)

We study the distribution of extensions of a number field k with fixed abelian Galois group G , from which a given finite set of elements of k are norms. In particular, we show the existence of such extensions. Along the way, we show that the Hasse norm principle holds for 100% of G -extensions of k , when ordered by conductor. The appendix contains an alternative purely geometric proof of our existence result.

Rationally connected rational double covers of primitive Fano varieties

We show that for a Zariski general hypersurface $V$ of degree $M+1$ in ${\mathbb P}^{M+1}$ for $M\geqslant 5$ there are no Galois rational covers $X\dashrightarrow V$ of degree $d\geqslant 2$ with an abelian Galois group, where $X$ is a rationally connected variety. In particular, there are no rational maps $X\dashrightarrow V$ of degree 2 with $X$ rationally connected. This fact is true for many other families of primitive Fano varieties as well and motivates a conjecture on absolute rigidity of primitive Fano varieties.

Arithmetic of p-adic curves and sections of geometrically abelian fundamental groups

Let X be a proper, smooth, and geometrically connected curve of genus g(X)ge 1 over a p-adic local field. We prove that there exists an effectively computable open affine subscheme Usubset X with the property that {text {period}}(X)=1, and {text {index}}(X) equals 1 or 2 (resp. {text {period}}(X)={text {index}}(X)=1, assuming {text {period}}(X)={text {index}}(X)), if (resp. if and only if) the exact sequence of the geometrically abelian fundamental group of Usplits. We compute the torsor of splittings of the exact sequence of the geometrically abelian absolute Galois group associated to X, and give a new characterisation of sections of arithmetic fundamental groups of curves over p-adic local fields which are orthogonal to {text {Pic}}^0 (resp. {text {Pic}}^{wedge }). As a consequence we observe that the non-geometric (geometrically pro-p) section constructed by Hoshi [3] is orthogonal to {text {Pic}}^0.

The conductor density of local function fields with abelian Galois group

We give an exact formula for the number of G-extensions of local function fields Fq((t)) for finite abelian groups G up to a conductor bound. As an application we give a lower bound for the corresponding counting problem by discriminant.

DENSITY RESULTS FOR SPECIALIZATION SETS OF GALOIS COVERS

AbstractWe provide evidence for this conclusion: given a finite Galois cover $f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$ of group $G$, almost all (in a density sense) realizations of $G$ over $\mathbb{Q}$ do not occur as specializations of $f$. We show that this holds if the number of branch points of $f$ is sufficiently large, under the abc-conjecture and, possibly, the lower bound predicted by the Malle conjecture for the number of Galois extensions of $\mathbb{Q}$ of given group and bounded discriminant. This widely extends a result of Granville on the lack of $\mathbb{Q}$-rational points on quadratic twists of hyperelliptic curves over $\mathbb{Q}$ with large genus, under the abc-conjecture (a diophantine reformulation of the case $G=\mathbb{Z}/2\mathbb{Z}$ of our result). As a further evidence, we exhibit a few finite groups $G$ for which the above conclusion holds unconditionally for almost all covers of $\mathbb{P}_{\mathbb{Q}}^{1}$ of group $G$. We also introduce a local–global principle for specializations of Galois covers $f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$ and show that it often fails if $f$ has abelian Galois group and sufficiently many branch points, under the abc-conjecture. On the one hand, such a local–global conclusion underscores the ‘smallness’ of the specialization set of a Galois cover of $\mathbb{P}_{\mathbb{Q}}^{1}$. On the other hand, it allows to generate conditionally ‘many’ curves over $\mathbb{Q}$ failing the Hasse principle, thus generalizing a recent result of Clark and Watson devoted to the hyperelliptic case.

On Galois group of factorized covers of curves

Let \mathcal{Y}\xrightarrow {{\psi}} \mathcal{X} \xrightarrow {\varphi} \mathbb{P}^{1} be a sequence of covers of compact Riemann surfaces. In this work we study and completely characterize the Galois group \mathfrak{G}(\varphi\circ\psi) of \varphi\circ\psi under the following properties: \varphi is a simple cover of degree m and \psi is a Galois unramified cover of degree n with abelian Galois group of type (n_1,n_2,\dots,n_s) . We prove that \mathfrak{G}(\varphi\circ\psi) \cong ({\mathbb Z}_{n_1} \times {\mathbb Z}_{n_2} \times \cdots \times {\mathbb Z}_{n_s})^{m-1} \rtimes {\bf S}_m . Furthermore, we give a natural geometric generator system of \mathfrak{G}(\varphi\circ\psi) obtained by studying the action on the compact Riemann surface \mathcal{Z} associated to the Galois closure of \varphi\circ\psi.

Факторизация полиномов над конечными полями

The laws of factorization of irreducible polynomials with integer coefficients over finite fields, a long-standing problem of number theory and algebra. The various reciprocity laws of number theory are connected with this problem. The Galois group of an irreducible polynomial $f(x)$ of degree n over the field of rational numbers, consider as a subgroup of the symmetric group $S_{n}$, actually describes possible types of factorization of $f(x)$ with respect to simple modules. The next problem is to describe prime numbers giving a certain type of factorization of the polynomial $f(x)$ in terms of invariants associated with this polynomial. For polynomials with Abelian Galois group this problem is solved in principle by a dap class field theory. For polynomials with a non-Abelian Galois group, little is known for certain classes of polynomials. In this paper we propose a method for solving this problem for irreducible over the field rational numbers of cubic polynomials.

Commuting-liftable subgroups of Galois groups II

Abstract Let n denote either a positive integer or ∞ {\infty} , let ℓ {\ell} be a fixed prime and let K be a field of characteristic different from ℓ {\ell} . In the presence of sufficiently many roots of unity in K, we show how to recover some of the inertia/decomposition structure of valuations inside the maximal ℓ n {\ell^{n}} -abelian Galois group of K using the maximal ℓ N {\ell^{N}} -abelian-by-central Galois group of K, whenever N is sufficiently large relative to n.

On the structure of the Galois group of the Abelian closure of a number field

Following a paper by Athanasios Angelakis and Peter Stevenhagen on the determination of imaginary quadratic fields having the same absolute Abelian Galois group A, we study this property for arbitrary number fields. We show that such a property is probably not easily generalizable, apart from imaginary quadratic fields, because of some p-adic obstructions coming from the global units. By restriction to the p-Sylow subgroups of A, we show that the corresponding study is related to a generalization of the classical notion of p-rational fields. However, we obtain some non-trivial information about the structure of the profinite group A, for every number field, by application of results published in our book on class field theory. This version corrects a technical error discovered by Peter Stevenhagen in a lemma of their first draft (arXiv:1209.6005) as well as in the previous versions of our paper reproducing this lemma; it will also be corrected in their final paper to appear in the proceedings volume of ANTS-X, San Diego 2012. This does not modify the nature of the results.

Imaginary quadratic fields with isomorphic abelian Galois groups

In 1976, Onabe discovered that, in contrast to the Neukirch-Uchida results that were proved around the same time, a number field $K$ is not completely characterized by its absolute abelian Galois group $A_K$. The first examples of non-isomorphic $K$ having isomorphic $A_K$ were obtained on the basis of a classification by Kubota of idele class character groups in terms of their infinite families of Ulm invariants, and did not yield a description of $A_K$. In this paper, we provide a direct `computation' of the profinite group $A_K$ for imaginary quadratic $K$, and use it to obtain many different $K$ that all have the same minimal absolute abelian Galois group.

Abelian covers of graphs and maps between outer automorphism groups of free groups

We explore the existence of homomorphisms between outer automorphism groups of free groups Out(F n ) → Out(F m ). We prove that if n > 8 is even and n ≠ m ≤ 2n, or n is odd and n ≠ m ≤ 2n − 2, then all such homomorphisms have finite image; in fact they factor through det : \({{\rm Out}(F_n) \to \mathbb{Z}/2}\) . In contrast, if m = r n(n − 1) + 1 with r coprime to (n − 1), then there exists an embedding \({{\rm Out}(F_n) \hookrightarrow {\rm Out}(F_m)}\) . In order to prove this last statement, we determine when the action of Out(F n ) by homotopy equivalences on a graph of genus n can be lifted to an action on a normal covering with abelian Galois group.

CONSTRUCTION OF SELF-DUAL INTEGRAL NORMAL BASES IN ABELIAN EXTENSIONS OF FINITE AND LOCAL FIELDS

Let F/E be a finite Galois extension of fields with abelian Galois group Γ. A self-dual normal basis for F/E is a normal basis with the additional property that Tr F/E(g(x), h(x)) = δg, h for g, h ∈ Γ. Bayer-Fluckiger and Lenstra have shown that when char (E) ≠ 2, then F admits a self-dual normal basis if and only if [F : E] is odd. If F/E is an extension of finite fields and char (E) = 2, then F admits a self-dual normal basis if and only if the exponent of Γ is not divisible by 4. In this paper, we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist. Now let K be a finite extension of ℚp, let L/K be a finite abelian Galois extension of odd degree and let [Formula: see text] be the valuation ring of L. We define AL/K to be the unique fractional [Formula: see text]-ideal with square equal to the inverse different of L/K. It is known that a self-dual integral normal basis exists for AL/K if and only if L/K is weakly ramified. Assuming p ≠ 2, we construct such bases whenever they exist.

Finite-order automorphisms of a certain torus

A classical result of Higman [H], [S, Exer. 6.3], asserts that the roots of unity in the group ring Z[Γ] of a finite commutative group Γ are the elements ±γ for γ ∈ Γ. One application of this result is the determination of the group of finite-order automorphisms of a torus T = ResS′/S(Gm) for a finite etale Galois covering f : S′ → S with S and S′ connected schemes and abelian Galois group Γ = Gal(S′/S) (e.g., an unramified extension of local fields). Indeed, the torus T has character group Z[Gal(S′/S)] with Gal(S′/S) acting through left translation, and an automorphism of T is “the same” as an automorphism of its character group X(T ) as a Galois module. Thus, to give an automorphism of T is to give a generator of Z[Gal(S′/S)] as a left module over itself. This in turn is just a unit in the group ring. Once we have the list of roots of unity, this gives the list of finite-order automorphisms. It follows that the group of finite-order automorphisms of the torus T is the finite group that is generated (as a direct product) by inversion and the action of Gal(S′/S). Now consider the problem (raised to the author by G. Prasad) of finding all finite-order automorphisms of the norm-1 subtorus T 1 in T when Γ is cyclic (e.g., the Galois group of an unramified extension of local fields). This subtorus is functorially described as the kernel of the determinant map

Remarks on normal bases

We prove that any Galois extension of a commutative ring with a normal basis and abelian Galois group of odd order has a self-dual normal basis. We apply this result to get a very simple proof of nonexistence of normal bases for certain extensions which a

Normal integral bases and the Spiegelungssatz of Scholz

Introduction. In Hilbert’s Zahlbericht one can find the first global result on the Galois module structure of rings of integers ([H], Satz 132). Slightly extended it states that if N/Q is a tame extension with abelian Galois group ∆, then oN , the ring of integers in N , is free as a module over the group ring Z∆; moreover, an explicit canonical algebraic integer a such that oN = Z∆a can be given. The numbers a (δ ∈ ∆), the algebraic conjugates of a, are said to form a normal integral basis of the field extension N/Q. This result has been the starting point for a modern development, which has led to the deep result that the structure of oN as a module over Z∆, for arbitrary tame Galois extensions of number fields N/K with Galois group ∆, is determined in terms of the symplectic root numbers of N/K (see [F] and [T1]). Frohlich’s book [F] also contains a detailed introduction and a rather complete list of references. For a more recent survey on Galois module theory we refer to [Ca–Ch–F–T]. In the special case that ∆ is of odd order the result mentioned above gives that oN is a free Z∆-module, but the proof does not provide an explicit basis. If one considers the richer oK∆-module structure of oN rather than the Z∆-module structure alone, then oN is expected to be “usually” not even free if K 6= Q. Results of Taylor show that by modifying both oN and oK∆ one can sometimes—if K and N are certain ray class fields over imaginary quadratic number fields—achieve the “ideal” of free modules with explicit generators (see [C–T]). However, if one decides not to modify the original classical problem of the determination of the oK∆-module structure of oN , then a natural question is to what extent, for given K and ∆, the realization of ∆ as a Galois group of a tame extension N/K is determined by the ramification of N/K together with the structure of oN as an oK∆-module. This point of view is worked out in [B2]. In a sense the core of the question is how rare unramified extensions which possess a normal integral basis are. We mention in passing that this question is equivalent to a special case of a problem considered by Taylor in recent work, that of determining the kernel

Elementary Abelianp-extensions of algebraic function fields

LetK be a field of characteristicp>0 andF/K be an algebraic function field. We obtain several results on Galois extensionsE/F with an elementary Abelian Galois group of orderpn. (a) E can be generated overF by some elementy whose minimal polynomial has the specific formTpn−T−z. (b) A formula for the genus ofE is given. (c) IfK is finite, then the genus ofE grows much faster than the number of rational points (as [E∶F] → ∞). (d) We present a new example of a function fieldE/K whose gap numbers are nonclassical.

On the Selmer Group of Twists of Elliptic Curves with Q-Rational Torsion Points

(1) The symbols p and q stand for prime numbers and throughout the paper we assume that p is fixed and contained in {3, 5, 7}. Let L be an algebraic number field (i.e., L is a finite extension of Q). Then prime divisors of L dividing p (resp. q) are denoted by (resp. ). The completion of L with respect to is denoted by . Let S be a finite set of prime numbers, and let M/L be a Galois extension with abelian Galois group of exponent p.Definition. M/L is said to be little ramified outside S if for primes q ∉ S and all one haswith k ∊ N and . Here ζp is a pth root of unity, u1, …, uk are elements in and is the normed valuation belonging to . In particular M/L is unramified at all divisors of primes q ∉ S ∪ {p}.

Taming Wild Extensions with Hopf Algebras

Let $K \subset L$ be a Galois extension of number fields with abelian Galois group $G$ and rings of integers $R \subset S$, and let $\mathcal {A}$ be the order of $S$ in $KG$. If $\mathcal {A}$ is a Hopf $R$-algebra with operations induced from $KG$, then $S$ is locally isomorphic to $\mathcal {A}$ as $\mathcal {A}$-module. Criteria are found for $\mathcal {A}$ to be a Hopf algebra when $K = {\mathbf {Q}}$ or when $L/K$ is a Kummer extension of prime degree. In the latter case we also obtain a complete classification of orders over $R$ in $L$ which are tame or Galois $H$-extensions, $H$ a Hopf order in $KG$, using a generalization of the discriminant.

Galois groups of number fields generated by torsion points of elliptic curves

Coates and Wiles [1] and B. Perrin-Riou (see [2]) study the arithmetic of an elliptic curve E defined over a number field F with complex multiplication by an imaginary quadratic field K by using p-adic techniques, which combine the classical descent of Mordell and Weil with ideas of Iwasawa’s theory of Zp-extensions of number fields. In a special case they consider a non-cyclotomic Zp-extension F∞ defined via torsion points of E and a certain Iwasawa module attached to E/F, which can be interpreted as an abelian Galois group of an extension of F∞. We are interested in the corresponding non-abelian Galois group and we want to show that the whole situation is quite analogous to the case of the cyclotomic Zp-extension (which is generated by torsion points of Gm).