On Galois structure of the integers in elementary abelian extensions of local number fields

Let p be an odd prime number and k a finite extension of Qp. Let K/k be a totally ramified elementary abelian Kummer extension of degree p² with Galois group G. We determine the isomorphism class of the ring of integers in K as an oG-module under some assumptions. The obtained results imply there exist extensions whose rings are ZpG-isomorphic but not oG-isomorphic, where Zp is the ring of p-adic integers. Moreover we obtain conditions that the rings of integers are free over the associated orders and give extensions whose rings are not free.     

On a ramification bound of torsion semi-stable representations over a local field

Let p be a rational prime, k be a perfect field of characteristic p, W=W(k) be the ring of Witt vectors, K be a finite totally ramified extension of Frac(W) of degree e and r be a non-negative integer satisfying r<p−1. In this paper, we prove the upper numbering ramification group for j>u(K,r,n) acts trivially on the pn-torsion semi-stable GK-representations with Hodge-Tate weights in {0,…,r}, where u(K,0,n)=0, u(K,1,n)=1+e(n+1/(p−1)) and u(K,r,n)=1−p−n+e(n+r/(p−1)) for 1<r<p−1.     

p-Rank and semi-stable reduction of curves

Let R be a discrete complete valuation ring, with field of fractions K, and with algebraically closed residue field k of characteristic p > 0. Let X be a germ of an R-curve at an ordinary double point. Consider a finite Galois covering f: Y → X, whose Galois group G is a p-group, such that Y is normal, and which is étale above X_k≔ x × _rk. Asume that Y has a semi-stable model :→ Y over R, and let y be a closed point of Y. If the inertia subgroup I(y) at y is cyclic of order pⁿ, we compute the p-rank of tf⁻¹ (y) by using a result of Raynaud. In particular, we prove that this p-rank is bounded by pⁿ −1.     

Explicit construction of self-dual integral normal bases for the square-root of the inverse different

Let K be a finite extension of Qp, let L/K be a finite abelian Galois extension of odd degree and let OL be the valuation ring of L. We define AL/K to be the unique fractional OL-ideal with square equal to the inverse different of L/K. For p an odd prime and L/Qp contained in certain cyclotomic extensions, Erez has described integral normal bases for AL/Qp that are self-dual with respect to the trace form. Assuming K/Qp to be unramified we generate odd abelian weakly ramified extensions of K using Lubin-Tate formal groups. We then use Dwork's exponential power series to explicitly construct self-dual integral normal bases for the square-root of the inverse different in these extensions.     

Potentially good reduction of Barsotti-Tate groups

Let R be a complete discrete valuation ring of mixed characteristic (0,p) with perfect residue field, K the fraction field of R. Suppose G is a Barsotti-Tate group (p-divisible group) defined over K which acquires good reduction over a finite extension K^′ of K. We prove that there exists a constant c?2 which depends on the absolute ramification index e(K^′/Qp) and the height of G such that G has good reduction over K if and only if G[pc] can be extended to a finite flat group scheme over R. For abelian varieties with potentially good reduction, this result generalizes Grothendieck's “p-adic Néron-Ogg-Shafarevich criterion” to finite level. We use methods that can be generalized to study semi-stable p-adic Galois representations with general Hodge-Tate weights, and in particular leads to a proof of a conjecture of Fontaine and gives a constant c as above that is independent of the height of G.     

On the vanishing of Selmer groups for elliptic curves over ring class fields

Let E/Q be an elliptic curve of conductor N without complex multiplication and let K be an imaginary quadratic field of discriminant D prime to N. Assume that the number of primes dividing N and inert in K is odd, and let Hc be the ring class field of K of conductor c prime to ND with Galois group Gc over K. Fix a complex character χ of Gc. Our main result is that if LK(E,χ,1)≠0 then Selp(E/Hc)χ_⊗W=0 for all but finitely many primes p, where Selp(E/Hc) is the p-Selmer group of E over Hc and W is a suitable finite extension of Zp containing the values of χ. Our work extends results of Bertolini and Darmon to almost all non-ordinary primes p and also offers alternative proofs of a χ-twisted version of the Birch and Swinnerton-Dyer conjecture for E over Hc (Bertolini and Darmon) and of the vanishing of Selp(E/K) for almost all p (Kolyvagin) in the case of analytic rank zero.     

Steinitz classes of tamely ramified Galois extensions of algebraic number fields

The Steinitz class of a number field extension K/k is an ideal class in the ring of integers Ok of k, which, together with the degree [K:k] of the extension determines the Ok-module structure of OK. We call Rt(k,G) the set of classes which are Steinitz classes of a tamely ramified G-extension of k. We will say that those classes are realizable for the group G; it is conjectured that the set of realizable classes is always a group. We define A^′-groups inductively, starting with abelian groups and then considering semidirect products of A^′-groups with abelian groups of relatively prime order and direct products of two A^′-groups. Our main result is that the conjecture about realizable Steinitz classes for tame extensions is true for A^′-groups of odd order; this covers many cases not previously known. Further we use the same techniques to determine Rt(k,Dn) for any odd integer n. In contrast with many other papers on the subject, we systematically use class field theory (instead of Kummer theory and cyclotomic descent).     

The Witt ring kernel for a fourth degree field extension and related problems

In the first part of this paper we compute the Witt ring kernel for an arbitrary field extension of degree 4 and characteristic different from 2 in terms of the coefficients of a polynomial determining the extension. In the case where the lower field is not formally real we prove that the intersection of any power n of its fundamental ideal and the Witt ring kernel is generated by n-fold Pfister forms.In the second part as an application of the main result we give a criterion for the tensor product of quaternion and biquaternion algebras to have zero divisors. Also we solve the similar problem for three quaternion algebras.In the last part we obtain certain exact Witt group sequences concerning dihedral Galois field extensions. These results heavily depend on some similar cohomological results of Positselski, as well as on the Milnor conjecture, and the Bloch-Kato conjecture for exponent 2, which was proven by Voevodsky.     

G-Closed fields and imbeddings of quadratic number fields

A field, K, that has no extensions with Galois group isomorphic to G is called G-closed. It is proved that a finite extension of K admits an infinite number of nonisomorphic extensions with Galois group G. A trinomial of degree n is exhibited with Galois group, the symmetric group of degree n, and with prescribed discriminant. This result is used to show that any quadratic extension of an A_n-closed field admits an extension with Galois group A_n.     

Sur l'indépendance l-adique de nombres algébriques

Let l a prime number and K a Galois extension over the field of rational numbers, with Galois group G. A conjecture is put forward on l-adic independence of algebraic numbers, which generalizes the classical ones of Leopoldt and Gross, and asserts that the l-adic rank of a G submodule of K^x depends only on the character of its Galois representation. When G is abelian and in some other cases, a proof is given of this conjecture by using l-adic transcendence results.     

Relative integral bases over a Hilbert class field

Let K be a number field, H its Hilbert class field and L a Galois extension of K containing H. In this paper, we prove that L|H has a relative integral basis (RIB) if the order of G=Gal(L|H) is odd or if the 2-Sylow subgroups of G are not cyclic. If the order of G is even and the 2-Sylow subgroups are cyclic we reduce the problem of the existence of a RIB to a quadratic extension of H.     

Ambiguous divisor classes in function fields

Let $\text{L}\text{K}$ be a Galois extension of algegraic function fields in one variable with Galois group G. Let J_K and J_L be the divisor classes of degree zero in K and L, respectively. A study is made of the kernel and cokernel of the natural map from J_K to $\text{J}{}_{\mathrm{L}}{}^{\mathrm{G}}$.     

Localizability of the embedding problem with symplectic kernel

In this paper we consider the question of how much information is supplied by local solutions to a global embedding problem for the special case in which the normal subgroup belonging to the given group extension is the projective symplectic group PSp(2m, q). It is proved that for suitable Galois extensions K of a given number field k (which constitute part of the data of the embedding problem), the local solutions in a sense determine whether or not an extension K ? K, Galois over k, with $\text{G(}\text{L}\text{K}\text{) ≈ PSp(2m, q)}$, represents a global solution to the embedding problem.     

Representability and Specht problem for G-graded algebras

Let W be an associative PI-algebra over a field F of characteristic zero, graded by a finite group G. Let idG(W) denote the T-ideal of G-graded identities of W. We prove: 1. [G-graded PI-equivalence] There exists a field extension K of F and a finite-dimensional Z/2Z×G-graded algebra A over K such that idG(W)=idG(A^∗) where A^∗ is the Grassmann envelope of A. 2. [G-graded Specht problem] The T-ideal idG(W) is finitely generated as a T-ideal. 3. [G-graded PI-equivalence for affine algebras] Let W be a G-graded affine algebra over F. Then there exists a field extension K of F and a finite-dimensional algebra A over K such that idG(W)=idG(A).     

Galois modular representation of associated Jacobians in the tamely ramified cyclic case

Let L/K be an ℓ-cyclic extension with Galois group G of algebraic function fields over an algebraically closed field k of characteristic p ≠  ℓ. In this paper, the -module structure of the ℓ-torsion of the Jacobian associated to L is explicitly determined.     

Ramification de certaines extensions de Lie p-adiques

Let G be a p-adic Lie group and let K be a finite extension of the p-adic number field ℚ _p . There are finitely many filtrations of G which could be ramification filtrations of totally ramified Galois extensions of K with Galois group G. Received: 19 October 1998     

Artin-Schreier extensions and Galois module structure

Let k be the power series field over a finite field of characteristic p>0. Let L be a cyclic extension over k of degree p. We give a necessary and sufficient condition for the integer ring O_L to be free over the associated order. Moreover, when O_L is free, we construct a free basis explicitly.     

Extensions of local fields and truncated power series

Let K be a finite tamely ramified extension of Q_p and let L/K be a totally ramified (Z/pⁿZ)-extension. Let π_L be a uniformizer for L, let σ be a generator for Gal(L/K), and let f(X) be an element of O_K[X] such that σ(π_L)=f(π_L). We show that the reduction of f(X) modulo the maximal ideal of O_K determines a certain subextension of L/K up to isomorphism. We use this result to study the field extensions generated by periodic points of a p-adic dynamical system.     

Equality of norm groups of subextensions of Sn(n?5) extensions of algebraic number fields

Let E/k be a Galois extension of algebraic number fields with the Galois group isomorphic to the symmetric group S_n on n?5 letters. For any field extensions k⊆K, L⊆E a necessary and a sufficient condition is given for the equality to hold, where is the group of norms from K to k of the elements of the multiplicative group K∗ of K.