The Compositum of Wild Extensions of Local Fields of Prime Degree

In this paper we present a general view of the totally and wildly ramified extensions of degree p of a p-adic field K. Our method consists in deducing the properties of the set of all extensions of degree p of K from the study of the compositum of all its elements. We show that in fact is the maximal abelian extension of exponent p of F = F(K), where F is the compositum of all cyclic extensions of K of degree dividing p − 1. By our method, it is fairly simple to recover the distribution of the extensions of K of degree p (and also of their isomorphism classes) according to their discriminant.     

Normal integral bases of ∞ -ramified abelian extensions of totally real number fields

LetF be a totally real number field and [ a product of some real primes ofF. J. Brinkhuis gave a necessary condition for a finite abelian extension ofF which is unramified outside [ to have a normal integral basis. We consider the converse of his result and give a necessary and sufficient condition. Furthermore, we concretely express it whenF is a real quadratic field or a cyclic cubic field.     

Application of Greenberg's conjecture in the capitulation problem

Let r be a positive integer. Assume Greenberg's conjecture for some totally real number fields, we show that there exists an infinite family of imaginary cyclic number fields F over the field of rational number field , with an elementary 2‐class group of rank equal to r that capitulates in an unramified quadratic extension over F. Also, we give necessary and sufficient conditions for the Galois group of the unramified maximal 2‐extension over F to be abelian.     

Parametrization of tamely ramified maximal tori using bounded subgroups

Let be a reductive group defined over a local complete field F with discrete valuation, and split over some unramified extension of F, and let G be its group of F-points. In this paper, we define a class of abelian “torus-like” subgroups in nonreductive groups, called pseudo-tori, which generalizes the notion of torus, and we establish a correspondence between conjugacy classes of tamely ramified maximal tori of G and association classes of maximal pseudo-tori of the quotients of parahorics of G by their second congruence subgroup, viewed as groups of k-points of algebraic groups defined over the residual field k of F.      

On the construction of normal wildly ramified extensions over Q 2

It is well known that a finite totally ramified extension of a local field can be generated by a uniformising element the minimal polynomial of which is also Eisenstein. The quadratic and the quartic normal totally ramified extensions of Q ₂ are well known and well characterized. In this note we characterize the Eisenstein polynomials of degree 4 with coefficients in Z ₂ that define normal totally ramified extensions of Q ₂. Furthermore we give some necessary conditions for the cyclic case of degree 2 ⁿ . Also examples are given.     

The Small Index Property for Free Nilpotent Groups

Let F be a relatively free algebra of infinite rank ?. We say that F has the small index property if any subgroup of Γ = Aut(F) of index at most ? contains the pointwise stabilizer Γ_(U) of a subset U of F of cardinality less than ?. We prove that every infinitely generated free nilpotent/abelian group has the small index property, and discuss a number of applications.     

Representation fields for quaternionic skew-hermitian forms

Classes of indefinite quadratic forms in a genus are in correspondence with the Galois group of an abelian extension called the spinor class field (Estes and Hsia, Japanese J. Math. 16, 341–350 (1990)). Hsia has proved (Hsia et al., J. Reine Angew. Math. 494, 129–140 (1998)) the existence of a representation field F with the property that a lattice in the genus represents a fixed given lattice if and only if the corresponding element of the Galois group is trivial on F. This far, the corresponding result for skew-hermitian forms was known only in some special cases, e.g., when the ideal (2) is square free over the base field. In this work we prove the existence of representation fields for quaternionic skew-hermitian forms in complete generality.     

Inertia groups and abelian surfaces

This paper classifies the finite groups that occur as inertia groups associated to abelian surfaces. These groups can be viewed as Galois groups for the smallest totally ramified extension over which an abelian surface over a local field acquires semistable reduction. The results extend earlier elliptic curves results of Serre and Kraus.     

A Valuation Theory for Nonassociative Quaternion Algebras

A nonassociative quaternion algebra over a field F is a 4-dimensional F-algebra A whose nucleus is a separable quadratic extension field of F. We define the notion of a valuation ring for A, and we also define a value function on A with values from a totally ordered group. We determine the structure of the set on which the function assumes non-negative values and we prove that, given a valuation ring of A, there is a value function associated to it if and only if the valuation ring is integral and invariant under proper F-automorphisms of A.     

Companion forms over totally real fields

We show that if F is a totally real field in which p splits completely and f is a mod p Hilbert modular form with parallel weight 2 < k < p, which is ordinary at all primes dividing p and has tamely ramified Galois representation at all primes dividing p, then there is a “companion form” of parallel weight k′ := p + 1 − k. This work generalises results of Gross and Coleman–Voloch for modular forms over Q.     

Normal integral bases and strict ray class groups modulo 4

Let F be a number field. We construct three tamely ramified quadratic extensions which are ramified at most at some given set of finite primes, such that K₃⊂K₁K₂, both K₁/F and K₂/F have normal integral bases, but K₃/F has no normal integral basis. Since Hilbert-Speiser's theorem yields that every finite and tamely ramified abelian extension over the field of rational numbers has a normal integral basis, it seems that this example is interesting (cf. [5] J. Number Theory 79 (1999) 164; Theorem 2). As we shall explain below, the previous papers (Acta Arith. 106 (2) (2003) 171-181; Abh. Math. Sem. Univ. Hamburg 72 (2002) 217-233) motivated the construction. We prove that if the class number of F is bigger than 1, or the strict ray class group of F modulo 4 has an element of order ?3, then there exist infinitely many triplets (K₁,K₂,K₃) of such fields.     

On holomorphic polydifferentials in positive characteristic

Let F/E be an abelian Galois extension of function fields over an algebraic closed field K of characteristic p > 0. Denote by G the Galois group of the extension F/E. In this paper, we study Ω(m), the space of holomorphic m‐(poly)differentials of the function field of F when G is cyclic or a certain elementary abelian group of order pⁿ; we give bases for each case when the base field is rational, introduce the Boseck invariants and give an elementary approach to the G module structure of Ω(m) in terms of Boseck invariants. The last computation is achieved without any restriction on the base field in the cyclic case, while in the elementary abelian case it is assumed that the base field is rational. Finally, an application to the computation of the tangent space of the deformation functor of curves with automorphisms is given.     

FORMAL GROUPS AND LOCAL CLASS FIELD THEORY

The purpose of this paper is to prove that every abelian extenison of a local field can be embedded into certain generalized Lubin-Tate extensions. As a consequence of the embedding theorem, a new proof of local class field theory is given, which looks more intuitive than Galois cohomology. Also the author gets a necessary and sufficient condition for a totally ramified extension of degree p to be normal in terms of the coefficients of its definition equation.     

Roots of Irreducible Polynomials in Tame Henselian Extension Fields

A class of irreducible polynomials 𝒫 over a valued field (F, v) is introduced, which is the set of all monic irreducible polynomials over F when (F, v) is maximally complete. A “best-possible” criterion is given for when the existence of an approximate root in a tamely ramified Henselian extension K of F of a polynomial f in 𝒫 guarantees the existence of an exact root of f in K.     

Square classes of totally positive units

In this article, we investigate some conditions for a real cyclic extension K over Q to satisfy the property that every totally positive unit of K is a square. As an application, we give a partial answer to Taussky's conjecture. We then extend our result to real abelian extensions of certain type.     

Extensions with Almost Maximal Depth of Ramification

The paper is devoted to classification of finite abelian extensions L/K which satisfy the condition [L:K]|D _L/K. Here K is a complete discretely valued field of characteristic 0 with an arbitrary residue field of prime characteristic p, D _L/K is the different of L/K. This condition means that the depth of ramification in L/K has its almost maximal value. The condition appeared to play an important role in the study of additive Galois modules associated with the extension L/K. The study is based on the use of the notion of independently ramified extensions, introduced by the authors. Two principal theorems are proven, describing almost maximally ramified extensions in the cases when the absolute ramification index e is (resp. is not) divisible by p-1. Bibliography: 7 titles.     

Weights in Serre’s conjecture for Hilbert modular forms: The ramified case

Let F be a totally real field and p ≥ 3 a prime. If ρ : is continuous, semisimple, totally odd, and tamely ramified at all places of F dividing p, then we formulate a conjecture specifying the weights for which ρ is modular. This extends the conjecture of Diamond, Buzzard, and Jarvis, which required p to be unramified in F. We also prove a theorem that verifies one half of the conjecture in many cases and use Dembélé’s computations of Hilbert modular forms over to provide evidence in support of the conjecture. The author thanks the NSF for a Graduate Research Fellowship that supported him during part of this work.     

n-Torsion of Brauer groups as relative Brauer groups of abelian extensions

It is now known [H. Kisilevsky, J. Sonn, Abelian extensions of global fields with constant local degrees, Math. Res. Lett. 13 (4) (2006) 599-607; C.D. Popescu, Torsion subgroups of Brauer groups and extensions of constant local degree for global function fields, J. Number Theory 115 (2005) 27-44] that if F is a global field, then the n-torsion subgroup of its Brauer group Br(F) equals the relative Brauer group Br(Ln/F) of an abelian extension Ln/F, for all n∈Z?1. We conjecture that this property characterizes the global fields within the class of infinite fields which are finitely generated over their prime fields. In the first part of this paper, we make a first step towards proving this conjecture. Namely, we show that if F is a non-global infinite field, which is finitely generated over its prime field and ?≠char(F) is a prime number such that μ?²⊆F^×, then there does not exist an abelian extension L/F such that . The second and third parts of this paper are concerned with a close analysis of the link between the hypothesis μ?²⊆F^× and the existence of an abelian extension L/F such that , in the case where F is a Henselian valued field.