We assume the validity of the equivariant Tamagawa number conjecture for a certain motive attached to an abelian extension K/k of number fields, and we calculate the Fitting ideal of the dual of clK− as a Galois module, under mild extra hypotheses on K/k. This builds on concepts and results of Tate, Burns, Ritter and Weiss. If k is the field of rational numbers, our results are unconditional.

Let E be a CM number field and let S be a finite set of primes of E containing the primes dividing a given prime number l and another prime u split above the maximal totally real subfield of E. If ES denotes a maximal algebraic extension of E which is unramified outside S, we show that the natural maps are injective. We discuss generalizations of this result.

We prove that the p-adic zeta-function constructed by Kubota and Leopoldt has the Dirichlet series expansion Where the convergence of the first summation is for the p-adic topology. The proof of this formula relates the values of p(–s, ω1+σ) for s ∈ Zp, with a branch of the ‘sth-fractional derivative’, of a suitable generating function.

In this paper we compute and continue meromorphically to the whole complex plane the zeta function for twisted modular curves. The twist of the modular curve is done by a mod p representation of the absolute Galois group.

This is a study of relations between pure cubic fields and their normal closures. Explicit formula shows how the discriminant, regulator and class number of the normal closure can be expressed in terms of the cubic field.

We study the distribution of principal ideals generated by irreducible elements in an algebraic number field.

In this paper, we generalize the Kučera's group-determinant formulae to obtain the real and relative class number formulae of any subfield of cyclotomic function fields with arbitrary conductor in the form of a product of determinants. From these formulae, we generalize the relative class number formula of Rosen and Bae-Kang and obtain analogous results of Tsumura and Hirabayashi for an intermediate field in the tower of cyclotomic function fields with prime power conductor.

The set ℳ* of numbers which occur as Mahler measures of integer polynomials and the subset ℳ of Mahler measures of algebraic numbers (that is, of irreducible integer polynomials) are investigated. It is proved that every number α of degree d in ℳ* is the Mahler measure of a separable integer polynomial of degree at most with all its roots lying in the Galois closure F of ℚ(α), and every unit in ℳ is the Mahler measure of a unit in F of degree at most over ℚ This is used to show that some numbers considered earlier by Boyd are not Mahler measures. The set of numbers which occur as Mahler measures of both reciprocal and nonreciprocal algebraic numbers is also investigated. In particular, all cubic units in this set are described and it is shown that the smallest Pisot number is not the measure of a reciprocal number.

Let L/K be an extension of number fields and let be the subgroup of the unit group consisting of the elements that are roots of units of . Denote by (L/K, B) the number of points in with relative height in the sense of Bergé-Martinet at most B. Here ℙ1(L) stands for the one-dimensional projective space over L. In this paper is proved the formula (L/K, B) = CB2 + O(B2−1/[L:Q]), where C is a constant given in terms of invariants of L/K such as the regulators, class number and discriminant.

Let L be a Galois extension of the number field K. Set n = nK = deg K/ℚ, nL = deg L/ℚ and nL/K = deg L/K. Let I = IL/K denote the group of fractional ideals of K whose prime decomposition contains no prime ideals that ramify in L, and let P = {(α)ΣI: αΣK*, α>0}. Following Hecke [9}, let (λ1, λ2, …, λn − 1) be a basis for the torsion-free characters on P that satisfy λi(α) = 1 (1≤i≤n − 1) for all units α>0 in , the ring of integers of K. Fixing an extension of each λi to a character on I, then λi,(α) (1 ≤i≤n − 1) are defined for all ideals α of K that do not ramify in L. So, for such ideals, we can define . Then the small region of K referred to above is

for 0<l<½ and with the notation that, for any α ∈ ℝ, we set β. where β is the unique real satisyfing .

In this paper we derive a relation between character sums and partial sums of Dirichlet series.

We find a basis for the universal punctured even distribution and then a basis for the cyclotomic units over function fields.

The following theorem, due to Artin [1], is one of a number of statements often referred to as Chebotarev density theorem.

By a fundamental theorem of Brauer every irreducible character of a finite group G can be written in the field Q(εm) of mth roots of unity where m is the exponent of G. Is it always possible to find a matrix representation over its ring Z[εm] of integers? In the present paper it is shown that this holds true provided it is valid for the quasisimple groups. The reduction to such groups relies on a useful extension theorem for integral representations. Iwasawa theory on class groups of cyclotomic fields gives evidence that the answer is at least affirmative when the exponent is replaced by the order, and provides for a general qualitative result.

We study the genus theory for function fields which is the analogue of the classical genus theory developed by Hasse and Fröhlich.

Let K be a number field of degree nK = r1 + 2r2, a fixed integral ideal and the group of fractional ideals of K whose prime decomposition contains no prime factors of . Let

and be an arbitrary Groessencharaktere mod f as defined in [15]. Then and, for

where {λi} forms a basis for the torsion–free characters on whose value on any depends only on the that exists such that a = (α). Note that because of the choice in such an α we have that 1 for all units ε in K satisfying (mod ), ε>0. Also, x is a narrow ideal class character mod , that is, a character on

Let R⊂S be two orders in a number field, and let ER and ES be the respective groups of units in each ring. Then ES/ER and S/R are both finite. We consider the problem of bounding the order of ES/ER in terms of the index of R in S. In this paper we solve this problem in the special case that S/R is cyclic as a module over Z.

In the case of F-isotropic groups for a global field F, Moore [Mo] computed the metaplectic kernel using crucially his theorem of uniqueness of reciprocity laws. For F-anisotropic G, a variant of Moore's theorem is, therefore, needed to compute the metaplectic kernel. Such a variant was announced by G. Prasad [GP1] (in 1986) and here we give the details.

Let 1 ≤ M ≤ N − 1 be integers and K be a convex, symmetric set in Euclidean N-space. Associated with K and M, Mahler identified the Mth compound body of K, (K)m, in Euclidean (MN)-space. The compound body (K)M is describable as the convex hull of a certain subset of the Grassmann manifold in Euclidean (MN)-space determined by K and M. The sets K and (K)M are related by a number of well-known inequalities due to Mahler.

Here we generalize this theory to the geometry of numbers over the adèle ring of a number field and prove theorems which compare an adelic set with its adelic compound body. In addition, we include a comparison of the adelic compound body with the adelic polar body and prove an adelic general transfer principle which has implications to Diophantine approximation over number fields.

Let K be an algebraic number field, [ K: ] = KΣ. Most of what we shall discuss is trivial when K = , so that we assume that K ≥ 2 from now onwards. To describe our results, we consider the classical device [2] of Minkowski, whereby K is embedded (diagonally-) into the direct product MK of its completions at its (inequivalent) infinite places. Thus MK is -algebra isomorphic to , and is to be regarded as a topological -algebra, dimRMK = K, in which K is everywhere dense, while the ring Zx of integers of K embeds as a discrete -submodule of rank K. Following the ideas implicit in Hecke's fundamental papers [6] we may measure the “spatial distribution” of points of MK (modulo units of κ) by means of a canonical projection onto a certain torus . The principal application of our main results (Theorems I–III described below) is to the study of the spatial distribution of the which have a fixed norm n = NK/Q(α). In §2 we shall show that, with suitable interpretations, for “typical” n (for which NK/Q(α) = n is soluble), these α have “almost uniform” spatial distribution under the canonical projection onto TK. Analogous questions have been considered by several authors (see, e.g., [5, 9, 14]), but in all cases, they have considered weighted averages over such n of a type which make it impossible to make useful statements for “typical” n.