Leopoldt's Conjecture Research Articles

PROJECTIVE EXTENSIONS OF FIELDS

A field K admits proper projective extensions, i.e. Galois extensions where the Galois group is a nontrivial projective group, unless K is separably closed or K is a pythagorean formally real field without cyclic extensions of odd degree. As a consequence, it turns out that almost all absolute Galois groups decompose as proper semidirect products. We show that each local field has a unique maximal projective extension, and that the same holds for each global field of positive characteristic. In characteristic 0, we prove that Leopoldt’s conjecture for all totally real number fields is equivalent to the statement that, for all totally real number fields, all projective extensions are cyclotomic. So the realizability of any non-procyclic projective group as Galois group over Q produces counterexamples to the Leopoldt conjecture, while the non-realizability may produce counterexamples to the classical inverse Galois problem.

Exchanging the places p and [formula omitted] in the Leopoldt conjecture

The Leopoldt conjecture is concerned with the image of the global units in the local units at the primes dividing p. In the definition of the global units the infinite place is distinguished. Exchanging p and infinity in the formulation one gets a new conjecture. It predicts that certain vectors should be linearly independent over the reals whose components are arguments of conjugates of Weil numbers. Using Baker's result on linear forms in logarithms we prove part of this new conjecture in certain abelian situations.

A sufficient condition for Leopoldt's conjecture

In this paper, by using an analogue of theorems of Iwasawa (Kenkichi Iwasawa Collected Papers, vol. 2, Springer, Berlin, 2001, pp. 862–870) we give a sufficient condition for Leopoldt's conjecture (J. Reine Angew. Math. 209 (1962) 54) on the non-vanishing of the p-adic regulator of an algebraic number field. Using this sufficient condition we are able to prove Leopoldt's conjecture for several non-Galois extensions over the rational number field Q .

Algèbres de Hecke quasi-ordinaires universelles

Let G be a connected reductive group over Q whose adjoint group admits discrete series representations. In this article, we construct a p-adic interpolation of the set of spaces of topological automorphic forms and we study the required properties to construct p-adic families of Hecke eigensystems. We impose a nearly-ordinarity assumption at p and we work with a local system of regular weight. Assuming Leopoldt conjecture, in the case of a unitary group in three variables associated with a CM extension of a totally real field F, we thus get p-adic families in 1+3[F: Q] variables.

K-groups with finite coefficients and arithmetic

In this paper we prove rank formulas for the even K-groups of number rings and relate Leopoldt's conjecture to K-theory. These results follow from a computation of the higher K-groups with finite coefficients.

Sur le rang d’une extension pro-$p$-libre d’un corps de nombres

For an algebraic number field $k$ and a prime number $p$ (if $p=2$, we assume that $\mu_4\subset k$), we study the maximal rank $\rho_k$ of a free pro-$p$-extension of $k$. We give various interpretations of $1+r_2(k)-\rho_k$. The first uses Iwasawa theory, the second uses the envelope of a module and the third is local-global. These expressions confirm that $1+r_2-\rho_k$ is related to the torsion of a certain Iwasawa module, hence to the dualizing module of a certain Galois group (under Leopoldt's conjecture).

Greenberg's conjecture and Leopoldt's conjecture

Let $p$ be an odd prime number. We show that the Iwasawa invariants of a certain non-abelian $p$-extension fields of $\mathbf{Q}$ vanish. And we construct non-abelian $p$-extensions over some imaginary quadratic fields satisfying Leopoldt's conjecture on the $p$-adic regulator.

The generic fiber of the universal deformation space associated to a tame Galois representation

We will study the generic fiber over of the universal deformation ring R Q , as defined by Mazur, for deformations unramified outside a finite set of primes Q of a given Galois representation , E a number field, k a finite field of characteristic l. The main result will be that, if ¯ρ is tame and absolutely irreducible, and if one assumes the Leopoldt conjecture for the splitting field E 0 of , then defines a smooth l-adic analytic variety, near the trivial lift ρ0 of ¯ρ, whose dimension is given by cohomological constraints and as predicted by Mazur. As a corollary it follows that, in the cases considered here, R Q is a quotient of by an ideal I generated by exactly m equations, where and . Under the above assumptions for and ¯ρ odd, using ideas of Coleman, Gouvea and Mazur it should now be possible to show that modular points are Zariski-dense in the component of , that contains the trivial lift ρ0, provided this lift satisfies the Artin conjecture and E 0 satisfies the Leopoldt conjecture. Furthermore, in the Borel case, we show that the Krull dimension of R Q can exceed any given number, provided Q is chosen appropriately. At the same time, we present some evidence that despite this fact, one might however expect that the dimension of the generic fiber is given by the same cohomological formula as in the tame case.

The structure of Selmer groups.

The purpose of this article is to describe certain results and conjectures concerning the structure of Galois cohomology groups and Selmer groups, especially for abelian varieties. These results are analogues of a classical theorem of Iwasawa. We formulate a very general version of the Weak Leopoldt Conjecture. One consequence of this conjecture is the nonexistence of proper Lambda-submodules of finite index in a certain Galois cohomology group. Under certain hypotheses, one can prove the nonexistence of proper Lambda-submodules of finite index in Selmer groups. An example shows that some hypotheses are needed.

A note on free pro-p-extensions of algebraic number fields II

For an algebraic number field k and a prime p, define the number p to be the maximal number d such that there exists a Galois extension of k whose Galois group is a free pro-p-group of rank d. The Leopoldt conjecture implies 1~ p ~ r2 +1, (r2 denotes the number of complex places of k). Some examples of k and p with 03C1 = r2 + 1 have been known so far. In this note, the invariant 03C1 is studied, and among other things some examples with p r2 + 1 are given.

A note on free pro-$p$-extensions of algebraic number fields

For an algebraic number field k and a prime p, define the number p to be the maximal number d such that there exists a Galois extension of k whose Galois group is a free pro-p-group of rank d. The Leopoldt conjecture implies 1~ p ~ r2 +1, (r2 denotes the number of complex places of k). Some examples of k and p with 03C1 = r2 + 1 have been known so far. In this note, the invariant 03C1 is studied, and among other things some examples with p r2 + 1 are given.

Some remarks on Leopoldt's conjecture

Some remarks on Leopoldt's conjecture

Leopoldt's conjecture for imaginary Galois number fields

In studying Leopoldt's conjecture for Galois number fields a sufficient condition is proposed which includes the known criteria and moreover refers only to the character table of the Galois group in question. Hence it may easily be checked. Tables of the computations are given. New examples, if only few, of imaginary number fields are exhibited for which Leopoldt's conjecture is proved to be true for all primes p . Some of them are covered by some kind of “Verschiebungssatz” for Leopoldt's conjecture.

Remarks on connections between the Leopoldt conjecture, p-class groups and unit groups of algebraic number fields

Remarks on connections between the Leopoldt conjecture, p-class groups and unit groups of algebraic number fields

Formula omitted]p-Extensions of complex multiplication fields

Suppose, K is a number field, and p is an odd prime. Setting R K = D K [ p −1 ] with D K being the ring of integers of K, the isomorphism classes of Z p -extensions of R K form a Z p -module H 1 ( R K , Z p ). Leopoldt's conjecture states that H 1 ( R K , Z p ) ≅ Z p r 2 + 1 . In this paper we define a Z p -submodule H( R K , Z p ) of H 1 ( R K , Z p ) consisting of those classes of Z p -extensions having a normal basis over R K . We prove for a complex multiplication field K that there is an isomorphism H( R K , Z p ) ≅ Z p r 2 + 1 .

Kummer's and Iwasawa's Version of Leopoldt's Conjecture

AbstractWe present a refinement of Iwasawa's approach to Leopoldt's conjecture on the non-vanishing of the p-adic regulator of an algebraic number field K. As an application, the conjecture for K implies the conjecture for a solvable extension L of degree g over K if g is relatively prime to p — 1 and p does not divide g, the discriminant of K, and the quotient of class numbers where is a primitive pth root of unity. This can be viewed as generalizing a theorem of Kummer on cyclotomic units.

Leopoldt's Conjecture in Parameterized Families

Leopoldt's Conjecture in Parameterized Families

An algorithm for testing Leopoldt's conjecture

Based on an elementary version of Leopoldt's conjecture due to Iwasawa and Sands we develop an algorithm for testing this conjecture in an arbitrary algebraic number field for any prime p. Using this algorithm we are able to prove Leopoldt's conjecture for several pure fields of degree 5 and 7. We also discuss relations with class numbers.

On the Leopoldt conjecture on the p-adic regulators

The Leopoldt conjecture for ( K, p) is investigated under the assumption that the Leopoldt conjecture for ( k, p) is valid, where k is a finite algebraic number field and K k is a cyclic extension of prime degree p. A lower bound for the p-adic absolute value of the p-adic regulator R p ( K) of K is also given when k = Q.

Indépendance Linéaire sur [formula omitted] de logarithmes P-adiques de nombres algébriques et rang P-adique du groupe des unités d'un corps de nombres

Following Ax's method a lower bound for the p-adic rank of the group of units in the general case of a Galois number field over the rationals is given. In some nontrivial special cases the result gives Leopoldt's conjecture. The same method is also applied to the case of p-units.