Cohomology of the Weil group of a p-adic field

We establish duality and vanishing results for the cohomology of the Weil group of a p-adic field. Among them is a duality theorem for finitely generated modules, which implies Tate–Nakayama Duality. We prove comparison results with Galois cohomology, which imply that the cohomology of the Weil group determines that of the Galois group. When the module is defined by an abelian variety, we use these comparison results to establish a duality theorem analogous to Tate?s duality theorem for abelian varieties over p-adic fields.     

Circular units in a bicyclic field

For a real abelian field with a non-cyclic Galois group of order l², l being an odd prime, the index of the Sinnott group of circular units is computed.     

On Galois representations for abelian varieties with complex and real multiplications

In this paper, we study the image of l-adic representations coming from Tate module of an abelian variety defined over a number field. We treat abelian varieties with complex and real multiplications. We verify the Mumford-Tate conjecture for a new class of abelian varieties with real multiplication.     

Abelian varieties over finitely generated fields and the conjecture of Geyer and Jarden on torsion

In this paper we prove the Geyer‐Jarden conjecture on the torsion part of the Mordell‐Weil group for a large class of abelian varieties defined over finitely generated fields of arbitrary characteristic. The class consists of all abelian varieties with big monodromy, i.e., such that the image of Galois representation on ?‐torsion points, for almost all primes ?, contains the full symplectic group.     

Extensions of abelian varieties defined over a number field

We study the arithmetic aspects of the finite group of extensions of abelian varieties defined over a number field. In particular, we establish relations with congruences between modular forms and special values of L-functions.     

On reduction maps and support problem in K-theory and abelian varieties

In this paper we consider orders of images of nontorsion points by reduction maps for abelian varieties defined over number fields and for odd dimensional K-groups of number fields. As an application we obtain the generalization of the support problem for abelian varieties and K-groups.     

Deformations of locally abelian Galois representations and unramified extensions

We study the deformation theory of Galois representations whose restriction to every decomposition subgroup is abelian. As an application, we construct unramified non-solvable extensions over the field obtained by adjoining all p-power roots of unity to the field of rational numbers.     

Fully maximal and fully minimal abelian varieties

We introduce and study a new way to categorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed or fully minimal. The type of A depends on the normalized Weil numbers of A and its twists. We analyze these types for supersingular abelian varieties and curves under conditions on the automorphism group. In particular, we present a complete analysis of these properties for supersingular elliptic curves and supersingular abelian surfaces in arbitrary characteristic, and for a one-dimensional family of supersingular curves of genus 3 in characteristic 2.     

Galois module structure of unramified covers

We use the theory of n-cubic structures to study the Galois module structure of the coherent cohomology groups of unramified Galois covers of varieties over the integers. Assuming that all the Sylow subgroups of the covering group are abelian, we show that the invariant that measures the obstruction to the existence of a “virtual normal integral basis” is annihilated by a product of certain Bernoulli numbers with orders of even K-groups of Z. We also show that the existence of such a basis is closely connected to the truth of the Kummer-Vandiver conjecture for the prime divisors of the degree of the cover. Partially supported by NSF grants # DMS05-01049 and # DMS01-11298 (via the Institute for Advanced Study).     

How often can a finite group be realized as a Galois group over a field?

Let G be any finite group and any class of fields. By we denote the minimal number of realizations of G as a Galois group over some field from the class . For G abelian and the class of algebraic extensions of ℚ we give an explicit formula for . Similarly we treat the case of an abelian p-group G and the class which is conjectured to be the class of all fields of characteristic ≠p for which the Galois group of the maximal p-extension is finitely generated. For non-abelian groups G we offer a variety of sporadic results. Received: 27 October 1998 / Revised version: 3 February 1999     

The distance between superspecial abelian varieties with real multiplication

Let L be a totally real field of strict class number one and let OL be its ring of integers. Let p be a rational prime which is unramified in L. We consider the distance between two superspecial abelian varieties with real multiplication in characteristic p, where by “distance” we mean the minimal degree of an OL-isogeny. We give upper and lower bounds on the distance between superspecial abelian varieties with real multiplication by L in characteristic p in terms of p and the degree and discriminant of L.     

Computing the Selmer group of an isogeny between Abelian varieties using a further isogeny to a Jacobian

This article describes an algorithm for computing the Selmer group of an isogeny between abelian varieties. This algorithm applies when there is an isogeny from the image abelian variety to the Jacobian of a curve. The use of an auxiliary Jacobian simplifies the determination of locally trivial cohomology classes. An example is presented where the rational solutions to x⁴+(y²+1)(x+y)=0 are determined.     

Pure Anderson motives over finite fields

In the arithmetic of function fields Drinfeld modules play the role that elliptic curves take on in the arithmetic of number fields. As higher dimensional generalizations of Drinfeld modules, and as the appropriate analogues of abelian varieties, G. Anderson introduced pure t-motives. In this article we study the arithmetic of the latter. We investigate which pure t-motives are semisimple, that is, isogenous to direct sums of simple ones. We give examples for pure t-motives which are not semisimple. Over finite fields the semisimplicity is equivalent to the semisimplicity of the endomorphism algebra, but also this fails over infinite fields. Still over finite fields we study the Zeta function and the endomorphism rings of pure t-motives and criteria for the existence of isogenies. We obtain answers which are similar to Tate's famous results for abelian varieties.     

Galois representations,Mumford-Tate groups and good reduction of abelian varieties

Let K be a number field and A an abelian variety over K. We are interested in the following conjecture of Morita: if the Mumford-Tate group of A does not contain unipotent -rational points then A has potentially good reduction at any discrete place of K. The Mumford-Tate group is an object of analytical nature whereas having good reduction is an arithmetical notion, linked to the ramification of Galois representations. This conjecture has been proved by Morita for particular abelian varieties with many endomorphisms (called of PEL type). Noot obtained results for abelian varieties without nontrivial endomorphisms (Mumfords example, not of PEL type). We give new results for abelian varieties not of PEL type.An erratum to this article can be found at     

On the Galois module structure over CM-fields

In this paper we make a contribution to the problem of the existence of a normal integral basis. Our main result is that unramified realizations of a given finite abelian group Δ as a Galois group Gal (N/K) of an extensionN of a givenCM-fieldK are invariant under the involution on the set of all realizations of Δ overK which is induced by complex conjugation onK and by inversion on Δ. We give various implications of this result. For example, we show that the tame realizations of a finite abelian group Δ of odd order over a totally real number fieldK are completely characterized by ramification and Galois module structure.     

Links between associated additive Galois modules and computation of H for local formal group modules

Using the methods described in the papers (Documenta Math. 5 (2000) 657; Local Leopoldt's problem for ideals in p-extensions of complete discrete valuation fields, to appear), we prove that a cocycle for a formal group in a Galois p-extension of a complete discrete valuation field is a coboundary if and only if the corresponding group algebra elements increase valuations by a number that is sufficiently large. We also calculate the valuation of the splitting element of a coboundary. A special case of the main theorem allows us to determine when a p-extension of a complete discrete valuation fields contains a root of a Kummer equation for a formal group. The theorem of Coates-Greenberg for formal group modules in deeply ramified extensions is generalized to noncommutative formal groups. Some results concerning finite torsion modules for formal groups are obtained.     

Counting MSTD sets in finite abelian groups

In an abelian group G, a more sums than differences (MSTD) set is a subset A⊂G such that |A+A|>|A−A|. We provide asymptotics for the number of MSTD sets in finite abelian groups, extending previous results of Nathanson. The proof contains an application of a recently resolved conjecture of Alon and Kahn on the number of independent sets in a regular graph.     

Torsion bounds for elliptic curves and Drinfeld modules

We derive upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L:K]. Our main tool is the adelic openness of the image of Galois representations, due to Serre, Pink and Rütsche. Our approach is to prove a general result for certain Galois modules, which applies simultaneously to elliptic curves and to Drinfeld modules.     

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.