Torsion points of elliptic curves over large algebraic extensions of finitely generated fields

The following Theorem is proved:Let K be a finitely generated field over its prime field. Then, for almost all e-tuples (σ)=(σ ₁, …,σ _e)of elements of the abstract Galois group G(K)of K we have:

1. If e=1,then E _tor(K(σ))is infinite. Morover, there exist infinitely many primes l such that E(K(σ))contains points of order l.
2. If e≧2,then E _tor(K(σ))is finite.
3. If e≧1,then for every prime l, the group E(K(σ))contains only finitely many points of an l-power order.

HereK(σ) is the fixed field in the algebraic closureK ofK, ofσ ₁, …,σ _e, and “almost all” is meant in the sense of the Haar measure ofG(K).     

Difference algebraic subgroups of commutative algebraic groups over finite fields

We study the question of which torsion subgroups of commutative algebraic groups over finite fields are contained in modular difference algebraic groups for some choice of a field automorphism. We show that if G is a simple commutative algebraic group over a finite field of characteristic p, ? is a prime different from p, and for some difference closed field (?, σ) the ?-primary torsion of G(?) is contained in a modular group definable in (?, σ), then it is contained in a group of the form {x∈G(?) :σ(x) =[a](x) } with a∈ℕ\p ^ℕ. We show that no such modular group can be found for many G of interest. In the cases that such equations may be found, we recover an effective version of a theorem of Boxall. Received: 28 May 1998 / Revised version: 20 December 1998     

Simple algebraic and semialgebraic groups over real closed fields

We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In Definably simple groups in o-minimal structures, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic language, is that every such group is either bi-interpretable with an algebraically closed field of characteristic zero (when the group is stable) or with a real closed field (when the group is unstable). It follows that every abstract isomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. We discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.

     

Abelian varieties over large algebraic fields with infinite torsion

Let A be a non-zero abelian variety defined over a number field K and let \(\overline K \) be a fixed algebraic closure of K. For each element σ of the absolute Galois group \({\text{Gal}}(\overline K /K)\), let \(\overline K (\sigma )\) be the fixed field in \(\overline K \) of σ. We show that the torsion subgroup of \(A(\overline K (\sigma ))\) is infinite for all \(\sigma \in {\text{Gal}}(\overline K /K)\) outside of some set of Haar measure zero. This proves the number field case of a conjecture of W.-D. Geyer and M. Jarden.     

Questions of rationality for solvable algebraic groups over nonperfect fields

Sunto Dato un gruppo algebrico risolubile, insieme con un suo campo di definizione, si dimostra che è possibile trovare certi sottogruppi di tipi importanti, p. e. tori massimi, senza introdurre alcuna inseparabilità

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This research was supported by the Air Force Office of Scientific Research.     

On abstract homomorphisms of anisotropic algebraic groups over real-closed fields

Abstract homomorphisms with Zariski-dense image of the group of rational points of anisotropic almost simple algebraic groups over real closed fields into other almost simple algebraic groups are described. The result states, in particular, that the kernel of such homomorphism is a congruence subgroup of the original group.     

Structure of groups of rational points of classical algebraic groups over number fields

Dedicated to the memory of A. I. Mal'tsev.     

Discrete subgroups of algebraic groups over local fields of positive characteristics

It is shown in this paper that ifG is the group ofk-points of a semisimple algebraic groupG over a local fieldk of positive characteristic such that all itsk-simple factors are ofk-rank 1 and Γ ⊂G is a non-cocompact irreducible lattice then Γ admits a fundamental domain which is a union of translates of Siegel domains. As a consequence we deduce that ifG has more than one simple factor, then Γ is finitely generated and by a theorem due to Venkataramana, it is arithmetic.     

Theory of spherical functions on reductive algebraic groups over p-adic fields

Publications mathématiques de l'IHÉS -     

Sharply transitive linear algebraic groups

We study Zariski-closed linear groupsG GL _n (k) over fieldsk of characteristic 0 which act sharply transitively on the non-zero vectors ofk ⁿ . For square-freen, orn15, or ifk has cohomological dimension 1 we obtain a complete classification (i.e. a reduction to questions about associative division algebras). The main tools are representation theory of Lie algebras over algebraically closed and non-closed fields, and results about simple associative algebras in order to control the interplay between linear Lie algebras and the associative algebras generated by them. The relation to nearfields and left-symmetric division algebras is also discussed.     

Whittaker-Shintani functions for general linear groups over p-adic fields

In this paper, we recreate the Whittaker-Shintani functions for general linear groups over nonarchimedean fields given by Kato et al. Those generalized spherical functions naturally arise from global zeta integrals of automorphic L-functions. More explicitly, this formula plays a fundamental role in the local calculation over the split places of tensor product L-functions defined by Jiang and Zhang(2014) and the twisted automorphic descents introduced by Jiang and Zhang(2015).     

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields II

Let k be a separably closed field. Let G be a reductive algebraic k-group. We study Serre’s notion of complete reducibility of subgroups of G over k. In particular, using the recently proved center conjecture of Tits, we show that the centralizer of a k-subgroup H of G is G-completely reducible over k if it is reductive and H is G-completely reducible over k. We show that a regular reductive k-subgroup of G is G-completely reducible over k. We present examples where the number of overgroups of irreducible subgroups and the number of G(k)-conjugacy classes of k-anisotropic unipotent elements are infinite.     

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields III

Let G be a reductive group over a nonperfect field k. We study rationality problems for Serre’s notion of complete reducibility of subgroups of G. In our previous work, we constructed examples of subgroups H of G that are G-completely reducible but not G-completely reducible over k (and vice versa). In this article, we give a theoretical underpinning of those constructions. Then using Geometric Invariant Theory, we obtain a new result on the structure of G(k)-(and G-) orbits in an arbitrary affine G-variety. We discuss several related problems to complement the main results.