Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field
103
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let k be a global field and let k_v be the completion of k with respect to v, a non-archimedean place of k. Let mathbf{G} be a connected, simply-connected algebraic group over k, which is absolutely almost simple of k_v-rank 1. Let G=mathbf{G}(k_v). Let Gamma be an arithmetic lattice in G and let C=C(Gamma) be its congruence kernel. Lubotzky has shown that C is infinite, confirming an earlier conjecture of Serre. Here we provide complete solution of the congruence subgroup problem for Gamm$ by determining the structure of C. It is shown that C is a free profinite product, one of whose factors is hat{F}_{omega}, the free profinite group on countably many generators. The most surprising conclusion from our results is that the structure of C depends only on the characteristic of k. The structure of C is already known for a number of special cases. Perhaps the most important of these is the (non-uniform) example Gamma=SL_2(mathcal{O}(S)), where mathcal{O}(S) is the ring of S-integers in k, with S={v}, which plays a central role in the theory of Drinfeld modules. The proof makes use of a decomposition theorem of Lubotzky, arising from the action of Gamma on the Bruhat-Tits tree associated with G.
rate research
Read More
Let $n$ be a positive integer and let $f_1, ldots, f_r$ be polynomials in $n^2$ indeterminates over an algebraically closed field $K$. We describe an algorithm to decide if the invertible matrices contained in the variety of $f_1, ldots, f_r$ form a subgroup of $GL(n,K)$; that is, we show how to decide if the polynomials $f_1, ldots, f_r$ define a linear algebraic group.
In this paper we study local-global principles for tori over semi-global fields, which are one variable function fields over complete discretely valued fields. In particular, we show that for principal homogeneous spaces for tori over the underlying discrete valuation ring, the obstruction to a local-global principle with respect to discrete valuations can be computed using methods coming from patching. We give a sufficient condition for the vanishing of the obstruction, as well as examples were the obstruction is nontrivial or even infinite. A major tool is the notion of a flasque resolution of a torus.
Fixing an arithmetic lattice $Gamma$ in an algebraic group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $Delta$ with $[Gamma : Gamma cap Delta] [Delta: Gamma cap Delta] = n$. This growth function gives a new setting where methods of F. Grunewald, D. Segal, and G. C. Smiths Subgroups of finite index in nilpotent groups apply to study arithmetic lattices in an algebraic group. In particular, we show that for any unipotent algebraic $mathbb{Z}$-group with arithmetic lattice $Gamma$, the Dirichlet function associated to the commensurability growth function satisfies an Euler decomposition. Moreover, the local parts are rational functions in $p^{-s}$, where the degrees of the numerator and denominator are independent of $p$. This gives regularity results for the set of arithmetic lattices in $G$.
Let $A$ be the ring of elements in an algebraic function field $K$ over a finite field $F_q$ which are integral outside a fixed place $infty$. In an earlier paper we have shown that the Drinfeld modular group $G=GL_2(A)$ has automorphisms which map congruence subgroups to non-congruence subgroups. Here we prove the existence of (uncountably many) normal genuine non-congruence subgroups, defined to be those which remain non-congruence under the action of every automorphism of $G$. In addition, for all but finitely many cases we evaluate $ngncs(G)$, the smallest index of a normal genuine non-congruence subgroup of $G$, and compare it to the minimal index of an arbitrary normal non-congruence subgroup.
We prove that the automorphism group of the braid group on four strands acts faithfully and geometrically on a CAT(0) 2-complex. This implies that the automorphism group of the free group of rank two acts faithfully and geometrically on a CAT(0) 2-complex, in contrast to the situation for rank three and above.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
