## Dirichlet's Theorem in projective general linear groups and the Absolute Siegel's Lemma

##### Abstract

This dissertation addresses two problems in diophantine number theory: (1) an
analogue of classical Dirichlet’s Theorem in a projective general linear group over a
local field and (2) a sharp bound on the conjugate products of successive minima
in the geometry of numbers over the adeles. For the first problem, we show that
if k is a local field and P GL(N, k) is endowed with a natural norm Φ, then unless
A ∈ P GL(N, k) is conjugate to an isometry, the orbit of A, {A, A2
, . . .} is bounded
away from the identity element 1N ; otherwise, we quantify min1≤m≤M Φ(Am) in
terms of M. We use classical techniques for the case k = C and group-theoretic
techniques for k nonarchimedean; as a side fact, we show that the maximal order
of an element in P GL(N, Fq) is (q
N − 1)/(q − 1). We also discuss a more general
vii
group-theoretic approach due to Vaaler that is suitable for any compact abelian
group and apply it to P GL(N, k). For the second problem, let k be a number field,
let A ∈ GL(N, kA) be an automorphism of the adeles kA, and let µn be the n-th
absolute successive minimum of A. We show that for 1 ≤ n ≤ N, the conjugate
proudct µn(A)µN−n+1(A∗
) is bounded above by e
(N−1)/2 where A∗ = (AT
)
−1
is the
dual of A. As a corollary, we deduce the (already known) Absolute Siegel’s Lemma,
i.e., the fact that QN
n=1 µn(A) ≤ e
N(N−1)/4
. We use induction, a symmetric algebra
argument due to Roy and Thunder, and new duality arguments to derive our results.

##### Department

##### Description

text