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.