Arithmetic equivalence and successive minima

Two number fields are said to be arithmetically equivalent if they have the same Dedekind zeta function. The central question about arithmetic equivalence is to determine how "similar" arithmetically equivalent number fields are. That is, we would like to determine which arithmetic invariants, such as the degree, discriminant, signature, units, class number, etc., are the same, and which ones can differ. A key result about arithmetic equivalence is Gassmann's theorem, which allows one to answer such questions using Galois theory and representation theory. I will give a general introduction to arithmetic equivalence, discussing some of the main results such as Gassmann's theorem and giving examples. I will then introduce the successive minima of a number field, and show that arithmetically equivalent number fields have approximately the same successive minima. "