Chapter IV: Euclidean Lattices, Theta Invariants, and Thermodynamic Formalism

Abstract

These are the notes of lectures delivered at Grenoble’s summer school on Arakelov Geometry and Diophantine Applications, in June 2017. They constitute an introduction to the study of Euclidean lattices and of their invariants defined in terms of theta series.

Recall that a Euclidean lattice is defined as a pair $\overline {E}:= (E, \Vert .\Vert )$ where E is some free $\mathbb {Z}$-module of finite rank E and ∥.∥ is some Euclidean norm on the real vector space $E_{\mathbb {R}} := E \otimes \mathbb {R}$. The most basic of these invariants is the non-negative real number:

$$\displaystyle h^0_\theta ({\overline E}) := \log \sum _{v \in E} e^{- \pi \Vert v \Vert ^2}. $$

In these notes, we explain how such invariants naturally arise when one investigates basic questions concerning classical invariants of Euclidean lattices, such as their successive minima, their covering radius, or the number of lattice points in balls of a given radius.

We notably discuss their significance from the perspective of Arakelov geometry and of the analogy between number fields and function fields, their role (discovered by Banaszczyk) in the derivation of optimal transference estimates, and their interpretation in terms of the formalism of statistical thermodynamics.

These notes have been primarily written for an audience of arithmetic geometers, but should also be suited to a wider circle of mathematicians and theoretical physicists with some interest in Euclidean lattices or in the mathematical foundations of statistical physics.