  In this memoir, we study the even unimodular lattices of rank at most 24, as
well as a related collection of automorphic forms of the orthogonal, symplectic
and linear groups of small rank. Our guide is the question of determining the
number of p-neighborhoods, in the sense of M. Kneser, between two isometry
classes of such lattices. We prove a formula for this number, in which occur
certain Siegel modular forms of genus 1 and 2. It has several applications,
such as the proof of a conjecture of G. Nebe and B. Venkov about the linear
span of the higher genus theta series of the Niemeier lattices, the computation
of the p-neighborhoods graphs of the Niemeier lattices (the case p = 2 being
due to Borcherds), or the proof of a congruence conjectured by G. Harder.
Classical arguments reduce the problem to the description of the automorphic
representations of a suitable integral form of the Euclidean orthogonal group
of R^24 which are unramified at each finite prime and trivial at the
archimedean prime. The recent results of J. Arthur suggest several new
approaches to this type of questions. This is the other main theme that we
develop in this memoir. We give a number of other applications, for instance to
the classification of Siegel modular cuspforms of weight at most 12 for the
full Siegel modular group.
