Statistics for Kneser p-neighbors

TL;DR

This paper derives asymptotic formulas for counting Kneser p-neighbors of integral Euclidean lattices, explaining recent classification observations and connecting automorphic representations to orthogonal groups.

Contribution

It provides new asymptotic formulas for Kneser p-neighbors of lattices, including cases with additional subgroup constraints, linking lattice theory with automorphic forms.

Findings

Asymptotic formulas for Kneser p-neighbors as p approaches infinity

Explanation of numerical classifications of unimodular lattices

Connection between automorphic representations and orthogonal groups

Abstract

Let L and L' be two integral Euclidean lattices in the same genus. We give an asymptotic formula for the number of Kneser p-neighbors of L which are isometric to L', when the prime p goes to infinity. In the case L is unimodular, and if we fix furthermore a subgroup A of L, we also give an asymptotic formula for the number of p-neighbors of L containing A and which are isomorphic to L'. These statements explain numerical observations in the recent classifications of unimodular lattices of rank 26, 27 and 28, by B. Allombert and the author. In an Appendix, O. Ta\"ibi shows how to deduce from Arthur's results the existence of global parameters associated to automorphic representations of definite orthogonal groups over the rationals.