Applications

Abstract

In a first part we explain how Theorem E of the introduction leads to a solution of the p-neighbor problem which involves in particular the four integers τ j,k (p) introduced in Chap. 9, which are “genus 2 analogs” of the Ramanujan τ(p). Using the analysis made in Chap. 3 of neighborhoods of the Leech lattice, we determine τ j,k (p) for p ≤ 113 (Theorem H), hence obtain an explicit numerical solution of the p-neighbor problem for those primes. In a second part we deduce from Theorem E numerous congruences for the τ j,k (p) which may be thought of as “genus 2 analogs” of the famous mod 691 congruence satisfied by τ(p); one of these congruences had been conjectured by Günter Harder (Theorem I). This part notably involves the theory of Galois representations.