math.NT — Number Theory
The classic enumeration of genera of even definite lattices
We revisit the enumeration of a genus of even positive definite lattices by Kneser neighbours, certified by the mass. We work at primes dividing the determinant, where the projective spaces are smallest, schedule the exploration of the neighbour graph by a random walk and by orders based on a Hecke bookkeeping, and compute canonical forms through the Weyl chamber of the root system. Genera of rank up to 19 with up to 40,617 classes are enumerated in minutes.
A mix of human-written and AI-generated textHuman understanding: all partsHecke operatorsKneser neighbourscanonical formseven latticesgenus enumeration