Enumerating maximal definite quadratic forms of bounded class number over Z in n >= 3 variables

TL;DR

This paper presents an algorithm to enumerate all primitive positive definite maximal quadratic forms over integers in three or more variables with bounded class number, and provides a complete list for class number one.

Contribution

The paper introduces a novel algorithm based on the mass formula to finitely enumerate such quadratic forms, including an open-source implementation.

Findings

Exactly 115 primitive positive definite maximal quadratic forms of class number one in n >= 3 variables.

Algorithm successfully enumerates forms with bounded class number using local invariants.

Provides a complete list of these forms for class number one.

Abstract

In this paper we give an algorithm for enumerating all primitive (positive) definite maximal Z-valued quadratic forms Q in n >= 3 variables with bounded class number h(Q) <= B. We do this by analyzing the exact mass formula [GHY], and bounding all relevant local invariants to give only finitely many possibilities. We also briefly describe an open-source implementation of this algorithm we have written in Python/Sage which explicitly enumerates all such quadratic forms of bounded class number in n >= 3 variables. Using this we determine that there are exactly 115 primitive positive definite maximal Z-valued quadratic forms in n >= 3 variables of class number one, and produce a list of them. In a future paper we will complete this chain of ideas by extending these algorithms to allow the enumeration of all primitive maximal totally definite O_F-valued quadratic lattices of rank n >= 3, where O_F is the ring of integers of any totally real number field F.