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

Jonathan Hanke|arXiv (Cornell University)|Oct 9, 2011

Algebraic Geometry and Number Theory15 references3 citations

TL;DR

This paper presents an algorithm to enumerate all primitive, positive definite, maximal Z-valued quadratic forms in n ≥ 3 variables with class number at most B, using the exact mass formula and bounds on local invariants. The key result is the explicit enumeration of exactly 115 such forms of class number one, providing a complete list for dimensions 3 through 10.

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.

Motivation & Objective

• To develop an algorithm for enumerating primitive, positive definite, maximal Z-valued quadratic forms in n ≥ 3 variables with bounded class number h(Q) ≤ B.
• To extend the understanding of small class number forms beyond binary and ternary cases, particularly for higher-rank forms where representation theory is less effective.
• To provide a complete and explicit list of all such forms with class number one, resolving a long-standing gap in the classification of definite forms.
• To lay the foundation for future enumeration of maximal totally definite OF-lattices over totally real number fields.

Proposed method

• The algorithm uses the exact mass formula from [GHY01] to constrain the possible invariants of quadratic forms with bounded class number.
• It bounds all relevant local invariants (e.g., over R and Z_p) to ensure only finitely many candidates remain.
• The method applies geometric and algebraic number theory techniques, including analysis of discriminants and representation conditions over local rings.
• The algorithm is implemented in Python/Sage, enabling explicit computation and verification of the forms.
• It leverages the fact that class number one implies that local representation implies global representation, simplifying the search.
• The algorithm systematically checks all possible forms satisfying the derived bounds and verifies maximality and primitivity.

Experimental results

Research questions

• RQ1 How many primitive, positive definite, maximal Z-valued quadratic forms in n ≥ 3 variables have class number one?
• RQ2 What is the complete list of such forms, and what are their invariants (e.g., determinants, forms, and representations)?
• RQ3 Can the class number one enumeration be extended beyond ternary forms using a systematic, algorithmic approach?
• RQ4 What are the structural and arithmetic constraints that limit the number of such forms in higher dimensions?
• RQ5 Can the algorithm be generalized to OF-lattices over totally real number fields?

Key findings

• The paper explicitly enumerates exactly 115 primitive, positive definite, maximal Z-valued quadratic forms in n ≥ 3 variables with class number one.
• The list includes forms in dimensions 3 through 10, with the number of such forms decreasing from 64 in dimension 3 to 1 in dimension 10.
• The algorithm successfully identifies all forms of class number one by bounding local invariants and applying the exact mass formula.
• The implementation in Python/Sage confirms the completeness of the list and enables verification of all 115 forms.
• The results confirm that no such forms exist in dimensions above 10 with class number one, consistent with prior bounds.
• The paper provides a complete table of all 115 forms, including their determinants, discriminants, and explicit quadratic expressions.