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Type number for orders of level (N_1,N_2)

Yifan Luo, Haigang Zhou

TL;DR

The paper derives an explicit formula for the type number $T_{N_1,N_2}$ of definite quaternion orders of level $(N_1,N_2)$ with $N_1$ consisting of odd powers of distinct primes and $(N_1,N_2)=1$, unifying previous squarefree and prime-power cases. It does so by extending the modified Hurwitz class numbers to $H^{(N_1,N_2)}(D)$, establishing a bijection between orders and ternary quadratic forms, and applying the Siegel-Weil formula together with local density calculations to express $T_{N_1,N_2}$ as a weighted sum of representation numbers. The approach yields new computational results (including all levels with $N_1N_2\le 100$ and a complete list of 27 type-number-one levels) and a classification of type number one levels beyond the squarefree regime. The work advances understanding of arithmetic invariants of quaternion orders and provides a robust framework for evaluating type numbers via automorphic and genus-theoretic tools with explicit local factors.

Abstract

We establish an explicit formula for the type number of quaternion orders of level $(N_1, N_2)$, where $N_1 = p_1^{2u_1+1} \cdots p_w^{2u_w+1}$ (with $u_i \geq 0$ and $w$ odd) and $\gcd(N_1, N_2) = 1$. Our main result generalizes Pizer's work on Eichler orders (where $N_1$ is squarefree) and Boyd's formula (where $N_1 = p^{2u+1}$) to the general case with arbitrary prime powers in $N_1$. The proof introduces a generalization of the modified Hurwitz class number $H^{(N_1,N_2)}(D)$, originally defined by Li, Skoruppa and the second author for squarefree levels. Through a bijection between quaternion orders and ternary quadratic forms, we express the type number as a weighted sum of representation numbers, which we evaluate explicitly via the Siegel-Weil formula and local density computations. We compute type numbers for all levels with $N_1 N_2 \leq 100$ and correct four entries in Boyd's 1994 table. As a further application, we classify all 27 pairs $(N_1, N_2)$ having type number $1$, extending the list of 9 squarefree pairs found by Boylan, Skoruppa and the second author.

Type number for orders of level (N_1,N_2)

TL;DR

The paper derives an explicit formula for the type number of definite quaternion orders of level with consisting of odd powers of distinct primes and , unifying previous squarefree and prime-power cases. It does so by extending the modified Hurwitz class numbers to , establishing a bijection between orders and ternary quadratic forms, and applying the Siegel-Weil formula together with local density calculations to express as a weighted sum of representation numbers. The approach yields new computational results (including all levels with and a complete list of 27 type-number-one levels) and a classification of type number one levels beyond the squarefree regime. The work advances understanding of arithmetic invariants of quaternion orders and provides a robust framework for evaluating type numbers via automorphic and genus-theoretic tools with explicit local factors.

Abstract

We establish an explicit formula for the type number of quaternion orders of level , where (with and odd) and . Our main result generalizes Pizer's work on Eichler orders (where is squarefree) and Boyd's formula (where ) to the general case with arbitrary prime powers in . The proof introduces a generalization of the modified Hurwitz class number , originally defined by Li, Skoruppa and the second author for squarefree levels. Through a bijection between quaternion orders and ternary quadratic forms, we express the type number as a weighted sum of representation numbers, which we evaluate explicitly via the Siegel-Weil formula and local density computations. We compute type numbers for all levels with and correct four entries in Boyd's 1994 table. As a further application, we classify all 27 pairs having type number , extending the list of 9 squarefree pairs found by Boylan, Skoruppa and the second author.

Paper Structure

This paper contains 30 sections, 39 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Quaternion algebras and orders
  3. Quaternion algebras over local and global fields
  4. Basic definitions
  5. Global quaternion algebras
  6. Reduced trace and norm
  7. Orders and ideals
  8. Lattices and orders
  9. Local structure of maximal orders
  10. Orders of level (N1, N2)
  11. Left and right ideals
  12. Two-sided ideals and normalizers
  13. Two-sided principal ideals
  14. Counting two-sided principal ideals
  15. Ternary quadratic forms and quaternion orders
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $Q_{N_0}$ be a definite quaternion algebra over $\mathbb{Q}$ ramifying only at primes $p \mid N_0$. The class number of orders of level $(N_1,N_2)$ in $Q_{N_0}$ is and the type number is

Theorems & Definitions (74)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Definition 2.1: Piz76
  • Remark 3
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Lemma 2.5: Piz77
  • Lemma 2.6
  • ...and 64 more