Type number for orders of level (N_1,N_2)

TL;DR

This paper derives an explicit formula for the type number of quaternion orders with levels $(N_1, N_2)$, generalizing previous results and providing comprehensive computations and classifications for these algebraic structures.

Contribution

It generalizes existing formulas for quaternion order type numbers to arbitrary prime power levels and introduces a new class number generalization, expanding understanding of quaternion orders.

Findings

Explicit formula for type numbers of quaternion orders with level $(N_1, N_2)$

Computed type numbers for all levels with $N_1 N_2 \,\leq\, 100$

Classified all pairs $(N_1, N_2)$ with type number 1

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.