Type number for orders of level (N_1,N_2)
Yifan Luo, Haigang Zhou

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 , 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…
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Taxonomy
TopicsMathematics and Applications
