Type number for orders of level (N_1,N_2)
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912867271311360 |
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| author | Luo, Yifan Zhou, Haigang |
| author_facet | Luo, Yifan Zhou, Haigang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10882 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Type number for orders of level (N_1,N_2) Luo, Yifan Zhou, Haigang Number Theory 11E20, 11R52, 11F37, 11E41 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. |
| title | Type number for orders of level (N_1,N_2) |
| topic | Number Theory 11E20, 11R52, 11F37, 11E41 |
| url | https://arxiv.org/abs/2410.10882 |