Weak Siegel-Weil formula for $\mathbb {M}_{2}(\mathbb {Q})$ and arithmetic on quaternions
Author:
Tuoping Du
Journal:
Trans. Amer. Math. Soc. 374 (2021), 3397-3426
MSC (2020):
Primary 11R52, 11G18, 11F32, 11F41, 11S23
DOI:
https://doi.org/10.1090/tran/8324
Published electronically:
February 11, 2021
MathSciNet review:
4237951
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Abstract | References | Similar Articles | Additional Information
Abstract: We prove a weak version of the Siegel-Weil formula on $\operatorname {SL}_2$ for the dual pair $(\operatorname {SL}_2, O_{2, 2})$, where $O_{2, 2}$ is the split orthogonal group. By this formula and the Siegel-Weil formula, we give explicit formulas for Hecke correspondence’s degree and average representation numbers over genus associated to Eichler orders. At last, we give explicit formulas for representations of a number as sums of three squares and four squares by local Whittaker functions, and it turns out that these functions are exactly the local factors of Hardy’s singular series.
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Additional Information
Tuoping Du
Affiliation:
Department of Mathematics, Southeast University, Nanjing, 211189, People’s Republic of China
Email:
dtp1982@163.com
Received by editor(s):
May 16, 2018
Received by editor(s) in revised form:
May 19, 2018, July 6, 2018, December 29, 2019, and July 31, 2020
Published electronically:
February 11, 2021
Article copyright:
© Copyright 2021
American Mathematical Society