## Weak Siegel-Weil formula for $\mathbb {M}_{2}(\mathbb {Q})$ and arithmetic on quaternions

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## 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.## References

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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
- © Copyright 2021 American Mathematical Society
- 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
- MathSciNet review: 4237951