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Weak Siegel-Weil formula for 𝕄₂(ℚ) and arithmetic on quaternions
Transactions of the American Mathematical Society ( IF 1.2 ) Pub Date : 2021-02-11 , DOI: 10.1090/tran/8324 Tuoping Du
Transactions of the American Mathematical Society ( IF 1.2 ) Pub Date : 2021-02-11 , DOI: 10.1090/tran/8324 Tuoping Du
Abstract:We prove a weak version of the Siegel-Weil formula on 
for the dual pair 
, where 
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.
中文翻译：
𝕄²（ℚ）的弱Siegel-Weil公式和四元数的算术运算
摘要：我们证明了对偶对的Siegel-Weil公式的一个弱版本，其中是分裂的正交基团。通过该公式和Siegel-Weil公式，我们给出了与Eichler阶相关的Hecke对应程度和平均表示数的明确公式。最后，我们给出了用局部Whittaker函数将数字表示为三个平方和四个平方之和的显式公式，结果证明这些函数正是Hardy奇异级数的局部因子。
更新日期：2021-04-02
for the dual pair , where 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. 中文翻译：
𝕄²（ℚ）的弱Siegel-Weil公式和四元数的算术运算
摘要：我们证明了对偶对的Siegel-Weil公式的一个弱版本，其中是分裂的正交基团。通过该公式和Siegel-Weil公式，我们给出了与Eichler阶相关的Hecke对应程度和平均表示数的明确公式。最后，我们给出了用局部Whittaker函数将数字表示为三个平方和四个平方之和的显式公式，结果证明这些函数正是Hardy奇异级数的局部因子。
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