Waldspurger’s formula gives an identity between the norm of a torus period and an $L$ -function of the twist of an automorphic representation on GL(2). For any two Hecke characters of a fixed quadratic extension, one can consider the two torus periods coming from integrating one character against the automorphic induction of the other. Because the corresponding $L$ -functions agree, (the norms of) these periods—which occur on different quaternion algebras—are closely related. In this paper, we give a direct proof of an explicit identity between the torus periods themselves.

中文翻译：

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四元数代数上 CM 形式的周期恒等式

Waldspurger 公式给出了圆环周期范数和周期范数之间的同一性 $L$ -GL(2) 上自守表示的扭曲函数。对于固定二次扩展的任意两个 Hecke 字符，可以考虑将一个字符积分与另一个字符的自守归纳得到的两个环面周期。因为对应 $L$ -函数一致，这些周期（出现在不同的四元数代数上）（的规范）密切相关。在本文中，我们直接证明了环面周期本身之间的明确身份。