论文标题
关于Motohashi的公式
On Motohashi's formula
论文作者
论文摘要
我们补充并提供了一个新的观点，证明了Motohashi-Type公式的证据，该公式将$ l $ functions的第四刻与$ \ mathrm {gl} _1 $的第三刻$ \ mathrm {gl} _1 $与$ \ mathrm {gl} _2 _2 _2 _2 _2 _2 _2 $ by Micheel-venkatekatekatekatekateshheSheSheshheSelsonsone一起研究。我们的主要工具是一种在$ \ mathrm {M} _2（\ Mathbb {a}）$上，而不是$ \ Mathrm {gl} _2（\ Mathbb {a}）上的新型的带有测试功能的新型预轨迹公式。这也是Bruggeman-Motohashi的其他公式证明的概括。在划分四元组代数而不是$ \ mathrm {m} _2 $的情况下，我们给出了方法的变体，产生了新的频谱互惠，我们不确定它是否在Michel-venkatesh给出的时期形式上。我们还指出了进一步的概括，这似乎超出了时期方法所能提供的。
We complement and offer a new perspective of the proof of a Motohashi-type formula relating the fourth moment of $L$-functions for $\mathrm{GL}_1$ with the third moment of $L$-functions for $\mathrm{GL}_2$ over number fields, studied earlier by Michel-Venkatesh and Nelson. Our main tool is a new type of pre-trace formula with test functions on $\mathrm{M}_2(\mathbb{A})$ instead of $\mathrm{GL}_2(\mathbb{A})$, on whose spectral side the matrix coefficients are replaced by the standard Godement-Jacquet zeta integrals. This is also a generalization of Bruggeman-Motohashi's other proof of Motohashi's formula. We give a variation of our method in the case of division quaternion algebras instead of $\mathrm{M}_2$, yielding a new spectral reciprocity, for which we are not sure if it is within the period formalism given by Michel-Venkatesh. We also indicate a further possible generalization, which seems to be beyond what the period method can offer.
