Fourier–Jacobi cycles and arithmetic relative trace formula (with an appendix by Chao Li and Yihang Zhu)
Fourier–Jacobi cycles and arithmetic relative trace formula (with an appendix by Chao Li and Yihang Zhu)
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DOI：
10.4310/cjm.2021.v9.n1.a1
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发表时间：
2021-02
期刊：
影响因子：
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通讯作者：
Yifeng Liu
中科院分区：
文献类型：
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作者：
Yifeng Liu
本文给出了一对等秩酉群的傅里叶-雅可比周期积分的算术模拟。我们构造了所谓的Fourier-Jacobi环，它们是酉Shimura簇和交换簇的乘积上的代数环。对于这些循环，我们提出了与某些Rankin-Selberg$L$-函数的中心导数有关的算术Gan-Gross-Prasad猜想，并对这个猜想发展了一个相对的迹公式方法。作为一个必要条件，我们提出了相应的算术基本引理猜想，并对至多2阶酉群和极小情形证明了猜想。
In this article, we develop an arithmetic analogue of Fourier--Jacobi period integrals for a pair of unitary groups of equal rank. We construct the so-called Fourier--Jacobi cycles, which are algebraic cycles on the product of unitary Shimura varieties and abelian varieties. We propose the arithmetic Gan--Gross--Prasad conjecture for these cycles, which is related to central derivatives of certain Rankin--Selberg $L$-functions, and develop a relative trace formula approach toward this conjecture. As a necessary ingredient, we propose the conjecture of the corresponding arithmetic fundamental lemma, and confirm it for unitary groups of rank at most two and for the minuscule case.