Non‐vanishing of symmetric cube L$L$‐functions
Non‐vanishing of symmetric cube L$L$‐functions
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对称立方 L$L$ 函数的不消失
DOI：
10.1112/jlms.12683
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发表时间：
2023
期刊：
影响因子：
--
通讯作者：
Lee, Min
中科院分区：
文献类型：
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作者：
Hoffstein, Jeff;Jung, Junehyuk;Lee, Min
我们证明在场 Q[−3]$\mathbb {Q}[\sqrt {-3}]$ 上存在无限多个 Maass-Hecke 尖形，使得相应的对称立方体 L$L$‐系列不会在临界带的中心消失。这是通过使用 Ginzburg、Jiang 和 Rallis 的结果来完成的，该结果表明，如果 Q[−3]$\mathbb {Q}[\sqrt {-3}]$ 上涉及尖点形式和三次 theta 函数的某个三重积积分不消失，则对称立方中心值不会消失。我们使用谱理论和三次 theta 函数的性质来证明，对于无限多个尖点形式，这种三重积的不消失会发生。我们还提出了关于三重乘积的绝对值平方的含义的猜想，这让人想起沃森的恒等式。
We prove that there are infinitely many Maass–Hecke cuspforms over the field Q[−3]$\mathbb {Q}[\sqrt {-3}]$ such that the corresponding symmetric cube L$L$‐series does not vanish at the center of the critical strip. This is done by using a result of Ginzburg, Jiang and Rallis which shows that if a certain triple product integral involving the cusp form and the cubic theta function on Q[−3]$\mathbb {Q}[\sqrt {-3}]$ does not vanish then the symmetric cube central value does not vanish. We use spectral theory and the properties of the cubic theta function to show that the non‐vanishing of this triple product occurs for infinitely many cusp forms. We also formulate a conjecture about the meaning of the absolute value squared of the triple product which is reminiscent of Watson's identity.