AI 中文总结
研究超越尖点谱的邦普 - 弗里德伯格型周期，通过全纯惠特克型泽塔积分刻画周期扩展，引入\(\textnormal{GL}_{2n + 1}\)上周期并对某些艾森斯坦级数求值，结果与全局数值猜想切空间预测相符。
AI 中文摘要
本文研究了几个超越尖点谱的邦普 - 弗里德伯格型周期。首先考虑了\(\textnormal{GL}_{2n}\)上的扭曲邦普 - 弗里德伯格周期以及\(\textnormal{GL}_1\times \textnormal{GL}_{2n}\)上的一个变体，在适当的尖点数据正则性条件下，这些周期连续扩展到一致适度增长的自守函数，通过全纯惠特克型泽塔积分来刻画。接着引入了\(\textnormal{GL}_{2n + 1}\)上的一个邦普 - 弗里德伯格型周期，在子群\(\textnormal{SL}_{n + 1}\times \textnormal{GL}_n\)上积分。对于某些艾森斯坦级数，将此周期评估为\(L\)函数特殊值与归一化局部泽塔积分乘积的有限和。假设预期的全局朗兰兹对应，该和由扩展\(L\)参数在推测对偶簇上的不动点索引，所得\(L\)因子与本 - 兹维 - 萨凯拉里迪斯 - 文卡特什全局数值猜想的切空间预测一致。
英文摘要
In this article, we study several Bump--Friedberg type periods beyond the cuspidal spectrum. We first consider the twisted Bump--Friedberg period on $\textnormal{GL}_{2n}$, as well as a variant on $\textnormal{GL}_1\times \textnormal{GL}_{2n}$. Under suitable regularity conditions on the cuspidal datum, these periods extend continuously to automorphic functions of uniform moderate growth. Such extensions are characterized by entire Whittaker-type zeta integrals. We then introduce a Bump--Friedberg type period on $\textnormal{GL}_{2n+1}$, integrating over the subgroup $\textnormal{SL}_{n+1}\times \textnormal{GL}_n$. For certain Eisenstein series, we evaluate this period as a finite sum of products of special values of $L$-functions and normalized local zeta integrals. Assuming the expected global Langlands correspondence, the sum is indexed by the fixed points of the extended $L$-parameter on the conjectural dual variety, and the resulting $L$-factors agree with the tangent space prediction of the global numerical conjecture of Ben-Zvi-Sakellaridis-Venkatesh.
Comments39 pages. Comments are welcome!
