Correction: "Modularity of the Rankin-Selberg $L$-series, and multiplicity one for SL(2)"
Correction: "Modularity of the Rankin-Selberg $L$-series, and multiplicity one for SL(2)"
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DOI：
10.2307/2661379
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发表时间：
2000-07
影响因子：
4.9
通讯作者：
Dinakar Ramakrishnan
中科院分区：
文献类型：
--
作者：
Dinakar Ramakrishnan
设f，g分别是N，M阶上半平面H上的本原尖点型，全纯或非全纯，（酉正规化）L-函数L（s，f）= [equation]，L（s，g）= [equation].当p不能整除N时，M），逆根αp，βp（分别α′p，β′p）是非零的，其和ap（resp. bp）。对每一个p与NM互素，设Lp（s，f × g）= [（1 − αpα′pp− s）（1 − αpβ′pp−s）（1 − βpα′pp−s）（1 − βpβ′pp−s）]^−1。设L ∈（s，f × g）表示Lp（s，f × g）在不整除NM的所有p上的（不完全欧拉）积.这与卷积L-级数[sum over n≥1] a[sub]n B[sub] n n^−s密切相关，后者的神奇性质首先由兰金和塞尔伯格研究。
Let f, g be primitive cusp forms, holomorphic or otherwise, on the upper half-plane H of levels N,M respectively, with (unitarily normalized) L-functions L(s, f) = [equation] and L(s, g) = [equation]. When p does not divide N (resp. M), the inverse roots αp, βp (resp. α′p, β′p ) are nonzero with sum ap (resp. bp). For every p prime to NM, set Lp(s, f × g) = [(1 − αpα′pp−s)(1 − αpβ′pp−s)(1 − βpα′pp−s)(1 − βpβ′pp−s)]^−1. Let L∗(s, f × g) denote the (incomplete Euler) product of Lp(s, f × g) over all p not dividing NM. This is closely related to the convolution L-series [sum over n≥1] a[sub]n b[sub] n n^−s, whose miraculous properties were first studied by Rankin and Selberg.