Functoriality of automorphic <Emphasis Type="Italic">L</Emphasis>-functions through their zeros |
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Authors: | JianYa Liu YangBo Ye |
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Institution: | (1) School of Mathematics, Shandong University, Jinan, 250100, China;(2) Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, USA |
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Abstract: | Let E be a Galois extension of ℚ of degree l, not necessarily solvable. In this paper we first prove that the L-function L(s, π) attached to an automorphic cuspidal representation π of cannot be factored nontrivially into a product of L-functions over E.
Next, we compare the n-level correlation of normalized nontrivial zeros of L(s, π1)…L(s, π
k
), where π
j
, j = 1,…, k, are automorphic cuspidal representations of , with that of L(s,π). We prove a necessary condition for L(s, π) having a factorization into a product of L-functions attached to automorphic cuspidal representations of specific , j = 1,…,k. In particular, if π is not invariant under the action of any nontrivial σ ∈ Gal
E/ℚ, then L(s, π) must equal a single L-function attached to a cuspidal representation of and π has an automorphic induction, provided L(s, π) can factored into a product of L-functions over ℚ. As E is not assumed to be solvable over ℚ, our results are beyond the scope of the current theory of base change and automorphic
induction.
Our results are unconditional when m,m
1,…,m
k
are small, but are under Hypothesis H and a bound toward the Ramanujan conjecture in other cases.
The first author was supported by the National Basic Research Program of China, the National Natural Science Foundation of
China (Grant No. 10531060), and Ministry of Education of China (Grant No. 305009). The second author was supported by the
National Security Agency (Grant No. H98230-06-1-0075). The United States Government is authorized to reproduce and distribute
reprints notwithstanding any copyright notation herein |
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Keywords: | automorphic induction automorphic L-function functoriality zero correlation |
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