Let E be a Galois extension of ? 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 $ GL_m (E_\mathbb{A} ) $ canno...Let E be a Galois extension of ? 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 $ GL_m (E_\mathbb{A} ) $ 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 $ GL_{m_j } (\mathbb{Q}_\mathbb{A} ) $ , 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 $ GL_{m_j } (\mathbb{Q}_\mathbb{A} ) $ , j = 1,…,k. In particular, if π is not invariant under the action of any nontrivial σ ∈ Gal E/?, then L(s, π) must equal a single L-function attached to a cuspidal representation of $ GL_{m\ell } (\mathbb{Q}_\mathbb{A} ) $ and π has an automorphic induction, provided L(s, π) can factored into a product of L-functions over ?. As E is not assumed to be solvable over ?, 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.展开更多
基金supported by the National Basic Research Program of China, the National Natural Science Foundation of China (Grant No. 10531060)Ministry of Education of China (Grant No. 305009)+1 种基金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
文摘Let E be a Galois extension of ? 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 $ GL_m (E_\mathbb{A} ) $ 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 $ GL_{m_j } (\mathbb{Q}_\mathbb{A} ) $ , 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 $ GL_{m_j } (\mathbb{Q}_\mathbb{A} ) $ , j = 1,…,k. In particular, if π is not invariant under the action of any nontrivial σ ∈ Gal E/?, then L(s, π) must equal a single L-function attached to a cuspidal representation of $ GL_{m\ell } (\mathbb{Q}_\mathbb{A} ) $ and π has an automorphic induction, provided L(s, π) can factored into a product of L-functions over ?. As E is not assumed to be solvable over ?, 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.
