Functoriality of automorphic L-functions through their zeros

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.

Let E be a Galois extension of Q of degree , not necessarily solvable. In this paper we first prove that the L-function L(s,π) attached to an automorphic cuspidal representation π of GLm(EA) 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,πk), where πj, j = 1,...,k, are automorphic cuspidal representations of GLmj(QA), 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 GLmj(QA), j = 1,...,k. In particular, if π is not invariant under the action of any nontrivial σ∈ GalE/Q, then L(s,π) must equal a single L-function attached to a cuspidal representation of GLm (QA) 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,m1,...,mk are small, but are under Hypothesis H and a bound toward the Ramanujan conjecture in other cases.