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.

Correlation of zeros of automorphic L-functions

We compute the n-level correlation of normalized nontrivial zeros of a product of Lfunctions: L(s, π1) … L(s, π k ), where πj, j = 1, …, k, are automorphic cuspidal representations of GL mj (?A). Here the sizes of the groups GL mj (?A) are not necessarily the same. When these L(s, π j ) are distinct, we prove that their nontrivial zeros are uncorrelated, as predicted by random matrix theory and verified numerically. When L(s, π j ) are not necessarily distinct, our results will lead to a proof that the n-level correlation of normalized nontrivial zeros of the product L-function follows the superposition of Gaussian Unitary Ensemble (GUE) models of individual L-functions and products of lower rank GUEs. The results are unconditional when m 1, …, m k ? 4, but are under Hypothesis H in other cases.