Zeros of Automorphic L-Functions and Noncyclic Base Change

Abstract

Abstract.Let π be an automorphic irreducible cuspidal representation of GL m over a Galois (not necessarily cyclic) extension E of Q of degree . Assume that π is invariant under the action of the Galois group Gal E/Q . We computed the n-level correlation of normalized nontrivial zeros of L(s, π) and proved that it is equal to the n-level correlation of normalized nontrivial zeros of a product of distinct L-functions L(s, π 1 ) · · · L(s, π ) attached to cuspidal representations π 1 , . . . , π of GL m over Q. This is done unconditionally for m = 1, 2 and for m = 3, 4 with the degree having no prime factor ≤ (m 2 + 1)/2. In other cases, the computation is under a conjecture of bounds toward the Ramanujan conjecture over E, and a conjecture on convergence of certain series over prime powers (Hypothesis H over E and Q). The results provide an evidence that π should be (noncyclic) base change of distinct cuspidal representations π 1 , . . . , π of GL m (Q A ), if it is invariant under the Galois action. A technique used in this article is a version of Selberg orthogonality for automorphic L-functions (Lemma 6.2 and Theorem 6.4), which is proved unconditionally, without assuming π and π 1 , . . . , π being self-contragredient. Introduction.According to Langlands' functoriality conjecture, the L-function attached to an automorphic irreducible cuspidal representation π of GL m over a number field E should equal a product of L(s, π j ) for certain cuspidal representations π j of GL m j over Q. Arthur and Clozel [ArtClo] proved that this is indeed the case when E is a cyclic Galois extension of Q and π is stable under Gal E/Q . In fact in this case, π is the base change of exactly nonequivalent cuspidal representations π Q , π Q ⊗ η E/Q , . . . , π Q ⊗ η

Euler-Stieltjes constants for the Rankin-Selberg L-function and weighted Selberg orthogonality

“…When m = 3, Hypothesis H over E is proved in [22,Appendix], and in the case m = 4, it is a consequence of [17], as pointed out by Kim and Sarnak in [17,Appendix]. An immediate consequence of equation (2.5) and the definition of Λ π,π ′ (n) is the following lemma.…”

“…The Hypothesis H was generalized to the setting of Galois extensions E of degree l over Q by J. Liu and Y. Ye in [22]. We will refer to this hypothesis as Hypothesis H over E.…”

Correlation of zeros of automorphic L-functions

“…Note that here h 1 (γ 1 /T ) · · · h n (γ n /T ) is different from (1.6), as we will follow the computation in [1,6,7] and change back later. Using (4.3), we have…”

“…In its full generality, i.e. without assuming self-contragredient, Lemma 3.1 was proved in [7]. We will need a smoothly weighted version of this orthogonality, which can be deduced from Lemma 3.1 by partial summation.…”

“…In [6], the n-level correlation of zeros for a product L(s, π 1 ) · · · L(s, π k ) of distinct L-functions was computed, where π j are non-equivalent cuspidal representations of GL m (Q A ). In [7], we further computed the n-level of zeros of L(s, π) for π being a cuspidal representation of GL m (E A ), where E is a Galois extension field of Q.…”