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Zeros of Automorphic L-Functions and Noncyclic Base Change
Book chapter

Zeros of Automorphic L-Functions and Noncyclic Base Change

Jianya Liu and Yangbo Ye
Number Theory, pp.119-152
Developments in Mathematics, Springer US
2006
DOI: 10.1007/0-387-30829-6_10

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Abstract

Let π be an automorphic irreducible cuspidal representation of GLm over a Galois (not necessarily cyclic) extension E of ℚ of degree ℓ. We compute the n-level correlation of normalized nontrivial zeros of L(s, π). Assuming that π is invariant under the action of the Galois group Gal(E/ℚ), we prove that it is equal to the n-level correlation of normalized nontrivial zeros of a product of ℓ distinct L-functions L(s, π1) ... L(s, πℓ) attached to cuspidal representations π1, ..., πℓ of GLm over ℚ. This is done unconditionally for m = 1,2 and for m = 3,4 with the degree ℓ having no prime factor ≤ (m2 + 1)/2. In other cases, the computation is made 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 ℓ distinct cuspidal representations π1,..., πℓ of GLm (ℚA), 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 π1,..., πℓ being self-contragredient.
 Fourier Analysis Special Functions Approximations and Expansions Mathematics Number Theory

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