Journal article
Functoriality of automorphic L-functions through their zeros
Science in China. Series A, Mathematics, physics, astronomy, Vol.52(1), pp.1-16
09/29/2008
DOI: 10.1007/s11425-008-0088-x
Abstract
Let
E
be a Galois extension of ℚ 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
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
, π
1
)…
L
(
s
, π
k
), where π
j
,
j
= 1,…,
k
, are automorphic cuspidal representations of
, 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
,
j
= 1,…,
k
. In particular, if π is not invariant under the action of any nontrivial σ ∈ Gal
E
/ℚ
, then
L
(
s
, π) must equal a single
L
-function attached to a cuspidal representation of
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
,
m
1
,…,
m
k
are small, but are under Hypothesis H and a bound toward the Ramanujan conjecture in other cases.
Details
- Title: Subtitle
- Functoriality of automorphic L-functions through their zeros
- Creators
- JianYa Liu - Shandong UniversityYangBo Ye - University of Iowa
- Resource Type
- Journal article
- Publication Details
- Science in China. Series A, Mathematics, physics, astronomy, Vol.52(1), pp.1-16
- Publisher
- SP Science in China Press
- DOI
- 10.1007/s11425-008-0088-x
- ISSN
- 1006-9283
- eISSN
- 1862-2763
- Language
- English
- Date published
- 09/29/2008
- Academic Unit
- Mathematics
- Record Identifier
- 9984240865002771
Metrics
6 Record Views