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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Subconvexity for twisted $ L$-functions on $ \mathrm{GL}_3$ over the Gaussian number field


Author: Zhi Qi
Journal: Trans. Amer. Math. Soc. 372 (2019), 8897-8932
MSC (2010): Primary 11M41
DOI: https://doi.org/10.1090/tran/7892
Published electronically: August 22, 2019
MathSciNet review: 4029716
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Abstract: Let $ q \in \mathbb{Z} [i]$ be prime, and let $ \chi $ be the primitive quadratic Hecke character modulo $ q$. Let $ \pi $ be a self-dual Hecke automorphic cusp form for $ \mathrm {SL}_3 (\mathbb{Z} [i] )$, and let $ f$ be a Hecke cusp form for $ \Gamma _0 (q) \subset \mathrm {SL}_2 (\mathbb{Z} [i])$. Consider the twisted $ L$-functions $ L (s, \pi \otimes f \otimes \chi ) $ and $ L (s, \pi \otimes \chi )$ on $ \mathrm {GL}_3 \times \mathrm {GL}_2$ and $ \mathrm {GL}_3$. We prove the subconvexity bounds

$\displaystyle L \big (\tfrac 1 2, \pi \otimes f \otimes \chi \big ) \ll _{\, \v... ...hi \big ) \ll _{\, \varepsilon , \pi , t } \mathrm {N} (q)^{5/8 + \varepsilon }$    

for any $ \varepsilon > 0$.

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Additional Information

Zhi Qi
Affiliation: School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China
Email: zhi.qi@zju.edu.cn

DOI: https://doi.org/10.1090/tran/7892
Keywords: $L$-functions, subconvexity, the Gaussian number field
Received by editor(s): January 8, 2019
Received by editor(s) in revised form: April 27, 2019, and May 4, 2019
Published electronically: August 22, 2019
Article copyright: © Copyright 2019 American Mathematical Society