Subconvexity for twisted 𝐿-functions on 𝐺𝐿₃ over the Gaussian number field

Qi, Zhi

doi:10.1090/tran/7892

Subconvexity for twisted $L$-functions on $\mathrm {GL}_3$ over the Gaussian number field
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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 \begin{equation*} L \big (\tfrac 1 2, \pi \otimes f \otimes \chi \big ) \ll _{ \varepsilon , \pi , f } \mathrm {N} (q)^{5/4 + \varepsilon }, \quad L \big (\tfrac 1 2 + it, \pi \otimes \chi \big ) \ll _{ \varepsilon , \pi , t } \mathrm {N} (q)^{5/8 + \varepsilon } \end{equation*} for any $\varepsilon > 0$.

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