arXiv subjects
Publications and source records attributed to Xinchen Miao.
We demonstrate that for symmetric cubes of algebraic regular Hecke eigenforms on $\mathrm{GL}(2)$, a decorrelation phenomenon occurs for global Bessel periods of $\mathrm{SO}(5)\times \mathrm{SO}(2)$ averaged over imaginary quadratic fields, conditional on the Generalized Riemann Hypothesis.
In this paper, we focus on the strong subconvexity bounds for triple product L-functions in the cubic level aspect. Our proof on the Weyl-type bound synthesizes techniques from classical analytic number theory with methods in automorphic forms and representation theory. The methods include the refined Petersson trace formula for the newforms of cubic level, classical Voronoi summation formula, Jutila's circle method, Kuznetsov trace formula and the spectral large sieve inequality.
Let $F$ be a number field with adele ring $\mathbb{A}_F$, $π_1, π_2$ be two unitary cuspidal automorphic representations of $\mathrm{PGL}_2(\mathbb{A}_F)$ with finite analytic conductor. We study the twisted first moment of the triple product $L$-function $L(\frac{1}{2}, π\otimes π_1 \otimes π_2)$ and the Hecke eigenvalues $λ_π(\mathfrak{l})$, where $π$ is a unitary automorphic representation of $\mathrm{PGL}_2(\mathbb{A}_F)$ and $\mathfrak{l}$ is an integral ideal coprimes with the finite analytic conductor $C(π\otimes π_1 \otimes π_2)$. The estimation becomes a reciprocity formula between different moments of $L$-functions. Combining with the ideas and estimations established in [HMN23] and [MV10], we study the subconvexity problem for the triple product $L$-function in the level aspect and give a new explicit hybrid subconvexity bound for $L(\frac{1}{2}, π\otimes π_1 \otimes π_2)$, allowing joint ramifications and conductor dropping range.
Let $F$ be a number field with adele ring $\mathbb{A}_F$, $π_1, π_2$ be two fixed unitary automorphic representations of $\mathrm{PGL}_2(\mathbb{A}_F)$ with finite coprime analytic conductor $\mathfrak{u}$ and $\mathfrak{v}$, $\mathfrak{q},\mathfrak{l}$ be two coprime integral ideals with $(\mathfrak{q} \mathfrak{l}, \mathfrak{u} \mathfrak{v})=1$. Following [Zac20], we estimate the first moment of $L(\frac{1}{2}, π\otimes π_1 \otimes π_2)$ twisted by the Hecke eigenvalues $λ_π(\mathfrak{l})$, where $π$ runs over unitary automorphic representations of finite conductor dividing $\mathfrak{u}\mathfrak{v}\mathfrak{q}$. By applying the triple product integrals, spectral decomposition and Plancherel formula, we get a reciprocity formula links the twisted first moment of triple product $L$-functions to the spectral expansion of certain triple product periods over automorphic representations of finite conductor dividing $\mathfrak{l}$. As application, we study the subconvexity problem for the triple product $L$-function in the level aspect and give a subconvex bound for $L(\frac{1}{2}, π\otimes π_1 \otimes π_2)$ in terms of the norm of $\mathfrak{q}$.
This paper will focus on the proof of local integrability of Bessel functions for GL(n) (p-adic case) by using the relations between Bessel functions and local Kloosterman (orbital) integrals proved in several papers of E. M. Baruch [Ba03] [Ba04] [Ba05], the theory of the (relative) Shalika germs established by H. Jacquet and Y. Ye in [JY96] [JY99] and G. Stevens' approach [Ste87] on estimating certain GL(n) generalized Kloosterman sums.
We prove a new case of spectral reciprocity formulae for the product of $GL(n+1) \times GL(n)$ and $GL(n) \times GL(n-1)$ Rankin-Selberg $L$-functions ($n \geq 3$), which are first developed by Blomer and Khan in \cite{BK17} for degree 8 $L$-functions ($n=2$ case, the product of $GL(3) \times GL(2)$ and $GL(2) \times GL(1)$ Rankin-Selberg $L$-functions). Our result can be viewed as a generalization of Blomer and Khan's work to higher rank case. We will mainly follow the method developed in \cite{Nun20}. We will use the integral representations of Rankin-Selberg $L$-functions generalized by Ichino and Yamana \cite{IY15}, spectral theory of $L^2$ space and the language of automorphic representations.