相关论文: A universal lower bound for certain quadratic inte…

Let $\pi$ be a unitary cuspidal automorphic representation of $\mathrm{GL}_n$ over a number field, and let $\tilde{\pi}$ be contragredient to $\pi$. We prove effective upper and lower bounds of the correct order in the short interval prime…

We prove Deligne's conjecture for central critical values of certain automorphic $L$-functions for ${\rm GL}(3)\times {\rm GL}(2)$. The proof is base on rationality results for central critical values of triple product $L$-functions, which…

Let $G={\rm GL}_{2n}$ over a totally real number field $F$ and $n\geq 2$. Let $\Pi$ be a cuspidal automorphic representation of $G(\mathbb A)$, which is cohomological and a functorial lift from SO$(2n+1)$. The latter condition can be…

We generalize our recent construction of the zeros of the Riemann $\zeta$-function to two infinite classes of $L$-functions, Dirichlet $L$-functions and those based on level one modular forms. More specifically, we show that there are an…

If $(M,g)$ is a compact Riemannian surface then the integrals of $L^2(M)$-normalized eigenfunctions $e_j$ over geodesic segments of fixed length are uniformly bounded. Also, if $(M,g)$ has negative curvature and $\gamma(t)$ is a geodesic…

Let $\pi_1, \pi_2, \pi_3$ be three cuspidal automorphic representations for the group ${\rm SL}(2, \Bbb{Z})$, where $\pi_1$ and $\pi_2$ are fixed and $\pi_3$ has large conductor. We prove a subconvex bound for $L(1/2, \pi_1 \otimes \pi_2…

Let $(S^2,g)$ be a convex surface of revolution and $H \subset S^2$ the unique rotationally invariant geodesic. Let $\varphi^\ell_m$ be the orthonormal basis of joint eigenfunctions of $\Delta_g$ and $\partial_\theta$, the generator of the…

We introduce a $p$-adic $L$-function $\mathscr L_{A/L}$ associated to an ordinary elliptic curve $A$ over a global function field $K$ of characteristic $p$ together with a $\mathbb{Z}_{p}^{d}$-extension $L/K$, $d=0$ allowed, unramified…

Poincare-type series, such as Selberg's, are known to produce automorphic functions, in the hyperbolic half-plane, the decompositions of which into eigenfunctions (genuine or generalized) of the automorphic Laplacian contain all modular…

Let $F$ be a number field and $D$ a quaternion algebra over $F$. Take a cuspidal automorphic representation $\pi$ of $D_{\mathbb{A}}^\times$ with trivial central character and a cusp form $\phi$ in $\pi$. Using the prehomogeneous zeta…

From a spectral identity we obtain asymptotics with error term for the second integral moments of families of automorphic L-functions for GL(2) over an arbitrary number field according to twists by idele characters with arbitrary…

We establish the universality theorem for the first four symmetric power L-functions of automorphic forms and their associated Rankin-Selberg L-functions. This generalizes some results of Laurincikas & Matsumoto and Matsumoto respectively.

We obtain (conditional and unconditional) results on large values of $L$-functions $L(s,\chi)$ in the critical strip $1/2 \leq \Re s \leq 1$ when the character $\chi$ runs through a thin subgroup of all characters modulo an integer $q$.…

Let $q \in \mathbb{Z} [i]$ be prime and $\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 $f$ be a Hecke cusp form for $\Gamma_0 (q)…

We study lower bounds for the Riemann zeta function $\zeta(s)$ along vertical arithmetic progressions in the right-half of the critical strip. We show that the lower bounds obtained in the discrete case coincide, up to the constants in the…

Let $\mathfrak{F}_m$ be the set of all cuspidal automorphic representations of $\mathrm{GL}_m(\mathbb{A}_{\mathbb{Q}})$, and let $F(s,\boldsymbol{\pi})$ be a polynomial in the derivatives of $L$-functions associated with representations…

We obtain an upper bound for the dimension of the cuspidal automorphic forms for $\mathrm{GL}_2$ over a number field, whose archimedean local representations are not tempered. More precisely, we prove the following result. Let $F$ be a…

The infinity symmetric power $L$-functions play a fundamental role in Wan's groundbreaking work on Dwork's conjecture[16]. Building upon this foundation, Haessig[8] established the $p$-adic estimates for these $L$-functions in the case of…

Let $F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ be a Dirichlet series in the axiomatically defined class ${\mathfrak A}^{\#}$ . The class ${\mathfrak A}^{\#}$ is known to contain the extended Selberg class ${\mathcal S}^{\#}$, as well as all…

We prove three main results: all Langlands-Shahidi automorphic $L$-functions over function fields are rational; after twists by highly ramified characters our automorphic $L$-functions become polynomials; and, if $\pi$ is a globally generic…