Related papers: Central L-values and periods for GL(2)

In joint work with M. Hopkins and C. Teleman we find a new description of the Verlinde algebra associated to a compact Lie group. In this expository account we describe twisted K-theory, prove the theorem for the group SU(2), and motivate…

An expression for the Lie algebra cocycle corresponding to the central extension of the group GLres(H) is derived in the book Loop Groups [1]. It will be shown in this paper that some of the terms in this expression are ambiguous and in…

We prove an explicit integral representation -- involving the pullback of a suitable Siegel Eisenstein series -- for the twisted standard $L$-function associated to a holomorphic vector-valued Siegel cusp form of degree $n$ and arbitrary…

A general construction is found for `topological' singular vectors of the twisted N=2 superconformal algebra. It demonstrates many parallels with the known construction for sl(2) singular vectors due to Malikov--Feigin--Fuchs, but is…

The Glasgow reweighting method is evaluated for SU(2) lattice gauge theory at nonzero \mu and finite T. We establish that the ' overlap problem' of SU(3) measurements, in which the transition points determined from thermodynamic observables…

In this article, we give explicit formulas of archimedean Whittaker functions on $GL(3)$ and $GL(2)$. Moreover, we apply those to the calculation of archimedean zeta integrals for $GL(3)\times GL(2)$, and show that the zeta integral for…

Michael HARRIS defined the arithmetic automorphic periods for certain cuspidal representations of $GL_{n}$ over quadratic imaginary fields in his Crelle paper 1997. He also showed that critical values of automorphic L-functions for…

Let $G$ be a non-compact connected semisimple real Lie group with finite center. Suppose $L$ is a non-compact connected closed subgroup of $G$ acting transitively on a symmetric space $G/H$ such that $L\cap H$ is compact. We study the…

We study the automorphic period associated to a $G$-Hamiltonian variety $M$ whose dual is $\check{M} = T^*(\check{G}/\check{L})$, where $\check{G}$ is a general linear group and $\check{L}$ is a Levi subgroup. For certain cuspidal…

The purpose of this semi-expository article is to give another proof of a classical theorem of Shimura on the critical values of the standard L-function attached to a Hilbert modular form. Our proof is along the lines of previous work of…

Using explicit constructions of the Weierstrass mock modular form, we offer a closed formula for generating the values of shifted convolution $L$-values for certain elliptic curves that can be computed to arbitrary precision. These…

We prove an asymptotic formula for the second moment of the $\mathrm{GL}(n)\times\mathrm{GL}(n+1)$ Rankin--Selberg central $L$-values $L(1/2,\Pi\otimes\pi)$, where $\pi$ is a fixed cuspidal representation of $\mathrm{GL}(n)$ that is…

Let $f$ be a normalized newform of weight 2 on $\Gamma_0(N)$ whose coefficients lie in $\mathbb{Q}$ and let $\chi_M$ be a primitive quadratic Dirichlet character with conductor $M$. In this paper, under mild assumptions on $M$, we give a…

We show that a pair of newforms $(f,g)$ can be uniquely determined by the product of the central $L$-values of their twists. To achieve our goal, we prove an asymptotic formula for the average of the product of the central values of two…

In this paper, over imaginary quadratic fields, we consider the family of $L$-functions $L (s, f)$ for an orthonormal basis of spherical Hecke--Maass forms $f$ with Archimedean parameter $t_f$. We establish asymptotic formulae for the…

In this paper we give an explicit formula for the twisted Alexander polynomial of any torus link and show that it is a locally constant function on the $SL(2, \mathbb C)$-character variety. We also discuss similar things for the higher…

We propose a geometric framework to produce a formula relating higher period integrals to higher central derivatives of $L$-functions over function fields, extending the framework of relative Langlands duality \`a la…

We prove hybrid subconvexity bounds twisted L-functions $L(s,f\times \chi)$ at the central point using a fourth moment estimate, including a new instance of the Burgess subconvexity bound.

We prove a spectral reciprocity formula for automorphic forms on $\mathrm{GL}(2)$ over a number field that is remininscent of the one found by Blomer and Khan. Our approach uses period representations of $L$-functions and the language of…

Given $c,$ a positive integer, we give an explicit formula and an asymptotic formula for \[ \sum\chi(c)|L(1,\,\chi)|^{2}, \] where $\chi$ is the non-trivial Dirichlet character mod $f$ with $f>c.$