Spinor L functions, theta correspondence, and Bessel coefficients
Ramin Takloo-Bighash
本文证明了全实数域上群GSp（4）的某些一般表示的旋量L函数的完备性。
In this paper we prove the entireness of the Spinor L function of certain generic representations of the group GSp(4) over a totally real field.
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We construct the anticyclotomic $p$-adic $L$-function that interpolates a square root of central values of twisted spinor $L$-functions of a quadratic base change of a Siegel cusp form of genus $2$ with respect to a paramodular group of square-free level, assuming the Böcherer conjecture for the central $L$-values with anticyclotomic twists.
We explicitly give an inner product formula and a Bessel period formula for theta series on GSp(4), which was studied by Harris, Soudry and Taylor. As a consequence, we prove a non-vanishing of the theta series of small weights and we give a criterion for the non-vanishing of the theta series modulo a prime.
Lattice sums of $I$-Bessel functions, theta functions, linear codes and heat equations
该研究推广了格上$I$-贝塞尔函数和的恒等式，建立其与theta函数变换、线性码完全重量枚举子及格上热方程解的关联，并得到$\boldsymbol{\rm Z}^n$上带线性码初始条件的热方程显式解。
A new generalization of the modified Bessel function of the second kind $K_{z}(x)$ is studied. Elegant series and integral representations, a differential-difference equation and asymptotic expansions are obtained for it thereby anticipating a rich theory that it may possess. The motivation behind introducing this generalization is to have a function which gives a new pair of functions reciprocal in the Koshliakov kernel $\cos \left( {πz} \right){M_{2z}}(4\sqrt {x} ) - \sin \left( {πz} \right){J_{2z}}(4\sqrt {x} )$ and which subsumes the self-reciprocal pair involving $K_{z}(x)$. Its applicati…
Theta series of ternary quadratic lattice cosets
本文证明三元格陪集 theta 级数的三部分自然分解分别对应亏格、spinor 亏格和类不变量，验证相关猜想，并推广 $p$-邻元以构造类代表元算法。
Bessel models for $GSp(4)$
利用theta对应的方法分析了局部和全局Bessel模型的$GSp(4)$，证明了Gross和Prasad的一个猜想，该猜想在局部情况下用局部因子来描述这些模型，而在全局情况下中心临界$L$值不为零.
We establish the Fourier-Jacobi case of the local Gross-Prasad conjecture for unitary groups, by using local theta correspondence to relate the Fourier-Jacobi case with the Bessel case established by Beuzart-Plessis. To achieve this, we prove two conjectures of D. Prasad on the precise description of the local theta correspondence for (almost) equal rank unitary dual pairs in terms of the local Langlands correspondence.
We consider a new integral representation for $L(s_1, Π\times τ_1) L(s_2, Π\times τ_2),$ where $Π$ is a globally generic cuspidal representation of $GSp_4,$ and $τ_1$ and $τ_2$ are two cuspidal representations of $GL_2$ having the same central character. As and application, we find a new period condition for two such $L$ functions to have a pole simultaneously. This points to an intriguing connection between a Fourier coefficient of a residual representation on $GSO(12)$ and a theta function on $\widetilde{Sp}(16).$ A similar integral on $GSO(18)$ fails to unfold completely, but in a way that …
Explicit Formulas for the Waldspurger and Bessel Models
本文研究非阿基米德局部域上可裂特殊正交群$SO(2n+1)$的不可约容许表示的某些模型。如果是$n=1$，Waldspurger考虑了这些模型。如果$n=2$，他们被认为是由Novodvorsky和Piatetski-Shapiro \cite{NP}，谁叫他们{ \it贝塞尔模型}。它们来自于各种Rankin-Selberg积分，本文的结果自然也会应用于L-函数的研究。他们也出现在研究的θ对应之间的$SO(2n+1)$和双盖的$Sp(2n)$，因此，他们将是重要的概括工作Waldspurger。作为整体应用，我们考虑了$GL(n)$上尖点自守表示$π$构成的$SO(2n+1)$上的Eisenstein级数，证明了它的Bessel周期（6.2）实质上是L-级数的乘积.这概括了Böcherer和Mizumoto的工作。
We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by Dirichlet series $$\sum_{n=1}^{\infty}\frac{a(n)}{λ_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{μ_n^{s}},$$ satisfying a familiar functional equation involving the gamma function $Γ(s)$. Two general identities are established. The first involves the modified Bessel function $K_μ(z)$, and can be thought of as a 'modular' or 'theta' relation wherein modified Bessel functions, instead of exponential functions, appear. Appearing in the second identity are $K_μ(z)$, the Bessel functions of imaginary argument…
The tower property on the genericity of global theta lifts
研究通过分析L函数的解析性质与特殊Bessel和傅里叶-雅可比周期的关系，证明双余元群间全局theta提升首次出现时保持通用性，并应用此结果验证SO_{2n+1}×SO_2的全球Gan-Gross-Prasad猜想。
Let $f(q)=1+\sum_{n=1}^{\infty} α(n)q^n$ be the well-known third order mock theta of Ramanujan. In 1964, George Andrews proved an asymptotic formula of the form $$α(n)= \sum_{c\leq\sqrt{n}} ψ(n)+O_ε\left(n^ε\right),$$ where $ψ(n)$ is an expression involving generalized Kloosterman sums and the $I$-Bessel function. Andrews conjectured that the series converges to $α(n)$ when extended to infinity, and that it does not converge absolutely. Bringmann and Ono proved the first of these conjectures. Here we obtain a power savings bound for the error in Andrews' formula, and we also prove the second o…
Certain Extensions of Results of Siegel, Wilton and Hardy
本文通过引入含贝塞尔函数的 theta 变换公式，将 Hardy 定理推广至满足 Hecke 函数方程的 Dirichlet 级数，并得到解析数论经典恒等式的新推广。
A generalized modular relation of the form $F(z, w, α)=F(z, iw,β)$, where $αβ=1$ and $i=\sqrt{-1}$, is obtained in the course of evaluating an integral involving the Riemann $Ξ$-function. It is a two-variable generalization of a transformation found on page $220$ of Ramanujan's Lost Notebook. This modular relation involves a surprising generalization of the Hurwitz zeta function $ζ(s, a)$, which we denote by $ζ_w(s, a)$. While $ζ_w(s, 1)$ is essentially a product of confluent hypergeometric function and the Riemann zeta function, $ζ_w(s, a)$ for $0<a<1$ is an interesting new special function. …
Foundations of Zeta Space: Geometry, Spectral States, and Fourier Mellin Duality
本文为zeta空间纲领奠定几何基础，通过接缝、叶片与边界构造统一谱态与Fourier-Mellin对偶，证明热迹与Mellin变换恢复theta与zeta函数，并建立对偶组装定理。
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