Stability of symmetric cube gamma factors for GL(2)

Beyond endoscopy for the Symmetric Cube L-function

This paper is a first attempt at getting information on a symmetric power representation of a $GL_2$ automorphic form via a trace formula that is beyond endoscopic techniques. In particular, we study the symmetric third power representation for all forms over $\Q$ adjoined the cube roots of unity. A key tool needed in the study is an identity relating cubic exponential sums to Kloosterman sums. This very same identity is crucial to the fundamental lemma of a trace formula comparison in work of Mao and Rallis.

Co-periods and central symmetric cube L-values

In this article, we study the co-period integral attached to an automorphic form on $\GL(2)$ and two exceptional theta series on the cubic Kazhdan-Patterson cover of $\GL(2)$. In the local aspect, we show the $\Hom$-space is always of one dimension and conduct the unramified calculations. In the global aspect, we give the Euler decomposition for the co-period integrals of Eisenstein series and propose an Ichino-Ikeda type conjecture relating the co-period integrals of cuspidal forms to the central critical value of symmetric cube $L$-functions. We also deduce from the local multiplicity one re…

对模形式对称立方转移的两个性质研究

Two properties of symmetric cube transfers of modular forms

本文研究对称立方转移的自守表示的导数和局部自守表示类型，通过ε因子的变化来分类可能的转移类型，重点分析p=3的情况。

Non-vanishing of symmetric cube $L$-functions

We prove that there are infinitely many Maass--Hecke cuspforms over the field $\mathbb{Q}[\sqrt{-3}]$ such that the corresponding symmetric cube $L$-series does not vanish at the center of the critical strip. This is done by using a result of Ginzburg, Jiang and Rallis which shows that the symmetric cube non-vanishing happens if and only if a certain triple product integral involving the cusp form and the cubic theta function on $\mathbb{Q}[\sqrt{-3}]$ does not vanish. We use spectral theory and the properties of the cubic theta function to show that the non-vanishing of this triple product oc…

关于 $\mathrm{GL}(2)$ 对称立方体的 $\mathrm{SO}(5)\times \mathrm{SO}(2)$ Bessel 周期的去相关

Decorrelation of $\mathrm{SO}(5)\times \mathrm{SO}(2)$ Bessel periods for symmetric cubes of $\mathrm{GL}(2)$

本文在广义黎曼假设下证明，对虚二次域平均时，GL(2) 代数正则 Hecke 特征形式的对称立方体的 SO(5)×SO(2) 全局 Bessel 周期呈现去相关现象。

Icosahedral Fibres of the Symmetric Cube and Algebraicity

For any number field F, call a cusp form πon GL(2)/F {\it special icosahedral}, or just s-icosahedral for short, if πis not solvable polyhedral, and for a suitable "conjugate" cusp form π' on GL(2)/F, sym^3(π) is isomorphic to sym^3(π'), and the symmetric fifth power L-series of πequals the Rankin-Selberg L-function L(s, sym^2(π') x π) (up to a finite number of Euler factors). Then the point of this Note is to obtain the following result: Let πbe s-icosahedral (of trivial central character). Then π_f is algebraic without local components of Steinberg type, π_\infty is of Galois type, and π_v i…

三次对称超几何系数的模 p⁴ 超同余

Order drop, Hecke descent, and a mod $p^4$ supercongruence for symmetric-cube hypergeometric coefficients

本文证明了三次对称超几何系数满足模 p⁴ 的超同余关系，通过 Ore 分解实现阶降，建立模算术字典并结合拉格朗日-布尔曼提取与三层指数截断，利用弗里克-海克 intertwining 方法消除缺陷形式。

Critical values of Rankin-Selberg L-functions for GL(n) x GL(n-1) and the symmetric cube L-functions for GL(2)

In a previous article we had proved an algebraicity result for the central critical value for L-functions for GL(n) x GL(n-1) over Q assuming the validity of a nonvanishing hypothesis involving archimedean integrals. The purpose of this article is to generalize that result for all critical values for L-functions for GL(n) x GL(n-1) over any number field F. Binyong Sun has recently proved that nonvanishing hypothesis and so the results of this article are unconditional. Using such results for the case of GL(3) x GL(2), new unconditional algebraicity results for the special values of symmetric c…

Some Symmetric Pyramids with trivial Dehn invariant

For some symmetric pyramids of $R^3$ , we find Galois obstruction for their Dehn invariant to be zero, i.e. for the pyramids to be scissor equivalent to a cube. These conditions are that some associated Kummer extensions of number fields must be abelian. This work can be viewed as a complement to [5]. Indeed, in [5], a classification of all rational tetrahedra is given, while we concentrate our attention on much more symmetric pyramids and prove that the only ones which are scissor equivalent to a cube are the rational ones.

On Greenberg's $L$-invariant of the symmetric sixth power of an ordinary cusp form

We derive a formula for Greenberg's $L$-invariant of Tate twists of the symmetric sixth power of an ordinary non-CM cuspidal newform of weight $\geq4$, under some technical assumptions. This requires a "sufficiently rich" Galois deformation of the symmetric cube which we obtain from the symmetric cube lift to $\GSp(4)_{/\QQ}$ of Ramakrishnan--Shahidi and the Hida theory of this group developed by Tilouine--Urban. The $L$-invariant is expressed in terms of derivatives of Frobenius eigenvalues varying in the Hida family. Our result suggests that one could compute Greenberg's $L$-invariant of all…

Comparison results for certain periods of cusp forms on GL(2n) over a totally real number field

This article grew out of my talk in "The Legacy of Srinivasa Ramanujan" conference where I spoke about some techniques to prove algebraicity results for the special values of symmetric cube L-functions attached to the Ramanujan Δ-function. If one wishes to compare these different techniques, then one needs to compare various automorphic periods attached to the symmetric cube transfer of Δ. Motivated by this problem, in this article we provide comparison results for Whittaker-Betti periods, Shalika-Betti periods and certain relative periods attached to a given cohomological cuspidal automorphic…

Integral cohomology of certain Picard modular surfaces

Let Gamma be a congruence subgroup of the Picard modular group of an imaginary number field k, and let D be the associated symmetric space. We describe a method to compute the integral cohomology of the locally symmetric space Gamma\D. The method is implemented for the case k=Q(i) and k=Q(sqrt(-3)), and the cohomology is computed for various Gamma.