Henry H. Kim, Freydoon Shahidi
The purpose of this paper is to prove the long awaited holomorphy of the third symmetric power L-functions attached to nonmonomial cusp forms of GL_2 over an arbitrary number field on the whole complex plane.
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We give a new proof of the stability of the symmetric cube gamma factor as defined by the Langlands-Shahidi method.
Let $F$ be a cuspidal eigenform of even weight and trivial nebentypus, let $p$ be a prime not dividing the level of $F$, and let $ρ_F$ be the $p$-adic Galois representation attached to $F$. Assume that the $L$-function attached to the symmetric cube of $ρ_F$ vanishes to odd order at its central point. Then under some mild hypotheses, and conditional on certain consequences of Arthur's conjectures, we construct a nontrivial element in the Bloch--Kato Selmer group of an appropriate twist of the symmetric cube of $ρ_F$, in accordance with the Bloch--Kato conjectures. Our technique is based on the…
On the Bloch--Kato conjecture for the symmetric cube
本文针对1级模形式的对称三次方，证明了Iwasawa主猜想的一个包含关系及解析秩0下的Bloch-Kato猜想。
In this paper we prove two new cases of Langlands functoriality. The first is a functorial product for cusp forms on $GL_2\times GL_3$ as automorphic forms on $GL_6$, from which we obtain our second case, the long awaited functorial symmetric cube map for cusp forms on $GL_2$. We prove these by applying a recent version of converse theorems of Cogdell and Piatetski-Shapiro to analytic properties of certain $L$-functions obtained from the method of Eisenstein series (Langlands-Shahidi method). As a consequence, we prove the bound 5/34 for Hecke eigenvalues of Maass forms over any number field a…
Suppose $π$, $π'$ are cusp forms on GL$(2)$, not of solvable polyhedral type, such that they have the same symmetric cubes. Then we show that either $π$, $π'$ are twist equivalent, or else a certain degree $36$ $L$-function associated to the pair has a pole at $s=1$. If we further assume that the symmetric fifth power of $π$ is automorphic, then in the latter case, $π$ is icosahedral in a suitable sense, agreeing with the usual notion when there is an associated Galois representation.
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.
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…
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。
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…
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。
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…
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。
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…
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.
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…
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…
We prove Deligne's conjecture for symmetric fifth $L$-functions of elliptic newforms of weight greater than $5$. As a consequence, we establish period relations between motivic periods associated to an elliptic newform and the Betti-Whittaker periods of its symmetric cube functorial lift to ${\rm GL}_4$.
The structure of sets with cube-avoiding sumsets
本文研究有限阿贝尔群中避立方和集的稠密子集结构，证明其本质上具有与 n 无关的有界维数，揭示了和集避开特定结构时集合的刚性特征。
A full $p^{4r}$ supercongruence tower for a level-three symmetric-cube hypergeometric sequence
本文证明了一个三级对称立方超几何序列的素数幂超同余塔 $A_{mp^r}\equiv A_{mp^{r-1}}\pmod{p^{4r}}$，通过模形式方法将已知的深度一模 $p^4$ 推广到所有素数幂。
Some zeta functions which are naturally attached to the locally homogeneous vector bundles over compact locally symmetric spaces of rank one are investigated. We prove that such functions can be expressed in terms of entire functions whose order is not larger than the dimension of the corresponding compact, even-dimensional, locally symmetric space.
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