Hiraku Atobe
We prove the local Gan-Gross-Prasad conjecture for the symplectic-metaplectic case under some assumptions. This is the last case of the local Gan-Gross-Prasad conjectures. We also prove two of Prasad's conjectures on the local theta correspondence in the almost equal rank case.
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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 prove the local Gross-Prasad conjecture for generic L-packets of representations of special orthogonal groups. The proof uses the same result for tempered L-packets proved in a preceding paper, and irreducibility results for the induced representations of whose the elements of the L-packets are Langlands quotients.
Under endoscopic assumptions about $L$-packets of unitary groups, we prove the local Gan-Gross-Prasad conjecture for tempered representations of unitary groups over $p$-adic fields. Roughly, this conjecture says that branching laws for $U(n-1)\subset U(n)$ can be computed using epsilon factors.
Spherical Characters in Families: the unitary Gan-Gross-Prasad case
本文研究了族中球面特征的变分，提出了其有理性与亚纯性质的猜想，并在酉 Gan-Gross-Prasad 情形下证明了这些猜想。
Twisted Gan-Gross-Prasad conjecture for unramified quadratic extensions
本文采用相对迹公式，在非分歧性与局部条件下证明了酉群嵌入一般线性群情形的扭曲全局 Gan-Gross-Prasad 猜想及其精细化。
The Local Gross-Prasad Conjecture over Archimedean Local Fields
该研究采用C. Moeglin和J.L.-Waldspurger的方法，完成了阿基米德局部域上局部Gross-Prasad猜想一般情形的证明，推进了相关数论问题的研究。
The local twisted Gan-Gross-Prasad conjecture for $\text{U}(V_K)/\text{U}(V)$
本文通过局部迹公式及其扭曲变体的几何展开，证明了在奇剩余特征非阿基米德域上对于缓增$L$-参数的$U(V_K)/U(V)$局部扭曲Gan-Gross-Prasad猜想。
We state and prove an extension of the global Gan-Gross-Prasad conjecture and the Ichino-Ikeda conjecture to the case of some Eisenstein series on unitary groups $U_n\times U_{n+1}$. Our theorems are based on a comparison of the Jacquet-Rallis trace formulas. A new point is the expression of some interesting spectral contributions in these formulas in terms of integrals of relative characters. As an application of our mains theorems, we prove the global Gan-Gross-Prasad and the Ichino-Ikeda conjecture for Bessel periods of unitary groups.
We introduce a new technique for isolating components on the spectral side of the trace formula. By applying it to the Jacquet--Rallis relative trace formula, we complete the proof of the global Gan--Gross--Prasad conjecture and its refinement Ichino--Ikeda conjecture for $\mathrm{U}(n)\times\mathrm{U}(n+1)$ in the stable case.
We formulate a global Gan-Gross-Prasad conjecture for general spin groups. That is, we formulate a conjecture on a relation between periods of certain automorphic forms on $GSpin_{n+1} \times GSpin_n$ along the diagonal subgroup $GSpin_n$ and some $L$-values. To support the conjecture, we show that the conjecture holds for $n=2$ and $3$ and for certain cases for $n=4$.
Through combining the work of Jean-Loup Waldspurger (\cite{waldspurger10} and \cite{waldspurgertemperedggp}) and Raphaël Beuzart-Plessis (\cite{beuzart2015local}), we give a proof for the tempered part of the local Gan-Gross-Prasad conjecture (\cite{ggporiginal}) for special orthogonal groups over any local fields of characteristic zero, which was already proved by Waldspurger over $p$-adic fields.
Xue proved an equational refinement of the unitary Shimura curve case of the arithmetic Gan-Gross-Prasad conjecture via the Gross-Zagier formula for quaternionic Shimura curves. On the other hand, Rapoport, Smithling and Zhang posed a variant of the conjecture, using modified PEL type Shimura varieties, which we call RSZ Shimura varieties. We reinterpret the result of Xue in terms of the modified Shimura curves. We then use the reinterpretation to prove a case of the variant of the conjecture. Our result combined with the work of Xue establishes a connection between the variant and the Gross-Z…
In this paper, we investigate the twisted GGP conjecture for certain tempered representations using the theta correspondence and establish some special cases, namely when the L-parameter of the unitary group is the sum of conjugate-dual characters of the appropriate sign.
In this paper, we prove, following earlier work of Waldspurger ([Wa1], [Wa4]), a sort of local relative trace formula which is related to the local Gan-Gross-Prasad conjecture for unitary groups over a local field $F$ of characteristic zero. As a consequence, we obtain a geometric formula for certain multiplicities $m(π)$ appearing in this conjecture and deduce from it a weak form of the local Gan-Gross-Prasad conjecture (multiplicity one in tempered L-packets). These results were already known over $p$-adic fields and thus are only new when $F=\mathbb{R}$.
We consider irreducible unitary representations $A_i$ of G=SO(n+1,1) with the same infinitesimal character as the trivial representation and representations $B_j$ of H=SO(n,1) with the same properties and discuss H-equivariant homomorphisms Hom_H($A_i,B_j$). For tempered representations our results confirm the predictions of conjectures by B. Gross and D. Prasad.
We prove a local Gan-Gross-Prasad conjecture on predicting the branching law for the non-tempered representations of general linear groups in the case of non-Archimedean field. We also generalize to Bessel and Fourier-Jacobi models and study a possible generalization to Ext-branching laws.
We prove the local Gross-Prasad conjecture for tempered representations of special orthogonal groups. Roughly speaking, the conjecture says that, if sigma is an irreducible representation of SO(n) and rho is an irreducible representation of SO(n-1), rho appears as quotient of the restriction of sigma to SO(n-1) with a multiplicity m(sigma,rho) that can be computed in terms of epsilon-factors. Our proof uses results of a previous papers which computes m(sigma,rho) and the epsilon-factors by integral formulas.
The global Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups I: Coarse expansions of the relative trace formulae
本文首次证明了单位群上Fourier-Jacobi周期的全局Gan-Gross-Prasad猜想，并建立了相对迹公式的大规模展开。
The local Gross-Prasad conjecture over $\mathbb{R}$: Epsilon dichotomy
本文基于Waldspurger的工作，证明了实数域上 tempered 局部 L-参数的局部格罗斯-普拉兹猜想的 ε 二分法部分。
Regularized Periods and the global Gan-Gross-Prasad conjecture : The case of $U(n+2r) \times U(n)$
该研究在非约化群上引入正则化三线性周期，既可为典型群、亚辛群定义两类正则化周期，又能证明斜埃尔米特酉群上Gan-Gross-Prasad猜想的一个方向。
The local twisted Gan-Gross-Prasad conjecture for $\text{GL(V)/U(V)}$
本文利用 Waldspurger 与 Beuzart-Plessis 的 Bessel 模型方法研究两个局部相对迹公式，通过谱展开与部分几何比较，在非阿基米德域上对 tempered 表示证明了 GL(V)/U(V) 的局部扭曲 Gan-Gross-Prasad 猜想。
Fourier-Jacobi models for real symplectic-metaplectic groups: the basic case
本文推广了Gan-Ichino和Atobe的方法，证明了傅里叶-雅可比模型下辛-元摄引群局部Gan-Gross-Prasad猜想的基本温顺情形。
We restrict a Siegel modular cusp form of degree 2 and square free level that is a Yoshida lifting (a lifting from the orthogonal group of a definite quaternion algebra) to the embedded product of two half planes and compute the Petersson product against the product of two elliptic cuspidal Hecke eigenforms. The square of this integral can be explicitly expressed in terms of the central critical value of an L-function attached to the situation. The result is related to a conjecture of Gross and Prasad about restrictions of automorphic representations of special orthogonal groups.
We study GL_2(F)-invariant periods on representations of GL_2(A), where F is a nonarchimedean local field and A/F a product of field extensions of total degree 3. For irreducible representations, a theorem of Prasad shows that the space of such periods has dimension at most 1, and is non-zero when a certain epsilon-factor condition holds. We give an extension of this result to a certain class of reducible representations (of Whittaker type), extending results of Harris--Scholl when A is the split algebra F x F x F.
The global Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups III: Proof of the main theorems
本文证明了关于单位ary群上的Fourier-Jacobi周期的Gan-Gross-Prasad猜想，并通过谱展开和零阶情况的减少完成了主要定理的证明。
Let $E/F$ be a quadratic extension of non-archimedean local fields of characteristic 0 and let $G=U(n)$, $H=U(m)$ be unitary groups of hermitian spaces $V$ and $W$. Assume that $V$ contains $W$ and that the orthogonal complement of $W$ is a quasisplit hermitian space (i.e. whose unitary group is quasisplit over $F$). Let $π$ and $σ$ be smooth irreducible representations of $G(F)$ and $H(F)$ respectively. Then Gan, Gross and Prasad have defined a multiplicity $m(π,σ)$ which for $m=n-1$ is just the dimension of $Hom_{H(F)}(π,σ)$. For $π$ and $σ$ tempered, we state and prove an integral formula f…
Residues of Rankin-Selberg Zeta integrals and the split non-tempered Gan-Gross-Prasad conjectures
本文通过 Zeta 积分残差正则化方法，解决了非温控 Gan-Gross-Prasad 猜想及其细化版本在局部域上的问题。
Central values of Asai L-functions and twisted Gan--Gross--Prasad conjecture
本文在相对 Langlands 纲领下研究了涉及 Weil 表示的新相对迹公式，通过几何分解与全局比较，在局部假设下证明了任意维度且允许数域分歧的扭曲 GGP 猜想。
On the Gan-Gross-Prasad conjecture and its refinement for $\left(\mathrm{U}\left(2n\right),\mathrm{U}\left(1\right)\right)$
本文一般地证明了酉群对 (U(2n),U(1)) 的 Gan-Gross-Prasad 猜想，并在一定假设下建立中心 L 值的 Ichino-Ikeda 型精细公式；同时证明其分裂类比 (GL_{2n},GL_1) 的相应猜想与精细形式。
In this paper, we prove the Gan-Gross-Prasad conjecture and the Ichino-Ikeda conjecture for unitary groups $U_n\times U_{n+1}$ in all the endoscopic cases. Our main technical innovation is the computation of the contributions of certain cuspidal data, called $*$-generic, to the Jacquet-Rallis trace formula for linear groups. We offer two different computations of these contributions: one, based on truncation, is expressed in terms of regularized Rankin-Selberg periods of Eisenstein series and Flicker-Rallis intertwining periods. The other, built upon Zeta integrals, is expressed in terms of fu…
Non-tempered Gan-Gross-Prasad conjecture for Archimedean general linear groups
本文针对阿基米德局部域上的一般线性群，证明了非tempered Gan-Gross-Prasad猜想中“周期蕴含相关”的方向，结合已有结果解决了该猜想。
In this paper, we investigate the local Gan-Gross-Prasad conjecture for some pair of representations of $U(3)\times U(2)$ involving a non-generic representation. For a pair of generic $L$-parameters of $(U(n),U(n-1))$, it is known that there is a unique pair of representations in their associateed Vogan $L$-packets which produces the unique Bessel model of these $L$-parameters. We showed that this is not ture for some pair of $L$-parameters involving a non-generic one. On the other hand, we give the precise local theta correspondence for $(U(1),U(3))$ not at the level of $L$-parameters but of …
Let F be a number field, A_F its ring of adeles, and let π_n and π_{n+1} be irreducible, cuspidal, automorphic representations of SO_n(A_F) and SO_{n+1}(A_F), respectively. In 1991, Benedict Gross and Dipendra Prasad conjectured the non-vanishing of a certain period integral attached to π_n and π_{n+1} is equivalent to the non-vanishing of L(1/2, π_n x π_{n+1}). More recently, Atsushi Ichino and Tamotsu Ikeda gave a refinement of this conjecture as well as a proof of the first few cases (n = 2,3). Their conjecture gives an explicit relationship between the aforementioned L-value and period int…
Duality for generalized Gan-Gross-Prasad relevant pairs for $p$-adic $\mathrm{GL}_n$
本文提出广义 GGP 相关对概念以描述 p 进一般线性群的商分支律，并基于导数与积分的交换关系，从表示论和组合角度证明相关对的对偶性。
On the Gross-Prasad conjecture with its refinement for $\left(\mathrm{SO}\left(5\right),\mathrm{SO}\left(2\right)\right)$ and the generalized Böcherer conjecture
本文通过组合多个theta对应，在（SO(5), SO(2)）情形下证明了Bessel周期的Ichino-Ikeda型公式，并由此将Böcherer猜想推广到非平凡环面特征情形。
Fourier-Jacobi periods and the non-tempered Gan--Gross--Prasad conjecture for $\Mp_{2n} \times \Sp_{2m}$
本文针对两类非缓增全局$A$-参数族，分别用正则化Fourier-Jacobi周期与全局theta对应等方法，建立了辛-亚辛对的非缓增全局GGP猜想的一个方向，还得到了缓增GGP猜想的相关结果与$L$-包二分性结论。
Rankin-Selberg integrals for $\mathrm{GSpin}$ groups with application to the global Gan-Gross-Prasad conjecture
本文利用 Bessel 模型为 $\mathrm{GSpin}$ 群构造了张量积部分 $L$-函数乘积的 Rankin-Selberg 积分，并以此证明了拟分裂 $\mathrm{GSpin}$ 群一般表示的全局 Gan-Gross-Prasad 猜想的一个方向。
We prove a conjectural formula relating the Bessel period of certain automorphic forms on $\mathrm{GSp}_4$ to a central $L$-value. This formula is proposed by Liu \cite{liu} as the refined Gan-Gross-Prasad conjecture for the groups $(\SO(5), \SO(2))$. The conjecture has been previously proved for certain automorphic forms on $\mathrm{GSp_4}$ from lifts. In this paper, we extend the formula to Siegel modular forms of $\Sp_4(\bZ)$.
The local Gan-Gross-Prasad conjecture of unitary groups, which is now settled by the works of Plessis, Gan and Ichino, says that for a pair of generic $L$-parameters of $(U(n+1),U(n))$, there is a unique pair of representations in their associated Vogan $L$-packets which produces the Bessel model. In this paper, we examined the conjecture for a pair of $L$-parameters of $\big(U(n+1), U(n)\big)$ as fixing a special non-generic parameter of $U(n+1)$ and varing tempered $L$-parameters of $U(n)$. We observed that there still exist a Gan-Gross-Prasad type formulae depending on the choice of $L$-par…
In this article, we compute the Gan-Gross-Prasad period integral of Klingen Eisenstein series over the unitary group $\mathrm{U}(m+1, n+1)$ with a cuspidal automorphic form over $\mathrm{U}(m+1, n)$, and show that it is related to certain special Rankin-Selberg $L$-values. We $p$-adically interpolate these Gan-Gross-Prasad period integrals as the Klingen Eisenstein series and the cuspidal automorphic form vary in Hida families. As a byproduct, we obtain a $p$-adic $L$-function of Rankin-Selberg type over $\mathrm{U}(m,n) \times \mathrm{U}(m+1, n)$. The ultimate motivation is to show the $p$-pr…
We prove a precise formula relating the Bessel period of certain automorphic forms on ${\rm GSp}_{4}(\mathbb{A}_{F})$ to a central $L$-value. This is a special case of the refined Gan--Gross--Prasad conjecture for the groups $({\rm SO}_{5},{\rm SO}_{2})$ as set out by Ichino--Ikeda and Liu. This conjecture is deep and hard to prove in full generality; in this paper we succeed in proving the conjecture for forms lifted, via automorphic induction, from ${\rm GL}_{2}(\mathbb{A}_{E})$ where $E$ is a quadratic extension of $F$. The case where $E=F\times F$ has been previously dealt with by Liu.
Special periods and some non-tempered cases of the Gan-Gross-Prasad conjecture
该研究建立了特殊周期与典型群自守表示特殊L值的关系，通过两个步骤结合相关准则，证明了非稳整体Gan-Gross-Prasad猜想的三个高阶秩族非稳情形。
We provide a construction of local and automorphic non-tempered Arthur packets of the group SO(3,2) and its inner form SO(4,1) associated with a certain Arthur's parameter and prove a multiplicity formula. We further study the restriction of the representations in these packets to the subgroup SO(3,1). In particular, we discover that the local Gross-Prasad conjecture, formulated for generic L-packets, does not generalize naively to a non-generic A-packet. We also study the non-vanishing of the automorphic SO(3,1)-period on the group SO(4,1) x SO(3,1) and SO(3,2) x SO(3,1) for the representatio…
An explicit refined Gan--Gross--Prasad identity for Fourier--Jacobi periods of degree 2 Siegel cusp forms
该研究针对Sp₄的傅里叶-雅可比周期，在新分歧情形下计算相关局部积分，构建了关联二次西格尔尖型形式与半整数权形式Petersson范数的显式猜想恒等式，并指出其在范数增长等方面的推论。
Let F be a non-archimedean local field, of characteristic 0. Let V be a finite dimensional vector space over F and q be a non-degenerate quadratic form on V. Denote d the dimension of V and G=SO(d) the special orthogonal group of (V,q). Let v_{0}\in V such that q(v_{0})\not=0, denote W the subspace of V orthogonal to v_{0} and H=SO(d-1) the special orthogonal group of W. Let π, resp. σ, an admissible irreducible representation of G(F), resp. H(F). Denote m(σ,π) the dimension of the complex space Hom_{H(F)}(π_{| H(F)},σ). By a theorem of Aizenbud, Gourevitch, Rallis and Schiffmann, we know that…
We prove explicit rationality-results for Asai- $L$-functions, $L^S(s,Π',{\rm As}^\pm)$, and Rankin-Selberg $L$-functions, $L^S(s,Π\timesΠ')$, over arbitrary CM-fields $F$, relating critical values to explicit powers of $(2πi)$. Besides determining the contribution of archimedean zeta-integrals to our formulas as concrete powers of $(2πi)$, it is one of the advantages of our approach, that it applies to very general non-cuspidal isobaric automorphic representations $Π'$ of ${\rm GL}_n(\mathbb A_F)$. As an application, this enables us to establish a certain algebraic version of the Gan--Gross--…
We prove a refinement of the global Gan-Gross-Prasad conjecture proposed by Ichino-Ikeda and N. Harris for unitary groups under some local conditions. We need to assume some expected properties of L-packets and some part of the local Gan-Gross-Prasad conjecture.
Refined global Gross-Prasad conjecture on special Bessel periods and Boecherer's conjecture
本文在特殊贝塞尔周期情形下证明了刘一峰提出的精细全局Gross-Prasad猜想，建立了中心L值的Ichino-Ikeda型显式公式，并由此证明了关于度为二的Hecke特征形式全纯Siegel尖点形式的Boecherer猜想。
Quotient branching law for $p$-adic $(\mathrm{GL}_{n+1}, \mathrm{GL}_n)$ I: generalized Gan-Gross-Prasad relevant pairs
本文针对非阿基米德局部域上的GL_{n+1}与GL_n，给出并证明了二者不可约光滑表示间同态非零的充要条件，解决了商分支律问题，还将结果推广到A型仿射Hecke代数并细化了分支律。
In this paper, we highlight and state precisely the local Langlands correspondence for quasi-split O_{2n} established by Arthur. We give two applications: Prasad's conjecture and Gross--Prasad conjecture for O_{n}. Also, we discuss the Arthur conjecture for O_{2n}, and establish the Arthur multiplicity formula for O_{2n}.
Gan--Gross--Prasad cycles and derivatives of $p$-adic $L$-functions
本文基于单位群上的算术 GGP 猜想，研究 p-adic L-函数的导数与算术对角周期的 p-adic 高度之间的关系，并推导出对动机的 p-adic Beilinson-Bloch-Kato 猜想的应用。
We consider cycles on a 3-dimensional Shimura varieties attached to a unitary group, defined over extensions of a CM field $E$, which appear in the context of the conjectures of Gan, Gross, and Prasad \cite{gan-gross-prasad}. We establish a vertical distribution relation for these cycles over an anticyclotomic extension of $E$, complementing the horizontal distribution relation of \cite{jetchev:unitary}, and use this to define a family of norm-compatible cycles over these fields, thus obtaining a universal norm construction similar to the Heegner $Λ$-module constructed from Heegner points.
Methods of theta correspondence are used to analyze local and global Bessel models for $GSp(4)$ proving a conjecture of Gross and Prasad which describes these models in terms of local epsilon factors in the local case, and the non-vanishing of central critical $L$-value in the global case.
The Miyawaki liftings are defined by the pullbacks of Ikeda liftings. Recently, Ikeda and Yamana extended the theory of Ikeda liftings. In this paper, using their results, we establish a theory of Miyawaki liftings, both locally and globally. In the local theory, we describe the Miyawaki liftings for almost tempered unitary representations explicitly. In the global theory, we discuss the non-vanishing of the Miyawaki liftings using seesaw identities and the global Gan-Gross-Prasad conjecture. As an application of local Miyawaki liftings, we prove a new case of the local Gan-Gross-Prasad conjec…
In this paper, we give an explicit determination of the non-vanishing of the theta liftings for unitary dual pairs (U(p,q), U(r,s)). Assuming the local Gan-Gross-Prasad conjecture, we determine when theta lifts of tempered representations are nonzero in terms of the local Langlands correspondence.
We define some formal moduli space of quasi-isogenies of isoclinic $p$-divisible groups with a non-reductive group as the "structure group". We then formulate new Arithmetic Fundamental Lemma conjectures for Bessel subgroups in the context of the arithmetic Gan-Gross-Prasad conjectures. Some (very limited) evidence is presented.
Semi-Lie arithmetic fundamental lemma for the full spherical Hecke algebra
本文针对Yifeng Liu提出的半Lie算术基本引理，将其推广到全球面Hecke代数并表述相应猜想，同时给出两类猜想中加权轨道积分的显式公式，证明了第一个非平凡情形。
We investigate the local descents for special orthogonal groups over p-adic local fields of characteristic zero, and obtain an explicit spectral decomposition of the local descents at the first occurrence index in terms of the local Langlands data via the explicit local Langlands correspondence and explicit calculations of relevant local root numbers. The main result can be regarded as a refinement of the local Gan-Gross-Prasad conjecture.
Whittaker-Shintani functions for Fourier-Jacobi models on unitary groups
本文陈述并证明了 p 进酉群和一般线性群上 Fourier-Jacobi 模型的 Whittaker-Shintani 函数公式，并以此证明了非分歧的 Ichino-Ikeda 猜想。
Local newforms for generic representations of unramified even unitary groups I: Even conductor case
本文针对非分歧偶酉群，在偶导子情形下构造紧开子群，并借助局部 Gan-Gross-Prasad 猜想、Rankin-Selberg 积分和 theta 对应建立不可约 tempered 通用表示的局部新形式理论。
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