For a representation of the absolute Galois group of the rationals over a finite field of characteristic $p$, we study the existence of a lift to characteristic zero that is geometric in the sense of the Fontaine-Mazur conjecture. For two-dimensional representations, Ramakrishna proved that under technical assumptions odd representations admit geometric lifts. We generalize this to higher dimensional orthogonal and symplectic representations. A key step is generalizing and studying a local deformation condition at $p$ arising from Fontaine-Laffaille theory.

2018-07-27math.NT

Orthogonal and symplectic n-level densities

In this paper we apply to the zeros of families of $L$-functions with orthogonal or symplectic symmetry the method that Conrey and Snaith used to calculate the $n$-correlation of the zeros of the Riemann zeta function. This method uses the Ratios Conjectures for averages of ratios of zeta or $L$-functions. Katz and Sarnak conjecture that the zero statistics of families of $L$-functions have an underlying symmetry relating to one of the classical compact groups $U(N)$, $O(N)$ and $USp(2N)$. Here we complete the work already done with $U(N)$ to show how new methods for calculating the $n$-level …

2018-06-21math.NT

除数函数和的辛猜想及正交情形的探索

Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime

本文提出了产生正交矩阵积分的类比问题，深入研究了辛和正交矩阵积分并将其与对称函数理论联系，进而导出了数域上的相关猜想。

2024-02-19math.NT

On the mean values of L-functions in orthogonal and symplectic families

Hybrid Euler-Hadamard products have previously been studied for the Riemann zeta function on its critical line and for Dirichlet L-functions in the context of the calculation of moments and connections with Random Matrix Theory. According to the Katz-Sarnak classification, these are believed to represent families of L-function with unitary symmetry. We here extend the formalism to families with orthogonal & symplectic symmetry. Specifically, we establish formulae for real quadratic Dirichlet L-functions and for the L-functions associated with primitive Hecke eigenforms of weight 2 in terms of …

2009-03-24math.NT

正交群和辛群的Taylor-Wiles方法的自守侧

Automorphic side of Taylor-Wiles method for orthogonal and symplectic groups

本文针对全实域上的定特殊正交群或辛群，构造Taylor-Wiles素数并发展其自守侧，证明极小R=ℤ[T]定理，推导伴随表示的Bloch-Kato猜想，为相关Langlands纲领提供新证据。

2025-08-19math.NT

The distribution of the logarithm in an orthogonal and a symplectic family of $L$-functions

We consider the logarithm of the central value $\log L(1/2)$ in the orthogonal family ${L(s,f)}_{f \in H_k}$ where $H_k$ is the set of weight $k$ Hecke-eigen cusp form for $SL_2(\mathbb{Z})$, and in the symplectic family ${L(s,χ_{8d})}_{d \asymp D}$ where $χ_{8d}$ is the real character associated to fundamental discriminant $8d$. Unconditionally, we prove that the two distributions are asymptotically bounded above by Gaussian distributions, in the first case of mean $-\frac{1}{2} \log \log k$ and variance $\log \log k$, and in the second case of mean $\frac{1}{2}\log \log D$ and variance $\log…

2014-11-24math.NT

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Arthur's multiplicity formula for certain inner forms of special orthogonal and symplectic groups

Let G be a special orthogonal group or an inner form of a symplectic group over a number field F such that there exists a non-empty set S of real places of F at which G has discrete series and outside of which G is quasi-split. We prove Arthur's multiplicity formula for automorphic representations of G having algebraic regular infinitesimal character at all places in S.

2015-10-28math.NTmath.RT

Some local-global non-vanishing results of theta lifts for symplectic-orthogonal dual pairs

Following the approach of B. Roberts, we characterize the non-vanishing of global theta lifts for symplectic-orthogonal dual pairs in terms of its local counterpart. In particular, we replace the temperedness assumption present in Robert's work by a certain weaker assumption, and apply our results to small rank similitude groups. Among our applications is a certain instance of Langlands functorial transfer of a (non-generic) cuspidal automorphic representation of $GSp(4)$ to GL(4).

2010-05-12math.NTmath.RT

Arithmetic properties of similitude theta lifts from orthogonal to symplectic groups

By adapting the work of Kudla and Millson we obtain a lifting of cuspidal cohomology classes for the symmetric space associated to GO(V) for an indefinite rational quadratic space V of even dimension to holomorphic Siegel modular forms on GSp_n(A). For n=2 we prove Thom's Lemma for hyperbolic 3-space, which together with results of Kudla and Millson imply an interpretation of the Fourier coefficients of the theta lift as period integrals of the cohomology class over certain cycles. This allows us to prove the p-integrality of the lift for a particular choice of Schwartz function for almost all…

2009-02-26math.NT

Petersson 内积与偶特殊正交群和辛群上的 Whittaker--Fourier 周期

Petersson Inner Products and Whittaker--Fourier Periods on Even Special Orthogonal and Symplectic Groups

本文在偶特殊正交群和辛群上建立 Whittaker--Fourier 系数平方范数与 Petersson 内积及 $L$-函数临界值的关系，并给出奇特殊正交群的类似结果。

2024-07-18math.NTmath.RT

Lower order terms for the moments of symplectic and orthogonal families of $L$-functions

We derive formulas for the terms in the conjectured asymptotic expansions of the moments, at the central point, of quadratic Dirichlet $L$-functions, $L(1/2,χ_d)$, and also of the $L$-functions associated to quadratic twists of an elliptic curve over $\Q$. In so doing, we are led to study determinants of binomial coefficients of the form $\det (\binom{2k-i-λ_{k-i+1}}{2k-2j})$.

2012-06-15math.COmath.NT

将实单位群和p-adic辛群和正交群的Arthur包联系起来

Relating Arthur packets of real unitary groups and $p$-adic symplectic and orthogonal groups

Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）；Tsinghua University（清华大学）

本文通过建立实单位群与p-adic辛群和正交群的Arthur包之间的显式对应关系，利用Zuckerman翻译 functor和Jacquet functor的关系，实现对实单位群Arthur包的计算，并关联Kazhdan-Lusztig多项式与微局部几何。

2026-03-17math.NTmath.RT

辛群与正交群的广义 Kloosterman 层的欧拉示性数

Euler characteristics of the generalized Kloosterman sheaves for symplectic and orthogonal groups

本文研究了 Yun 构造的广义 Kloosterman 层相关局部系统的单值性，计算了其在辛群与正交群标准表示下的欧拉示性数，为 Reeder-Yu 关于 epipelagic 表示 Langlands 参数预测的 Swan 导子猜想提供了证据。

2024-07-29math.AGmath.NTmath.RT

随机辛矩阵和正交矩阵特征多项式导数的联合矩

Joint moments of derivatives of characteristic polynomials of random symplectic and orthogonal matrices

该研究针对随机辛矩阵和正交矩阵系综，证明了其特征多项式导数联合矩的渐近公式，给出首阶系数的两种显式表达式，并基于结果提出$L$-函数导数联合矩的相关猜想。

2023-12-08math-phmath.MPmath.NT

Big symplectic or orthogonal monodromy modulo l

Let k be a field not of characteristic two and L be a set of almost all rational primes invertible in k. Suppose we have a variety X/k and strictly compatible system {M_ell -> X : ell in L} of constructible F_ell-sheaves. If the system is orthogonally or symplectically self-dual, then the geometric monodromy group of M_ell is a subgroup of a corresponding isometry group G_ell over F_ell, and we say it has big monodromy if it contains the derived subgroup DG_ell=[G_ell,G_ell]. We prove a theorem which gives sufficient conditions for M_ell to have big monodromy. We apply the theorem to explicit …

2007-03-11math.AGmath.NT

正特征情形下奇特殊正交群与辛群的局部逆定理

The local converse theorem for odd special orthogonal and symplectic groups in positive characteristic

本文在正特征非阿基米德局部域上，对奇特殊正交群和辛群证明了局部逆定理：具有相同中心特征和相同局部gamma因子的不可约可容许通用表示必同构，将扭曲秩从2r-1降至r并推广了特征零情形的结果。

2025-01-05math.NTmath.RT

On the Whittaker range of the generalized metaplectic theta lift

The classical theta correspondence, based on the Weil representation, allows one to lift automorphic representations on symplectic groups or their double covers to automorphic representations on special orthogonal groups. It is of interest to vary the orthogonal group and describe the behavior in this theta tower (the Rallis tower). In prior work, the authors obtained an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups that is based on the tensor product of the Weil representation with another small representation. …

2021-09-10math.NTmath.RT