Multiplicities of representations in spaces of modular forms
BinYong Hsie
本文证明了对于给定的$Γ/Γ_1$的不可约表示$ρ$，$k$的两个函数dim（$M_k(Γ_1,ρ)$）和dim（$S_k(Γ_1,ρ)$）几乎是线性函数.
This paper shows that for a given irreducible representation $ρ$ of $Γ/Γ_1$, the two functions dim($M_k(Γ_1,ρ)$) and dim($S_k(Γ_1,ρ)$) of $k$ are almost linear functions.
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The correlation between multiplicities of closed geodesics on the modular surface
证明了模曲面上闭测地线重数的相关函数的显式乘积表示。这使得严格的一部分的调查Bogomolny，Leyvraz和施密特的相关性的特征值的拉普拉斯的模块化表面。这个结果也可以被看作是Hardy-Littlewood孪生素数猜想的一个（严格证明的）类似物。
We study the irreducible constituents of the reduction modulo p of irreducible algebraic representations V of Res_{K/Q_p} GL_2 for K a finite extension of Q_p. We show that asymptotically, the multiplicity of each constituent depends only on the dimension of V and the central character of its reduction modulo p. As an application, we compute the asymptotic value of multiplicities that are the object of the Breuil-Mézard conjecture.
Multiplicities of Galois representations of weight one (with an appendix by Niko Naumann)
在这篇文章中，我们考虑模p的模块化伽罗瓦表示，这是unramified在p，使Frobenius元素在p的行为通过一个标量矩阵。主要结果表明，任何这样的表示的重数都大于1。
Sharp bounds for multiplicities of Bianchi modular forms
本文证明了非全实数域上 $\mathrm{SL}_2$ 的固定级、增长权上同调自守形式空间维数的一次节省界，建立了尖 Bianchi 模形式增长的尖锐界，并通过代数微局部化将问题转化为完备泛包络代数上的表示论问题予以解决。
The mean square of weighted multiplicities function
本文对有限面积黎曼曲面上给定长度的闭测地线定义了一个加权重数函数。这些加权的多重性自然出现在塞尔伯格迹公式，特别是他们的均方起着重要的作用，在研究统计的特征值的表面上的拉普拉斯算子。在模域的情况下，E. Bogomolny，F. Leyvraz和C.施米特给出了一个均方公式，M.彼得本文计算了有理四元数代数单位群的同余子群所对应的曲面的加权重数的均方，特别是模群的同余子群的加权重数的均方.值得注意的是，结果是模域均方的有理倍数。
Given a family $\mathcal{F}=\{A_1,\dots,A_s\}$ of subsets of $\mathbb{Z}_n$, define $Δ\mathcal{F}$ to be the multiset of all (cyclic) distances dist$(x,y)$, where $\{x,y\} \subset A_i$, $x \neq y$, for some $i=1,\dots,s$. Taking inspiration from a Euclidean distance problem of Erdős, we say that $\mathcal{F}$ is Erdős-deep if the multiplicities of distances that occur in $Δ\mathcal{F}$ are precisely $1,2,\dots,k-1$ for some integer $k$. In the case $s=1$, it is known that a modular arithmetic progression in $\mathbb{Z}_n$ achieves this property (under mild conditions); conversely, APs are the …
A Mean Value Theorem for Closed Geodesics on Congruence Surfaces
本文对有限面积黎曼曲面上给定长度的闭测地线定义了一个加权重数函数。这些加权的多重性自然出现在塞尔伯格迹公式，特别是他们的均方起着重要的作用，在研究统计的特征值的表面上的拉普拉斯算子。在模域的情况下，E. Bogomolny，F. Leyvraz和C.施米特给出了一个均方公式，M.彼得本文计算了有理四元数代数单位群的同余子群所对应的曲面的加权重数的均方，特别是模群的同余子群的加权重数的均方.值得注意的是，结果是模域均方的有理倍数。
Bogomolny-Leyvraz-Schmit (1996) and Peter (2002) proposed an asymptotic formula for the correlation of the multiplicities in length spectrum on the modular surface, and Lukianov (2007) extended its asymptotic formula to the Riemann surfaces derived from the congruence subgroup $\Gam_0(n)$ and the quaternion type co-compact arithmetic groups. The coefficients of the leading terms in these asymptotic formulas are described in terms of Euler products over prime numbers, and they appear in eigenvalue statistic formulas found by Rudnick (2005) and Lukianov (2007) for the Laplace-Beltrami operators …
A general proposition is proved relating multiplicities (of restriction of a representation of a group to a subgroup) under basechange, and used to calculate some multiplicities for cuspidal representations which become principal series representations under basechange for which multiplicities can be calculated by geometric methods.
Given a continuous ordinary Galois representation $\barρ:G_{\mathbb{Q}_p}\rightarrow\mathrm{GL}_n(\overline{\mathbb{F}}_p)$, Breuil and Herzig constructed an admissible smooth $\overline{\mathbb{F}}_p$-representation $Π(\barρ)^{\mathrm{ord}}$ of $\mathrm{GL}_n(\mathbb{Q}_p)$ and showed that it occurs in certain globally defined mod $p$ cohomology spaces. By applying Taylor-Wiles patching to spaces of ordinary automorphic representations we prove that the indecomposable pieces of $Π(\barρ)^{\mathrm{ord}}$ each occur with the same multiplicity at a well-chosen tame level.
Multiplicities of Representations in Algebraic Families
本文研究一般代数族中表示的重数，证明了同调重数的上半连续性和Euler-Poincare数的局部常值性，推广了Aizenbud-Sayag关于无挠扭曲族的主要结果。
Bounds for Multiplicities of Unitary Representations of Cohomological Type in Spaces of Cusp Forms
设$\Goo$是具有酉对偶$\Ghat$的半单实李群.本文的目的是给出上同调型表示$π\in \Ghat$在$\Goo$上的某些尖点形式空间中的多重性的新上界.
Let $K/F$ be a finite Galois extension of number fields and $σ$ be an absolutely irreducible, self-dual representation of $\mathrm{Gal}(K/F)$. Let $p$ be an odd prime and consider two elliptic curves $E_1, E_2$ with good, ordinary reduction at primes above $p$ and equivalent mod-$p$ Galois representations. In this article, we study the variation of the parity of the multiplicities of $σ$ in the representation space associated to the $p^\infty$-Selmer group of $E_i$ over $K$. We also compare the root numbers for the twist of $E_i/F$ by $σ$ and show that the $p$-parity conjecture holds for the t…
We prove a new upper bound for the dimension of the space of cohomological automorphic forms of fixed level and growing parallel weight on $\mathrm{GL}_2$ over a number field which is not totally real, improving the one obtained by Marshall. The main tool of the proof is the mod $p$ representation theory of $\mathrm{GL}_2(\mathbb{Q}_p)$ as started by Barthel-Livne and Breuil, and developed by Paskunas.
We study generating functions for Lusztig's $t$-analog of weight multiplicities associated to integrable highest weight representations of the simplest affine Lie algebra $A_1^{(1)}$. At $t=1$, these reduce to the {\em string functions} of $A_1^{(1)}$, which were shown by Kac and Peterson to be related to certain Hecke indefinite modular forms. Using their methods, we obtain a description of the general $t$-string function; we show that its values can be realized as radial averages of a certain extension of the Hecke indefinite modular form.
In this paper, we compute basis elements of certain spaces of weight 0 weakly holomorphic modular forms and consider the integrality of Fourier coefficients of the modular forms. We use the results to construct automorphic correction of the rank 2 hyperbolic Kac-Moody algebras H(a), a=4,5,6, through Hilbert modular forms explicitly given by Borcherds lifts of the weakly holomorphic modular forms. We also compute asymptotic of the Fourier coefficients as they are related to root multiplicities of the rank 2 hyperbolic Kac-Moody algebras. This work is a continuation of the previous paper, where …
We compute presentations of crystalline framed deformation rings of a two dimensional representation $\barρ$ of the absolute Galois group of $\mathbb{Q}_p$, when $\barρ$ has scalar semi-simplification, the Hodge-Tate weights are small and $p>2$. In the non-trivial cases, we show that the special fibre is geometrically irreducible, generically reduced and the Hilbert-Samuel multiplicity is either $1$, $2$ or $4$ depending on $\barρ$. We show that in the last two cases the deformation ring is not Cohen-Macaulay.
A Conjecture for Multiplicities of Strongly Tempered Spherical Varieties
本文针对无N型根的强tempered球面簇的tempered表示重数提出猜想，推广了Gan-Gross-Prasad与Wan-Zhang的ε二分猜想。
Representation of modular invariant function by generators of a modular function field
给出了用模函数域的生成元表示模不变函数的方法。
The Breuil-Mézard conjecture for potentially Barsotti-Tate representations
本文证明了对于p>2，绝对Galois群G_K的二维潜在Barsotti-Tate表示的Breuil-Mézard猜想，并应用于Serre猜想的权部分，包括Buzzard-Diamond-Jarvis猜想。
In this paper, we propose an improved algorithm for computing mod $\ell$ Galois representations associated to eigenforms of arbitrary levels prime to $\ell$. Precisely, we present a method to find the Jacobians of modular curves which have the smallest possible dimensions in a well-defined sense to realize the modular Galois representations. This algorithm also works without the assumption $\ell \ge k-1$, where $k$ are the weights of the eigenforms.
We establish an isomorphism between certain complex-valued and vector-valued modular form spaces of half-integral weight, generalizing the well-known isomorphism between modular forms for $Γ_0(4)$ with Kohnen's plus condition and modular forms for the Weil representation associated to the discriminant form for the lattice with Gram matrix $(2)$. With such an isomorphism, we prove the Zagier duality and write down the Borcherds lifts explicitly.
We prove the indecomposability of Galois representation restricted to the p-decomposition group attached to a non CM nearly p-ordinary weight two Hilbert modular form under mild conditions.
We prove the existence of Euler systems for adjoint modular Galois representations using deformations of Galois representations coming from Hilbert modular forms and relate them to $p$-adic $L$-functions under a conjectural formula for the Fitting ideals of some equivariant congruence modules for abelian base change.
In this note, we consider discriminant forms that are given by the norm form of real quadratic fields and their induced Weil representations. We prove that there exists an isomorphism between the space of vector-valued modular forms for the Weil representations that are invariant under the action of the automorphism group and the space of scalar-valued modular forms that satisfy some epsilon-condition, with which we translate Borcherds's theorem of obstructions to scalar-valued modular forms. In the end, we consider an example in the case of level 12.
The Weil representation and Hecke operators for vector valued modular forms
我们定义了Hecke算子的向量值模形式变换与Weil表示相关联的判别形式。我们描述了Hecke算子的相应代数的性质，并研究了它在模形式上的作用。
On the Modularity of Wildly Ramified Galois Representations
证明了在5处分歧的具有不可解射影象的奇复二维伽罗瓦表示的无限族是模的，从而验证了Artin猜想的一个新的例子.这样一个家庭包含原来的例子研究布勒。在这个过程中，我们通过研究$Λ$ -adic Hecke代数，证明了无穷族剩余模伽罗瓦表示是模的。
We construct plane models of the modular curve $X_H(\ell)$, and use their explicit equations to compute Galois representations associated to modular forms for values of $\ell$ that are significantly higher than in prior works.
In this work we provide a level raising theorem for $\mod λ^n$ modular Galois representations. It allows one to see such a Galois representation that is modular of level $N$, weight 2 and trivial Nebentypus as one that is modular of level $Np$, for a prime $p$ coprime to $N$, when a certain local condition at $p$ is satisfied. It is a generalization of a result of Ribet concerning $\mod \ell$ Galois representations.
Modular representations of p-adic groups
我将调查一些结果的理论模表示的还原$p$ -进群，在积极的特点$\ell \neq p$和$\ell=p$。
This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. Algorithms are described that compute the maximum integer modulo which two monic coprime integral polynomials have a root in common in a sense that is defined. These techniques are applied to the study of congruences of modular forms and modular Galois representations modulo prime powers. Finally, some computational results with implications on the (non-)liftability of modular forms modulo prime powers and possible generalisations of level raising are presented.
This paper uses previous results of the authors on vector-valued modular forms to study certain non-congruence modular forms. We prove that these forms have unbounded denominators, and in certain cases we verify congruences of Atkin--Swinnerton-Dyer type satisfied by the Fourier coefficients of these forms. Our results rest on group-theoretic facts about the modular group, a detailed study of its imprimitive three-dimensional representations, and the theory of their associated vector-valued modular forms. For the proof of the congruences we also make essential use of a result of Katz.
We show that the image of the adelic Galois representation attached to a non-CM modular form is open in the adelic points of a suitable algebraic group. We also show a similar result for the adelic Galois representation attached to a finite set of modular forms.
In this paper we study rank two symmetric hyperbolic Kac-Moody algebras H(a) and their automorphic correction in terms of Hilbert modular forms. We associate a family of H(a)'s to the quadratic field Q(p) for each odd prime p and show that there exists a chain of embeddings in each family. When p = 5, 13, 17, we show that the first H(a) in each family, i.e. H(3), H(11), H(66), is contained in a generalized Kac-Moody superalgebra whose denominator function is a Hilbert modular form given by a Borcherds product. Hence, our results provide automorphic correction for those H(a)'s. We also compute …
Given a number field we construct a canonical Galois gerbe over it and show that its cohomology provides a bridge between the refined local endoscopy introduced in arXiv:1304.3292 and classical global endoscopy. As particular applications, we express the canonical adelic transfer factor that governs the stabilization of the Arthur-Selberg trace formula as a product of normalized local transfer factors, we give an explicit construction of the pairing between an adelic L-packet and the corresponding S-group (based on the conjectural pairings in the local setting) that is the essential ingredient…
Multiplicities of eigenvalues and quadratic representations of integers
研究矩形或环面的拉普拉斯算子非零本征值的重数集M，证明当矩形边长比r²为有理数时M为正整数集，环面生成向量长度比r和夹角θ时，若r cosθ和r²均为有理数，则M为2N、4N或6N的集合。
Patching and multiplicities of p-adic eigenforms
本文利用与局部Galois表示三角簇相关的拼合技术及凝聚层比较，证明了在3变量定酉群上存在关联于crystalline但临界Galois表示的非经典p-adic自守特征形式。
If $ρ$ denotes a finite dimensional complex representation of $\textbf{SL}_2(\textbf{Z})$, then it is known that the module $M(ρ)$ of vector valued modular forms for $ρ$ is free and of finite rank over the ring $M$ of scalar modular forms of level one. This paper initiates a general study of the structure of $M(ρ)$. Among our results are absolute upper and lower bounds, depending only on the dimension of $ρ$, on the weights of generators for $M(ρ)$, as well as upper bounds on the multiplicities of weights of generators of $M(ρ)$. We provide evidence, both computational and theoretical, that a …
Monstrous moonshine relates distinguished modular functions to the representation theory of the monster. The celebrated observations that 196884=1+196883 and 21493760=1+196883+21296876, etc., illustrate the case of the modular function j-744, whose coefficients turn out to be sums of the dimensions of the 194 irreducible representations of the monster. Such formulas are dictated by the structure of the graded monstrous moonshine modules. Recent works in moonshine suggest deep relations between number theory and physics. Number theoretic Kloosterman sums have reappeared in quantum gravity, and …
We prove an integral formula computing multiplicities of square-integrable representations relative to Galois pairs over $p$-adic fields and we apply this formula to verify two consequences of a conjecture of Dipendra Prasad. One concerns the exact computation of the multiplicity of the Steinberg representation and the other the invariance of multiplicities by transfer among inner forms.
A finite Linear Dependence of Discrete Series Multiplicities
本文证明离散系列重数可由有限个重数决定，从而精细化强重数一结果，并应用于尖点形式空间维数的相等性判定。
We study a family of Siegel modular forms that are constructed using Jacobi forms that arise in Umbral moonshine. All but one of them arise as the Weyl-Kac-Borcherds denominator formula of some Borcherds-Kac-Moody (BKM) Lie superalgebras. These Lie superalgebras have a $\widehat{sl(2)}$ subalgebra which we use to study the Siegel modular forms. We show that the expansion of the Umbral Jacobi forms in terms of $\widehat{sl(2)}$ characters leads to vector-valued modular forms. We obtain closed formulae for these vector-valued modular forms. In the Lie algebraic context, the Fourier coefficients …
This paper contains two results concerning the spectral decomposition, in a broad sense, of the space of nondegenerate Hermitian matrices over a local field of characteristic zero. The first is an explicit Plancherel decomposition of the associated $L^2$ space thus confirming a conjecture of Sakellaridis-Venkatesh in this particular case. The second is a formula for the multiplicities of generic representations in the $p$-adic case that extends previous work of Feigon-Lapid-Offen. Both results are stated in terms of Arthur-Clozel's quadratic local base-change and the proofs are based on local …
The goal of this paper is to explain certain experimentally observed properties of the (cuspidal) spectrum and its associated automorphic forms (Maass waveforms) on the congruence subgroup $Γ_{0}(9)$. The first property is that the spectrum possesses multiplicities in the so-called new part, where it was previously believed to be simple. The second property is that the spectrum does not contain any "genuinely new" eigenvalues, in the sense that all eigenvalues of $Γ_{0}(9)$ appear in the spectrum of some congruence subgroup of lower level. The main theorem in this paper gives a precise decompo…
The modular properties of characters of representations of a family of W-superalgebras extending the affine Lie superalgebra of gl(1|1) are considered. Modules fall into two classes, the generic type and the non-generic one. Characters of non-generic modules are expressed in terms of higher-level Appell-Lerch sums. We compute the modular transformations of characters and interpret the Mordell integral as an integral over characters of generic representations. The \C-span of a finite number of non-generic characters together with an uncountable set of characters of the generic type combine into…
The low energy expansion of Type II superstring amplitudes at genus one is organized in terms of modular graph functions associated with Feynman graphs of a conformal scalar field on the torus. In earlier work, surprising identities between two-loop graphs at all weights, and between higher-loop graphs of weights four and five were constructed. In the present paper, these results are generalized in two complementary directions. First, all identities at weight six and all dihedral identities at weight seven are obtained and proven. Whenever the Laurent polynomial at the cusp is available, the f…
We describe the effect of the differential operators defined by Boecherer-Nagaoka, Flander-Ghitza and Yamauchi on the Galois representations (conjecturally) attached to Siegel modular eigenforms.
Let G be a locally analytic group. We use deformation theory and Emerton's 'augmented' Iwasawa modules to study the possible liftings of a given absolutely irreducible smooth modular representation of G to a unitary Banach space representation. Final version. Some minor items changed. Appears in Israel J. Math.
The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series in the space of level $1$ automorphic forms of a split classical group $G$ over $\mathbb{Z}$, and provide numerical applications in absolute rank $\leq 8$. Second, we prove a classification result for the level one cuspidal algebraic automorphic representations of ${\rm GL}_n$ over $\mathbb{Q}$ ($n$ arbitrary) whose motivic weight is $\leq 24$. In both cases, a key ingredient is a classical method based on the Weil explicit formula, which allows to disprove the existenc…
Conformal characters and the modular representation
给出了一个一般的程序，以确定，给定任何适当的表示的模群，所有可能的有理共形场论的特征，其相关的模表示是给定的。相关的思想和方法的两个非平凡的例子：杨-李和伊辛模型说明。
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