Sonja Žunar
We prove a vector-valued version of Muić's integral non-vanishing criterion for Poincaré series on the upper half-plane $ \mathcal H $. Moreover, we give an accompanying result on the construction of vector-valued modular forms in the form of Poincaré series. As an application of these results, we construct and study the non-vanishing of the classical and elliptic vector-valued Poincaré series.
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We prove that under suitable conditions, the Jacobi Poincaré series of exponential type of integer weight and matrix index does not vanish identically. For classical Jacobi forms, we construct a basis consisting of the "first" few Poincaré series and also give conditions both dependent and independent of the weight, which ensures non-vanishing of classical Jacobi Poincaré series. Equality of certain Kloosterman-type sums is proved. Also, a result on the non-vanishing of Jacobi Poincaré series is obtained when an odd prime divides the index.
We prove some non-vanishing results of Hilbert Poincaré series. We derive these results, by showing that the Fourier coefficients of Hilbert Poincaré series satisfy some nice orthogonality relations for sufficiently large weight as well as for sufficiently large level. To prove later results, we generalize a method of E. Kowalski et. al.
Non-vanishing of Poincaré Series on Average
本文研究了在平均意义下庞加莱级数非零性的条件，证明了当权重或索引在特定范围内变化时，适合参数的庞加莱级数几乎不为零。
We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré series of a given level $Γ_0(N)$ and weight $k\ge 2$ coincides with the field generated by the single-valued periods of a certain motive attached to $Γ_0(N)$. This clarifies the arithmetic nature of such Fourier coefficients and generalises previous formulas of Brown and Acres--Broadhurst giving explicit series expansions for the single-valued periods of some modular forms. Our proof is based on Bringmann--Ono's construction of harmonic lifts of Poincaré series.
We prove recursive formulas for the Taylor coefficients of cusp forms, such as Ramanujan's Delta function, at points in the upper half-plane. This allows us to show the non-vanishing of all Taylor coefficients of Delta at CM points of small discriminant as well as the non-vanishing of certain Poincaré series. At a "generic" point all Taylor coefficients are shown to be non-zero. Some conjectures on the Taylor coefficients of Delta at CM points are stated.
On the non-vanishing of Hilbert Poincaré series
本文研究 Hilbert Poincaré 级数的非零性，证明小范数条件下其不恒为零，并在数域上建立 Kloosterman 和的 Selberg 恒等式，进而在阿代尔框架中用 Hecke 算子加以解释。
Vanishing of Poincaré series for congruence subgroups
本文研究同余子群 $Γ_0(N)$ 的 Poincaré 级数消失问题，证明在若干权 $k$、指标 $m$ 和级 $N$ 的参数选择下 $P_{k,m,N}$ 非零，并改进 Rankin 与 Mozzochi 的既有结果。
In this paper, we study the $ K $-finite matrix coefficients of integrable representations of the metaplectic cover of $ \mathrm{SL}_2(\mathbb R) $ and give a result on the non-vanishing of their Poincaré series. We do this by adapting the techniques developed for $ \mathrm{SL}_2(\mathbb R) $ by Muić to the case of the metaplectic group.
Non-Vanishing of L-Functions of Vector-Valued Modular Forms
本文证明了全模群上向量值模形式尖点形式空间正交基的L-函数平均值非消没，并在特定条件下存在不消没的基元素。
Given a pair $(Γ,ρ)$ of a Fuchsian group of the first kind, and a unitary representation $ρ$ of $Γ$ of arbitrary rank, the problem of construction of vector-valued Poincaré series of weight 2 is considered. Implications in the theory of parabolic bundles are discussed. When the genus of the group is zero, it is shown how an explicit basis for the space of these functions can be constructed.
On the non-vanishing of Poincaré series on irreducible bounded symmetric domains
本文将Muić积分非零准则推广至向量值情形，研究不可约有界对称域上多项式型庞加莱级数对应的全纯自守形式非零性，并在SU(p,q)群情形下给出应用示例。
Construction and non-vanishing of a family of vector-valued Siegel Poincaré series
本文利用可积反全纯离散系列表示的矩阵系数构造了一族向量值Siegel Poincaré级数，为Siegel尖点形式空间提供生成集，并借助Muić积分判据的变体证明了这些级数的非消失性。
Results on the Non-Vanishing of Derivatives of L-Functions of Vector-Valued Modular Forms
本文证明了向量值模形式L函数导数平均值的非零性，并推广至Kohnen正空间与Jacobi形式。
We prove a strengthening of Muić's integral non-vanishing criterion for Poincaré series on unimodular locally compact Hausdorff groups and use it to prove a result on non-vanishing of L-functions associated to cusp forms of half-integral weight.
Let $ Γ$ be a congruence subgroup of $ \mathrm{Sp}_{2n}(\mathbb Z) $. Using Poincaré series of $ K $-finite matrix coefficients of integrable discrete series representations of $ \mathrm{Sp}_{2n}(\mathbb R) $, we construct a spanning set for the space $ S_m(Γ) $ of Siegel cusp forms of weight $ m\in\mathbb Z_{>2n} $. We prove the non-vanishing of certain elements of this spanning set using Muić's integral non-vanishing criterion for Poincaré series on locally compact Hausdorff groups. Moreover, using the representation theory of $ \mathrm{Sp}_{2n}(\mathbb R) $, we study the Petersson inner pro…
In this paper we prove that the $\mathcal{E}^\dagger_K$-valued cohomology, introduced in [9] is finite dimensional for smooth curves over Laurent series fields $k((t))$ in positive characteristic, and forms an $\mathcal{E}^\dagger_K$-lattice inside `classical' $\mathcal{E}_K$-valued rigid cohomology. We do so by proving a suitable version of the p-adic local monodromy theory over $\mathcal{E}^\dagger_K$, and then using an étale pushforward for smooth curves to reduce to the case of $\mathbb{A}^1$. We then introduce $\mathcal{E}^\dagger_K$-valued cohomology with compact supports, and again prov…
A vanishing theorem for vector-valued Siegel automorphic forms in characteristic $p$
本文证明了特征 p 下向量值 Siegel 自守形式在权处于显式轨迹之外时为零，从而验证了与 Goldring 提出的 Hodge 型 Shimura 簇一般猜想的一个特例。
For a rational valued periodic function, we associate a Dirichlet series and provide a new necessary and sufficient condition for the vanishing of this Dirichlet series specialized at positive integers. This question was initiated by Chowla, and carried out by Okada for a particular infinite sum. Our approach relies on the decomposition of the Dirichlet characters in terms of primitive characters. Using this, we find some new family of natural numbers for which a conjecture of Erdös holds. We also characterize rational valued periodic functions for which the associated Dirichlet series vanishe…
We address the question of non-vanishing of $L(1,f)$ where $f$ is an algebraic-valued, periodic arithmetical function. We do this by characterizing algebraic-valued, periodic functions $f$ for which $L(1,f)=0$. The case of odd functions was resolved by Baker, Birch and Wirsing in 1973. We apply a result of Bass to obtain a characterization for the even functions. We also describe a theorem of the first two authors which says that it is enough to consider only the even and the odd functions in order to obtain a complete characterization.
We give an explicit construction of vector-valued Yoshida lifts and derive a formula of the Bessel periods of Yoshida lifts, by which we prove the non-vanishing modulo a prime of Yoshida lifts attached to a pair of elliptic modular newforms. As a consequence, we obtain a new proof of the non-vanishing of Yoshida lifts.
An algebraic classification is given for spaces of holomorphic vector-valued modular forms of arbitrary real weight and multiplier system, associated to irreducible, T-unitarizable representations of the full modular group, of dimension less than six. For representations of dimension less than four, it is shown that the associated space of vector-valued modular forms is a cyclic module over a certain skew polynomial ring of differential operators. For dimensions four and five, a complete list of possible Hilbert-Poincare series is given, using the fact that the space of vector-valued modular f…
We discuss the problem of the vanishing of Poincaré series. This problem is known to be related to the existence of weakly holomorphic forms with prescribed principal part. The obstruction to the existence is related to the pseudomodularity of Ramanujan's mock theta functions. We embed the space of weakly holomorphic modular forms into the larger space of harmonic weak Maass forms. From this perspective we discuss the linear relations between Poincaré series and the connection to Ramanujan's mock theta functions.
We consider logarithmic vector- and matrix-valued modular forms of integral weight $k$ associated with a $p$-dimensional representation $ρ: SL_2(\mathbb{Z}) \to GL_p(\mathbb{C})$ of the modular group, subject only to the condition that $ρ(T)$ has eigenvalues of absolute value 1. The main result is the construction of meromorphic matrix-valued Poincaré series associated to $ρ$ for all large enough weights. The component functions are logarithmic $q$-series, i.e., finite sums of products of $q$-series and powers of $\log q$. We derive several consequences, in particular we show that the space $\…
$L^\infty$-sizes of the spaces Siegel cusp forms of degree $n$ via Poincaré series
本文通过波尔查级数研究了不同重量和层次下的Siegel尖点形式空间的$L^\infty$大小，证明了其大小与$k^{3n(n+1)/4}$相关，方法基于Bergman核的傅里叶展开。
We consider a certain linear combination $S(\mathbf{s},\mathbf{y};I;Δ)$ of zeta-functions of root systems, where $Δ$ is a root system of rank $r$ and $I\subset\{1,2,\ldots,r\}$. Showing two different expressions of $S(\mathbf{s},\mathbf{y};I;Δ)$, we find that a certain signed sum of zeta-functions of root systems is equal to a sum involving Bernoulli functions of root systems. This identity gives a non-trivial functional relation among zeta-functions of root systems, if the signed sum does not identically vanish. This is a genralization of the authors' previous result proved in \cite{KMTLondon…
In this work we use the Rankin-Selberg method to obtain results on the analytic properties of the standard $L$-function attached to vector valued Siegel modular forms. In particular we provide a detailed description of its possible poles and obtain a non-vanishing result of the twisted $L$-function beyond the usual range of absolute convergence. We remark that these results were known in this generality only in the case of scalar weight Siegel modular forms. As an interesting by-product of our work we establish the cuspidality of some theta series.
Non-Archimedean Poincaré series and geodesics on the Bruhat-Tits tree
本文在Drinfeld半平面上构造非阿基米德Poincaré级数，证明其收敛性与非零性，并用于生成模形式空间，推广了Kurihara的方法。
We prove that products of at most two vector valued Eisenstein series that originate in level 1 span all spaces of cusp forms for congruence subgroups. This can be viewed as an analogue in the level aspect to a result that goes back to Rankin, and Kohnen and Zagier, which focuses on the weight aspect. The main feature of the proof are vector valued Hecke operators. We recover several classical constructions from them, including classical Hecke operators, Atkin-Lehner involutions, and oldforms. As a corollary to our main theorem, we obtain a vanishing condition for modular forms reminiscent of …
Let $V$ be a representation of the modular group $Γ$ of dimension $p$. We show that the $\mathbb{Z}$-graded space $\mathcal{H}(V)$ of holomorphic vector-valued modular forms associated to $V$ is a free module of rank $p$ over the algebra $\mathcal{M}$ of classical holomorphic modular forms. We study the nature of $\mathcal{H}$ considered as a functor from $Γ$-modules to graded $\mathcal{M}$-lattices and give some applications, including the calculation of the Hilbert-Poincaré of $\mathcal{H}(V)$ in some cases.
Vector-valued orthogonal modular forms
本专著系统建立符号为(2,n)正交群的向量值模形式理论，重点研究第二个Hodge丛的性质与核心作用，并由此推导出小权向量值模形式的最优消失定理。
We show how one can use the representation theory of ternary quartics to construct all vector-valued Siegel modular forms and Teichmüller modular forms of degree 3. The relation between the order of vanishing of a concomitant on the locus of double conics and the order of vanishing of the corresponding modular form on the hyperelliptic locus plays an important role. We also determine the connection between Teichmüller cusp forms on \overline{M}_g and the middle cohomology of symplectic local systems on M_g. In genus 3, we make this explicit in a large number of cases.
Orthogonality relations for Poincaré series
本文证明了具有有限中心的连通半单 Lie 群上由 Poincaré 级数给出的尖点自守形式的内积公式，并以此推广了上半平面尖点形式的相关结果。
We use Poincaré series of $ K $-finite matrix coefficients of genuine integrable representations of the metaplectic cover of $ \mathrm{SL}_2(\mathbb R) $ to construct a spanning set for the space of cusp forms $ S_m(Γ,χ) $, where $ Γ$ is a discrete subgroup of finite covolume in the metaplectic cover of $ \mathrm{SL}_2(\mathbb R) $, $ χ$ is a character of $ Γ$ of finite order, and $ m\in\frac52+\mathbb Z_{\geq0} $. We give a result on the non-vanishing of the constructed cusp forms and compute their Petersson inner product with any $ f\in S_m(Γ,χ) $. Using this last result, we construct a Poin…
Igusa's map on modular forms vanishing on the hyperelliptic locus
本文将Igusa映射$ρ$扩展至Siegel上半空间超椭圆轨迹上消失的模形式，利用一般Thomae公式计算其最低非零导数作为向量值模形式，并以亏格$3$和$4$的例子进行说明。
The first half of this dissertation reviews the basic notion of vector-valued modular forms and its connection to differential equations. The main purpose of the dissertation is to classify spaces of vector-valued modular forms associated to irreducible, T-unitarizable representations of the full modular group, of dimension less than six. Given such a representation, it is shown that the associated graded complex linear space of vector-valued modular forms is a free module over the ring of integral weight modular forms for the full modular group, whose rank is equal to the dimension of the giv…
We continue the analysis of modular invariant functions, subject to inhomogeneous Laplace eigenvalue equations, that were determined in terms of Poincaré series in a companion paper. The source term of the Laplace equation is a product of (derivatives of) two non-holomorphic Eisenstein series whence the modular invariants are assigned depth two. These modular invariant functions can sometimes be expressed in terms of single-valued iterated integrals of holomorphic Eisenstein series as they appear in generating series of modular graph forms. We show that the set of iterated integrals of Eisenst…
We use covariants of binary sextics to describe the structure of modules of scalar-valued or vector-valued Siegel modular forms of degree 2 with character, over the ring of scalar-valued Siegel modular forms of even weight. For a modular form defined by a covariant we express the order of vanishing along the locus of products of elliptic curves in terms of the covariant.
While vector-valued automorphic forms can be defined for an arbitrary Fuchsian group $Γ$ and an arbitrary representation $R$ of $Γ$ in GL$(n,{\mathbb C})$, their existence has been established in the literature only when restrictions are imposed on both $Γ$ and $R$. In this paper, we prove the existence of $n$ linearly independent vector-valued automorphic forms for any Fuchsian group $Γ$ and any $n$-dimensional complex representation $R$ of $Γ$. To this end, we realize these automorphic forms as global sections of a special rank $n$ vector bundle built using solutions to the Riemann-Hilbert p…
The $D^6 R^4$ interaction as a Poincaré series, and a related shifted convolution sum
本文通过庞加莱级数构造弦论 $D^6 R^4$ 微分方程的自守解，并利用二重狄利克雷级数证明傅里叶展开中特定项的消失，证实了相关猜想。
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