Bernhard Heim, Atsushi Murase
Let q be an integral quadratic form of signature (2,m+2). We will show that the Siegel theta functions attached to q satisfies certain symmetries. As an application, we prove the symmetries for automorphic forms on the orthogonal group of q closely related to Heegener divisors (Borcherds lifts and automorphic Green functions).
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A truncated Siegel-Weil formula and Borcherds forms
本文利用Gan-Qiu-Takeda的正则化Siegel-Weil公式，推导出theta函数及Borcherds形式对数在截断模曲线上积分的表达式，明确了其渐近行为，并补充了Kudla在极限情形下的相关工作。
We classify the holomorphic Borcherds products of singular weight for all simple lattices of signature $(2,n)$ with $n \geq 3$. In addition to the automorphic products of singular weight for the simple lattices of square free level found by Dittmann, Hagemeier and the second author, we obtain several automorphic products of singular weight $1/2$ for simple lattices of signature $(2,3)$. We interpret them as Siegel modular forms of genus $2$ and explicitly describe them in terms of the ten even theta constants. In order to rule out further holomorphic Borcherds products of singular weight, we d…
Automorphic products that are singular modulo primes
本文利用O(n, 2)上的Rankin--Cohen括号，证明了反射Borcherds乘积的傅里叶系数通常满足模某些素数的同余关系。
Perfectoid overconvergent Siegel modular forms and the overconvergent Eichler--Shimura morphism
本文通过推广完美化方法构建了Siegel模形式的过收敛自守sheaf，其全局截面即为过收敛Siegel模形式，并建立了适用于Siegel模形式的过收敛Eichler—Shimura映射。
We relate the study of Landau-Siegel zeros to the ranks of Jacobians $J_0(q)$ of modular curves for large primes $q$. By a conjecture of Brumer-Murty, the rank should be equal to half of the dimension. Equivalently, almost all newforms of weight two and level $q$ have analytic rank $\leq 1$. We show that either Landau-Siegel zeros do not exist, or that almost all such newforms have analytic rank $\leq 2$. In particular, almost all odd newforms have analytic rank equal to one. Additionally, for a sparse set of primes $q$ we show the rank of $J_0(q)$ is asymptotically equal to the rank predicted…
Cheng constructed product expansions from twists of elliptic genera of symmetric powers of K3 surfaces that are related to M_24 moonshine. We study which of them are Siegel modular forms. If the predicted level is non-composite, they are modular, and their powers can be represented as products of rescaled Borcherds products.
The theta-block conjecture proposed by Gritsenko--Poor--Yuen in 2013 characterizes Siegel paramodular forms which are simultaneously Borcherds products and additive Jacobi lifts. In this paper, we prove this conjecture for two new infinite series of theta blocks of weights 2 and 3. The proof is based on Scheithauer's classification of reflective modular forms of singular weight.
We give a formula that relates the difference of the j-invariants with the Borcherds Phi-function, an automorphic form on the period domain for Enriques surfaces characterizing the discriminant divisor.
We provide a construction of the multiplicative Borcherds lift for unitary groups U(1,m), which takes weakly holomorphic elliptic modular forms and lifts them to meromorphic automorphic forms having infinite product expansions and taking their zeros and poles along Heegner divisors. The result is obtained through the transfer of Borcherds' theory to unitary groups via a suitable embedding of U(1,m) into O(2,2m). Further, we examine the values taken by the Borcherds products around the cusps of the symmetric domain of the unitary group. Finally, as an application of the lifting, we derive a mod…
In this paper, we consider automorphic forms on $\mathrm{Sp}_4(\mathbb{A}_\mathbb{Q})$ which generate large discrete series representations of $\mathrm{Sp}_4(\mathbb{R})$ as $(\mathfrak{sp}_4(\mathbb{R}),K_\infty)$-modules. We determine the cuspidal components and the structure of the space of such automorphic forms.
We continue our study of Yoshida's lifting, which associates to a pair of automorphic forms on the adelic multiplicative group of a quaternion algebra a Siegel modular form of degree 2. We consider here the case that the automorphic forms on the quaternion algebra correspond to modular forms of arbitrary even weights and square free levels; in particular we obtain a construction of Siegel modular forms of weight 3 attached to a pair of elliptic modular forms of weights 2 and 4.
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。
Constructing vector-valued automorphic forms on unitary groups
本文引入一种从标量值自守形式构造酉群上向量值自守形式的方法，并给出显式例子，该方法使用微分算子并克服了Siegel情形中未出现的挑战。
We prove Siegel-Walfisz type theorems (over long and short intervals) for the Fourier coefficients of certain automorphic $L$-functions and Rankin-Selberg $L$-functions over number fields.
In this paper, we decompose the space of nearly holomorphic Hilbert-Siegel automorphic forms as representations of the adele group under certain assumptions. We also give an application for classical holomorphic Hilbert-Siegel modular forms. In particular, we show the surjectivity of the global Siegel operator for certain congruence subgroups with large weights.
We extend Borcherds' singular theta lift in signature $(1,2)$ to harmonic Maass forms of weight $1/2$ whose non-holomorphic part is allowed to be of exponential growth at $i\infty$. We determine the singularities of the lift and compute its Fourier expansion. It turns out that the lift is continuous but not differentiable along certain geodesics in the upper half-plane corresponding to the non-holomorphic principal part of the input. As an application, we obtain a generalization to higher level of the weight $2$ modular integral of Duke, Imamoglu and Tóth. Further, we construct automorphic pro…
The cone conjecture for vector-valued Siegel automorphic forms
本文证明了特征$p$下秩为$n$的Siegel模簇上自守向量丛的有效锥由$G$-zip叠编码，证明了零阶上同调群在zip锥外消失，并在$n=3$时给出显式消失条件，同时定义了平凡型自守形式。
An Explicit Theta Lift to Siegel Paramodular Forms
研究数域实二次扩张下的Theta提升问题，构造从含Γ₀级Hilbert模形式的GL(2,E)尖点自守表示到含Siegel参模形式的GSp(4,L)自守表示的显式映射，给出除野分歧外有限位处生成参模不变向量的局部数据。
Let $π$ be a cuspidal automorphic representation of $\mathrm{GSp}_4(\mathbf{A_Q})$, whose archimedean component is a holomorphic discrete series or limit of discrete series representation. If $π$ is not CAP or endoscopic, then we show that its associated $\ell$-adic Galois representations are irreducible and crystalline for $100\%$ of primes $\ell$. If, moreover, $π$ is neither an automorphic induction nor a symmetric cube lift, then we show that, for $100\%$ of primes $\ell$, the image of its mod $\ell$ Galois representation contains $\mathrm{Sp}_4(\mathbf{F}_{\ell})$.
Let $F$ be a vector-valued real analytic Siegel cusp eigenform of weight $(2,1)$ with the eigenvalues $-\frac 5{12}$ and 0 for the two generators of the center of the algebra consisting of all $Sp_4(\R)$-invariant differential operators on the Siegel upper half plane of degree 2. Under the assumptions (1) the validity of the transfer of automorphic representations of $GSp_4$ to $GL_4$; (2) the existence of mod $\ell$ Galois representation attached to $F$ and its lift to characteristic zero; (3) rationality of the space consisting of any such $F$; and (4) the integrality of Hecke polynomials of…
Vanishing results for the coherent cohomology of automorphic vector bundles over the Siegel variety in positive characteristic
本文证明了正特征Siegel簇上自守向量丛的凝聚上同调在特定权条件下的消失结果，并提供了Sage算法以显式计算。
Twisted Siegel-Weil formulas for $\mathrm{GL}_2$ over non-Galois quartic CM fields
研究非伽罗瓦四次CM域上\(\mathrm{GL}_2\)的扭曲西格尔 - 韦伊公式，通过联系扭曲西塔积分与赫克积分的土居 - 永野提升，证明雅克比 - 朗兰兹对应基变换是同构，还得出博切尔兹形式扭曲CM值相关结论。
The goal of this paper is to construct infinite dimensional Lie algebras using infinite product identities, and to use these Lie algebras to reduce the generalized moonshine conjecture to a pair of hypotheses about group actions on vertex algebras and Lie algebras. The Lie algebras that we construct conjecturally appear in an orbifold conformal field theory with symmetries given by the monster simple group. We introduce vector-valued modular functions attached to families of modular functions of different levels, and we prove infinite product identities for a distinguished class of automorphic…
We establish a connection between motivic cohomology classes over the Siegel threefold and special values of the degree four $L$-function of some cuspidal automorphic representations of $\mathrm{GSp}(4)$. Our computation relies on our previous work [Le15] and on an integral representation of the $L$-function due to Piatetski-Shapiro.
We show that Landau-Siegel zeros for Dirichlet $L$-functions do not exist or quantum unique ergodicity for $\mathrm{GL}_2$ Hecke-Maass newforms holds with an effective rate of convergence. This follows from a more general result: Landau-Siegel zeros of Dirichlet $L$-functions repel the zeros of all other automorphic $L$-functions from the line $\mathrm{Re}(s)=1$.
We classify the simple even lattices of square free level and signature (2,n) for n > 3. A lattice is called simple if the space of cusp forms of weight 1+n/2 for the dual Weil representation of the lattice is trivial. For a simple lattice every formal principal part obeying obvious conditions is the principal part of a vector valued modular form. Using this, we determine all holomorphic Borcherds products of singular weight (arising from vector valued modular forms with non-negative principal part) for the simple lattices. We construct the corresponding vector valued modular forms by eta prod…
Using the relationship between Siegel cusp forms of degree $2$ and cuspidal automorphic representations of $\mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}})$, we derive some congruences involving dimensions of spaces of Siegel cusp forms of degree $2$ and the class number of $\mathbb{Q}(\sqrt{-p})$. We also obtain some congruences between the $4$-core partition function $c_4(n)$ and dimensions of spaces of Siegel cusp forms of degree $2$.
Let $π$ be the automorphic representation of $\GSp_4(\A)$ generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and $τ$ be an arbitrary cuspidal, automorphic representation of $\GL_2(\A)$. Using Furusawa's integral representation for $\GSp_4\times\GL_2$ combined with a pullback formula involving the unitary group $\GU(3,3)$, we prove that the $L$-functions $L(s,π\timesτ)$ are "nice". The converse theorem of Cogdell and Piatetski-Shapiro then implies that such representations $π$ have a functorial lifting to a cuspidal representation of $\GL_4(\A)$. Combined wit…
We prove a harmonically weighted equidistribution result for the $p$-th Satake parameters of the family of automorphic cuspidal representations of $\operatorname{PGSp}(2n)$ of fixed weight $\mathtt{k}$ and prime-to-$p$ level $N\to \infty$. The main tool is a new asymptotic Petersson formula for $\operatorname{GSp}(2n)$ in the level aspect.
The twisted elliptic genera of a $K3$ surface associated with the conjugacy classes of the Mathieu group $M_{24}$ are known to be weak Jacobi forms of weight $0$. In 2010, Cheng constructed formal infinite products from the twisted elliptic genera and conjectured that they define Siegel modular forms of degree two. In this paper we prove that for each conjugacy class of level $N_g$ the associated product is a meromorphic Borcherds product on the lattice $U(N_g)\oplus U \oplus A_1$ in a strict sense. We also compute the divisors of these products and determine for which conjugacy classes the pr…
A converse theorem for Borcherds products and the injectivity of the Kudla-Millson theta lift
本文证明了无平方因子级且判别群各向异性的格的乘法 Borcherds 提升逆定理，并利用 Kudla-Millson theta 提升的单射性得到满射性，推广了 Bruinier 和 BF1 的结果，同时改进了 Scheithauer 关于反射自守乘积不存在性的定理。
Dimension formulas for Siegel modular forms of level $4$
本文通过结合已知维数公式与 $p=2$ 处局部表示中固定向量空间的维数，推导出级为 $4$ 的若干 Siegel 模形式空间的新维数公式，尤其涵盖 Klingen 同余子群情形。
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 …
Landau-Siegel Zeros of Triple Product L-functions
本文研究三重积L函数的兰道-西格尔零点问题，证明在一般类型情况下无例外零点，并给出非一般类型情况下的零点可能条件。
We construct a p-adic Eisenstein measure with values in the space of vector-weight p-adic automorphic forms on certain unitary groups. This measure allows us to p-adically interpolate special values of certain vector-weight C-infinity automorphic forms, including Eisenstein series, as their weights vary. We also explain how to extend our methods to the case of Siegel modular forms and how to recover Nicholas Katz's p-adic families of Eisenstein series for Hilbert modular forms.
Harder's conjecture and Hermitian automorphic forms
研究Hermitian Klingen-Eisenstein提升与尖形式之间的同余关系，证明同余中的Hermitian尖特征形式是Siegel尖特征形式的Hermitian旋提升，从而得到Harder猜想预言的旋量L-多项式同余。
In this paper we study automorphic correction of the hyperbolic Kac-Moody algebra $E_{10}$, using the Borcherds product for O(10,2) attached to a weakly holomorphic modular form of weight -4 for $SL_2(\mathbb Z)$. We also clarify some aspects of automorphic correction for Lorentzian Kac-Moody algebras and give heuristic reasons for the expectation that every Lorentzian Kac-Moody algebra has an automorphic correction.
In his celebrated 1998 Inventiones paper, Borcherds constructed meromorphic automorphic forms Psi(F) for arithmetic subgroups associated to even integral lattices M of signature (n,2). The input to his construction is a vector valued weakly holomorphic modular form F of weight 1 - n/2, and the resulting Borcherds form has an explicit divisor on the arithmetic quotient X = Gamma_M\ D. Most remarkably, in the neighborhood of each cusp (= rational point boundary component), there is a beautiful product formula for Psi(F), reminiscent of the classical product formula for the Dedekind eta-function.…
The shifted convolution problem for Fourier coefficients of Siegel modular forms of degree $2$
本文提出了一种基于自守诠释的新方法，首次对更高秩群上的移位卷积和给出了非平凡估计，通过估计塞尔曼泊松级数的傅里叶系数实现。
The square-root of Siegel modular forms of CHL Z_N orbifolds of type II compactifications are denominator formulae for some Borcherds-Kac-Moody Lie superalgebras for N=1,2,3,4. We study the decomposition of these Siegel modular forms in terms of characters of two sub-algebras: one is a $\widehat{sl(2)}$ and the second is a Borcherds extension of the $\widehat{sl(2)}$. This is a continuation of our previous work where we studied the case of Siegel modular forms appearing in the context of Umbral moonshine. This situation is more intricate and provides us with a new example (for N=5) that did no…
We construct classes in the middle degree plus one motivic cohomology of the Siegel Shimura variety of almost any dimension. We compute their image by Beilinson's higher regulator in terms of Rankin-Selberg type automorphic integrals. Our construction generalises the one for $\mathrm{GSp}(4)$ and for $\mathrm{GSp}(6)$. For Siegel varieties associated to small genus symplectic groups, we also show how these integrals unfold.
Let F in S_k(Sp(2g, Z)) be a cuspidal Siegel eigenform of genus g with normalized Hecke eigenvalues mu_F(n). Suppose that the associated automorphic representation pi_F is locally tempered everywhere. For each c>0 we consider the set of primes p for which |mu_F(p)| >= c and we provide an explicit upper bound on the density of this set. In the case g=2, we also provide an explicit upper bound on the density of the set of primes p for which mu_F(p) >= c.
The fake monster Lie algebra is determined by the Borcherds function Phi_{12} which is the reflective modular form of the minimal possible weight with respect to O(II_{2,26}). We prove that the first non-zero Fourier-Jacobi coefficient of Phi_{12} in any of 23 Niemeier cusps is equal to the Weyl-Kac denominator function of the affine Lie algebra of the root system of the corresponding Niemeier lattice. This is an automorphic answer (in the case of the fake monster Lie algebra) on the old question of I. Frenkel and A. Feingold (1983) about possible relations between hyperbolic Kac-Moody algebra…
Computation of Fourier coefficients of automorphic forms of type $G_2$
本文介绍了$G_2$型自守形式Fourier系数的计算公式及其在SAGE中的实现，并推导了相关的理论推论。
The Fourier coefficients of the Siegel-Eisenstein series are p-adically continued for all primes p, as meromorphic functions, using the reciprocal of a product of L-functions. A construction of p-adic meromorphic families of such series is given in relation to the geometry of homogeneous spaces. Applications are given to p-adic L-functions, to Siegel's Mass Formula, to p-adic analytic families of automorphic representations. Based on author's talk for the Conference Automorphic Forms and Related Geometry, Assessing the Legacy of I.I. Piatetski-Shapiro
Many important analytic statements about automorphic forms, such as the analytic continuation of certain L-functions, rely on the well-known rapid decay of K-finite cusp forms on Siegel sets. We extend this here to prove a more general decay statement along sets much larger than Siegel sets, and furthermore state and prove the decay for smooth but not necessarily K-finite cusp forms. We also state a general theorem about the convergence of Rankin-Selberg integrals involving unipotent periods, closing a gap in the literature on L-functions. These properties serve as the analytic basis of a new …
An asymptotic expansion for a Lambert series associated to Siegel cusp forms
本文研究Siegel尖点形式相关的Lambert级数，发现其常数项可类似地用Riemann zeta函数的非平凡零点表达，推广了Hafner和Stopple关于Zagier猜想的结果。
We prove an algebraicity property for a certain ratio of Petersson norms associated to a Siegel cusp form of degree 2 (and arbitrary level) whose adelization generates a weak endoscopic lift. As a preparation for this, we explicate various features of the correspondence between scalar valued Siegel cusp forms of degree n and automorphic representations on GSp_{2n}.
We define the space of nearly holomorphic automorphic forms on a connected reductive group $G$ over $\mathbb{Q}$ such that the homogeneous space $G(\mathbb{R})^1/ K_\infty^\circ$ is a Hermitian symmetric space. By Pitale, Saha and Schmidt's study, there are the classification of indecomposable $(\mathfrak{g},K_\infty)$-modules which occur in the space of nearly holomorphic elliptic modular forms and Siegel modular forms of degree $2$. This paper studies global representations of the adele group $G(\mathbb{A}_\mathbb{Q})$ which occur in the space of nearly holomorphic Hilbert modular forms. In …
We derive a Voronoi-type series approximation for the local weighted mean of an arithmetical function that is associated to Dirichlet series satisfying a functional equation with gamma factors. The series is exploited to study the oscillation frequency with a method of Heath-Brown and Tsang [7]. A by-product is another proof for the well-known result of no element in the Selberg class of degree 0 \textless{} d \textless{} 1. Our major applications include the sign-change problem of the coefficients of automorphic L-functions for GL m , which improves significantly some results of Liu and Wu [1…
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…
We undertake a detailed study of the lowest weight modules for the Hermitian symmetric pair (G,K), where G=Sp_4(R) and K is its maximal compact subgroup. In particular, we determine K-types and composition series, and write down explicit differential operators that navigate all the highest weight vectors of such a module starting from the unique lowest-weight vector. By rewriting these operators in classical language, we show that the automorphic forms on G that correspond to the highest weight vectors are exactly those that arise from nearly holomorphic vector-valued Siegel modular forms of d…
Based on Furusawa's theory, we present an integral representation for the L-function L(s,π\times τ), where πis a cuspidal automorphic representation on GSp(4) related to a holomorphic Siegel modular form, and where τis an arbitrary cuspidal automorphic representation on GL(2). As an application, a special value result for this L-function in the spirit of Deligne's conjecture is proved.
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 …
A geometric Jacquet-Langlands correspondence for paramodular Siegel threefolds
本文研究仿模层级西格尔模三重fold的 Picard-Lefschetz 公式，证明权-单值猜想，并借助 Rapoport-Zink 一致化构造 GSp₄ 与定内形式间的几何 Jacquet-Langlands 对应，同时得到整数值结果与降层级定理。
A minimal modularity lifting theorem for Siegel modular forms
该研究在亏格为二的西格尔模形式下，证明极小模性提升定理，剩余表示源于稳定吉田提升。基于\(R = \mathbb{T}\)定理，建立通用极小普通伽罗瓦变形环自由性及希达族唯一性。
The classical Maass lift is a map from holomorphic Jacobi forms to holomorphic scalar-valued Siegel modular forms. Automorphic representation theory predicts a non-holomorphic and vector-valued analogue for Hecke eigenforms. This paper is the first part of a series of papers. In this series of papers, we provide an explicit construction of the non-holomorphic Maass lift that is linear and also applies to non-eigenforms. In this first part, we develop new techniques to study Fourier series expansions of Siegel modular forms, which allow us to construct a Maass lift from harmonic Maass-Jacobi fo…
In this note we study the symmetric powers of strongly modular icosahedral representations $ρ$ of ${\rm Gal} (\bar{F}/F)$, $F$ a number field, and their twisted $L$--functions. We prove that for such $ρ$, there exists a cuspidal automorphic representation $Π= Π_{\infty} \otimes Π_{f}$ of $GL_{6} (\mathbb{A}_{F})$ such that $L (s, {\rm sym}^{5} (ρ)) = L (s, Π_{f})$. One sees that ${\rm sym}^{5} (ρ)$ is twist equivalent to $ρ' \otimes {\rm sym}^{2} (ρ)$ for another modular icosahedral representation $ρ'$, and our theorem is a special case of a cuspidality criterion formulated and proved in this …
We give a Rankin-Selberg integral representation for the Spin (degree eight) $L$-function on $\mathrm{PGSp}_6$. The integral applies to the cuspidal automorphic representations associated to Siegel modular forms. If $π$ corresponds to a level one Siegel modular form $f$ of even weight, and if $f$ has a non-vanishing maximal Fourier coefficient (defined below), then we deduce the functional equation and finiteness of poles of the completed Spin $L$-function $Λ(π,Spin,s)$ of $π$.
Let $p$ be an odd prime, $N$ a square-free odd positive integer prime to $p$, $π$ a $p$-ordinary cohomological irreducible cuspidal automorphic representation of $\mathrm{GSp}_4(\mathbb{A}_\mathbb{Q})$ of principal level $N$ and Iwahori level at $p$. Using a $p$-integral version of Rallis inner product formula and modularity theorems for $\mathrm{GSp}_{4/\mathbb{Q}}$ and $\mathrm{U}_{4/\mathbb{Q}}$, we establish an identity between the $p$-part of the critical value at $1$ of the degree $5$ $L$-function of $π$ twisted by the non-trivial quadratic Dirichlet character $ξ$ associated to the exten…
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