R. W. Bruggeman, R. J. Miatello, M. I. Pacharoni
We give density results for automorphic representations of Hilbert modular groups. In particular, we show that there are infinitely many automorphic representations that have a prescribed discrete series factor at some (but not all) real places.
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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 show that the systems of prime-to-$p$ Hecke eigenvalues arising from automorphic forms$\pmod p$ for a good prime $p$ associated to an algebraic group $G/\mathbb Q$ of Hodge type are the same as those arising from algebraic modular forms$\pmod p$ associated to an inner form of $G$.
We generalize the first part of A. Connes paper (math/9811068) on the zeroes of the Riemann zeta function from a number field $k$ to any simple algebra $M$ over $k$. To a given automorphic representation $π$ of the reductive group $M^\times$ of invertible elements of $M$ we find a Hilbert space $H_π$ and an operator $D_π$ (Polya-Hilbert operator), which is the infinitesimal generator of a canonical flow such that the spectrum of $D_π$ coincides with the purely imaginary zeroes of the function $L(π,\rez{2} +z)$. As a byproduct we get holomorphicity of all automorphic $L$-functions, not only the…
In this paper, we study modular forms on two simply connected groups of type $D_4$ over ${\mathbb Q}$. One group, $G_s$ is a globally split group of type $D_4$, viewed as the group of isotopies of the split rational octonions. The other, $G_c$, is the isotopy group of the rational (non-split) octonions. We study automorphic forms on $G_s$, in analogy to the work of Gross, Gan, and Savin on $G_2$; namely we study automorphic forms whose component at infinity corresponds to a quaternionic discrete series representation. We study automorphic forms on $G_c$ using Gross's formalism of ``algebraic m…
We define two-parameter generalizations of Andrews' $(k+1)$-marked odd Durfee symbols and $2k$th symmetrized odd rank moments, and study the automorphic properties of some of their generating functions. When $k=0$ we obtain families of modular forms and mock modular forms. When $k \geq 1$, we find quasimodular forms and quasimock modular forms.
We consider modular functions (i.e., the Eisenstein series and Hecke-Maass forms) for the group PSL(2,Z). We fix a quadratic number field E. This gives rise to twisted (by a Hecke character of the field E) periods of a modular function along the torus corresponding to E. We prove meromorphic continuation for a Dirichlet series generated by these twisted periods.
We compare modular forms of characteristic $p>0$ (i.e. Drinfeld's modular forms) and automorphic forms. We prove that spaces of these modular forms (which are of characteristic $p$) can be described by function spaces of characteristic zero, close to those of automorphic forms.
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。
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 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 …
The purpose of this semi-expository article is to give another proof of a classical theorem of Shimura on the critical values of the standard L-function attached to a Hilbert modular form. Our proof is along the lines of previous work of Harder and Hida (independently). What is different is an organizational principle based on the period relations proved by Raghuram and Shahidi for periods attached to regular algebraic cuspidal automorphic representations. The point of view taken in this article is that one need only prove an algebraicity theorem for the most interesting L-value, namely, the c…
We prove a sharp bound for the average value of the triple product of modular functions for the Hecke subgroup Γ_0(N). Our result is an extension of the main result in {Bernstein&Reznikov-2004} to a fixed cuspidal representation of the adele group PGL(2,A).
We consider the genus of $20$ classes of unimodular Hermitian lattices of rank $12$ over the Eisenstein integers. This set is the domain for a certain space of algebraic modular forms. We find a basis of Hecke eigenforms, and guess global Arthur parameters for the associated automorphic representations, which recover the computed Hecke eigenvalues. Congruences between Hecke eigenspaces, combined with the assumed parameters, recover known congruences for classical modular forms, and support new instances of conjectured Eisenstein congruences for $\mathbb{U}(2,2)$ automorphic forms.
Counting Discrete, Level-$1$, Quaternionic Automorphic Representations on $G_2$
本文利用“hyperendoscopy”技术发展一般迹公式，研究例外群 $G_2$ 上的四元数自守表示，克服了离散级数非正则的技术困难，并计算出 level-$1$ 表示空间的维数。
We investigate modular embeddings for semi-arithmetic Fuchsian groups. First we prove some purely algebro-geometric or even topological criteria for a regular map from a smooth complex curve to a quaternionic Shimura variety to be covered by a modular embedding. Then we set up an adelic formalism for modular embeddings and apply our criteria to study the effect of abstract field automorphisms of $\mathbb{C}$ on modular embeddings. Finally we derive that the absolute Galois group of $\mathbb{Q}$ operates on the dessins d'enfants defined by principal congruence subgroups of (arithmetic or non-ar…
We construct infinitely many nonholomorphic automorphic forms and modular forms associated to a discrete subgroup of infinite covolume of $U(n, 1)$.
Construction of vector-valued adelic automorphic forms
本文构造了Scheithauer提升映射的adele版本，引入向量值adele自守形式空间并定义adele提升映射，证明其与经典映射在adelization下相容，将标量与向量值模形式的关系推广至adele框架。
In this paper we prove a general theorem about congruences between automorphic forms on a reductive group G which is compact at infinity modulo the center. If the rank is one, this essentially reduces to Ribet's level-raising theorem. We then specialize to the higher rank case where G is an inner form of GSp(4). Here we get congruences with automorphic forms having a generic local component. In particular, a Saito-Kurokawa form is congruent to a form which is not of Saito-Kurokawa type. We get similar results for U(3) at split primes.
Building on the works of S. Bochner on equivalence of modular relation with functional equation associated to the Dirichlet series, K. Chandrasekharan and R. Narasimhan obtained new equivalences between the functional equation and some arithmetical identities. Sister Ann M. Heath considered the functional equation in the Hawkins and Knopp context and showed its equivalence to two arithmetical identities associated with entire modular cusp integrals involving rational period functions for the full modular group. In this paper we use techniques of Chandrasekharan and Narasimhan and extend the re…
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.
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…
A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight $k$ on $Γ_0(N)$ to a simpler function of $k$ and $N$, showing that the two are equal whenever $N$ is squarefree. We prove the converse of this theorem (with one small exception), thus providing a characterization of squarefree integers. We also establish a similar characterization of prime numbers in terms of the number of Hecke newforms of weight $k$ on $Γ_0(N)$. It follows that a hypothetical fast algorithm for computing the number of such automorphic repre…
We consider the problem of explicitly computing dimensions of spaces of automorphic or modular forms in level one, for a split classical group $\mathbf{G}$ over $\mathbb{Q}$ such that $\mathbf{G}(\R)$ has discrete series. Our main contribution is an algorithm calculating orbital integrals for the characteristic function of $\mathbf{G}(\mathbb{Z}_p)$ at torsion elements of $\mathbf{G}(\mathbb{Q}_p)$. We apply it to compute the geometric side in Arthur's specialisation of his invariant trace formula involving stable discrete series pseudo-coefficients for $\mathbf{G}(\mathbb{R})$. Therefore we e…
We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger a result on the existence of modular lifts of specified type for Galois representations corresponding to Hilbert modular forms of parallel weight 2. We discuss some conjectures on the weights of $n$-dimensional mod $p$ Galois representations. Finally, we use recent work of Taylor to prove level raising and lo…
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 …
We obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region, for $\SL_2$ over a totally real number field $F$, with discrete subgroup of Hecke type $Γ_0(I)$ for a non-zero ideal $I$ in the ring of integers of $F$. The weights are products of Fourier coefficients. This implies in particular the existence of infinitely many cuspidal automorphic representations with multi-eigenvalues in various regions growing to infinity. For instance, in the quadratic case, the regions include floating boxes, floating balls, sectors, slanted strips and products of prescribe…
Siegel Modular Forms and Theta Series attached to quaternion algebras II
我们继续我们的研究吉田的提升，它与一对自守形式的adelic乘法群的四元数代数西格尔模形式的程度2。我们在这里考虑的情况下，自守形式的四元数代数对应于模形式的任意偶数重量和平方自由水平，特别是我们得到的西格尔模形式的重量3连接到一对椭圆模形式的重量2和4。
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.
The goal of this paper is to provide a complete and refined study of the standard $L$-functions $L(π,\operatorname{Std},s)$ for certain non-generic cuspidal automorphic representations $π$ of $G_2(\mathbb{A})$. For a cuspidal automorphic representation $π$ of $G_2(\mathbb{A})$ that corresponds to a modular form $φ$ of level one and of even weight on $G_2$, we explicitly define the completed standard $L$-function, $Λ(π,\operatorname{Std},s)$. Assuming that a certain Fourier coefficient of $φ$ is nonzero, we prove the functional equation $Λ(π,\operatorname{Std},s) = Λ(π,\operatorname{Std},1-s)$.…
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。
É.Ghys proved that the linking numbers of modular knots and the "missing" trefoil $K_{2,3}$ in $S^3$ coincide with the values of a highly ubiquitous function called the Rademacher symbol for ${\rm SL}_2\mathbb{Z}$. In this paper, we replace ${\rm SL}_2\mathbb{Z}=Γ_{2,3}$ by the triangle group $Γ_{p,q}$ for any coprime pair $(p,q)$ of integers with $2\leq p<q$. We invoke the theory of harmonic Maass forms for $Γ_{p,q}$ to introduce the notion of the Rademacher symbol $ψ_{p,q}$, and provide several characterizations. Among other things, we generalize Ghys's theorem for modular knots around any "…
We prove an automorphic analogue of Deligne's conjecture for symmetric fourth $L$-functions of Hilbert modular forms. We extend the result of Morimoto based on generalization and refinement of the results of Grobner and Lin to cohomological irreducible essentially conjugate self-dual cuspidal automorphic representations of ${\rm GL}_2$ and ${\rm GL}_3$ over CM-fields.
Automorphic Green functions on Hilbert modular surfaces
本文将 Hilbert 模曲面上的自守 Green 函数理论推广至任意理想，给出 Fourier 展开、正则化、尖点增长、局部 Borcherds 乘积及生成级数模性等结果。
For a fixed mod $p$ automorphic Galois representation, $p$-adic automorphic Galois representations lifting it determine points in universal deformation space. In the case of modular forms and under some technical conditions, Böckle showed that every component of deformation space contains a smooth modular point, which then implies their Zariski density when coupled with the infinite fern of Gouvêa-Mazur. We generalize Böckle's result to the context of polarized Galois representations for CM fields, and to two dimensional Galois representations for totally real fields. More specifically, under …
We give archimedean and non-archimedean constructions of Darmon points on modular abelian varieties attached to automorphic forms over arbitrary number fields and possibly non-trivial central character. An effort is made to present a unifying point of view, emphasizing the automorphic nature of the construction.
We extend Holowinsky and Soundararajan's proof of quantum unique ergodicity for holomorphic Hecke modular forms on SL(2,Z), by establishing it for automorphic forms of cohomological type on GL_2 over an arbitrary number field which satisfy the Ramanujan bounds. In particular, we have uncondtional theorems over totally real and imaginary quadratic fields. In the totally real case we show that our result implies the equidistribution of the zero divisors of holomorphic Hecke modular forms, generalising a result of Rudnick over Q.
Automorphic Integrals with Rational Period Functions and Arithmetical Identities
本文将Heath关于全模群上整模尖点积分中函数方程与算术恒等式等价的结果，推广到离散Hecke群上具有有理周期函数的整自守积分情形。
In this paper we study the compatible family of degree-4 Scholl representations $ρ_{\ell}$ associated with a space $S$ of weight $κ> 2$ noncongruence cusp forms satisfying Quaternion Multiplications over a biquadratic field $K$. It is shown that when either $K$ is totally real or $κ$ is odd, $ρ_\ell$ is automorphic, that is, its associated L-function has the same Euler factors as the L-function of an automorphic form for $GL_4(\mathbb Q)$. Further, it yields a relation between the Fourier coefficients of noncongruence cusp forms in $S$ and those of certain automorphic forms via the three-term …
This paper deals with a new analytic type of vector- and Clifford algebra valued automorphic forms in one and two vector variables. For hypercomplex generalizations of the classical modular group and their arithmetic congruence subgroups Eisenstein- and Poincaré type series that are annihilated by Dirac operators, and more generally, by iterated Dirac operators on the upper half-space of $R^n$ are discussed. In particular we introduce (poly-)monogenic modular forms on hypercomplex generalizations of the classical theta group.
In this paper, we prove Deligne's conjecture for symmetric sixth $L$-functions of Hilbert modular forms. We extend the result of Morimoto based on a different approach. We define automorphic periods associated to globally generic $C$-algebraic cuspidal automorphic representations of ${\rm GSp}_4$ over totally real number fields whose archimedean components are (limits of) discrete series representations. We show that the algebraicity of critical $L$-values for ${\rm GSp}_4 \times {\rm GL}_2$ can be expressed in terms of these periods. In the case of Kim-Ramakrishnan-Shahidi lifts of ${\rm GL}_…
We investigate the sixth moment of the family of $L$-functions associated to holomorphic modular forms on $GL_2$ with respect to a congruence subgroup $Γ_1(q)$. We improve on previous work and obtain an unconditional upper bound of the correct order of magnitude.
A general Voronoï summation formula for the (metaplectic) double cover of $\text{GL}_2$ is derived via the representation theoretic framework à la Ichino--Templier. The identity is also formulated classically and used to establish Voronoï summation formulae for half-integral weight modular forms and Maaß forms.
We study $p$-adic families of cohomological automorphic forms for ${\mathrm{GL}}(2)$ over imaginary quadratic fields and prove that families interpolating a Zariski-dense set of classical cuspidal automorphic forms only occur under very restrictive conditions. We show how to computationally determine when this is not the case and establish concrete examples of families interpolating only finitely many Bianchi modular forms.
We prove vanishing results for the $0$th cohomology of automorphic line bundles on the Ekedahl--Oort strata of the special fiber of Hilbert--Blumenthal Shimura varieties. This work vastly extends to all Ekedahl--Oort strata previous results of Diamond--Kassaei and Goldring--Koskivirta. Furthermore, our results also cover the case of unitary Shimura varieties of rank $\leq 2$ attached to an arbitrary CM extension. We also prove a conjecture of Goldring--Koskivirta on the set of possible weights of automorphic forms.
Mazur's principle gives a criterion under which an irreducible mod $\ell$ Galois representation arising from a modular form of level $Np$ (with $p$ prime to $N$) can also arise from a modular form of level $N.$ We prove an analogous result showing that a mod $\ell$ Galois representation arising from a stable cuspidal automorphic representation of the unitary similitude group $G=\mathrm{GU}(1,2)$ which is Steinberg at an inert prime $p$ can also arise from an automorphic representation of $G$ that is unramified at $p$.
In this paper, we investigate the non-modular solutions to the Schwarz differential equation $\{f,τ\}=sE_4(τ)$ where $E_4(τ)$ is the weight 4 Eisenstein series and $s$ is a complex parameter. In particular, we provide explicit solutions for each $s=2π^2(n/6)^2$ with $n\equiv 1\mod 12$. These solutions are obtained as integrals of meromorphic weight 2 modular forms. As a consequence, we find explicit solutions to the differential equation $\displaystyle y''+\frac{π^2n^2}{36}\,E_4\,y=0$ for each $n\equiv 1\mod 12$ generalizing the work of Hurwitz and Klein on the case $n=1$. Our investigation re…
In this paper, we prove the existence of certain lifts of Hilbert cusp forms to general odd spin groups. We then use those lifts to provide evidence for a conjecture of Gross on the modularity of abelian varieties not of ${\rm GL}_2$-type.
For triangle groups, the (quasi-)automorphic forms are known just as explicitly as for the modular group SL$(2,\bbZ)$. We collect these expressions here, and then interpret them using the Halphen differential equation. We study the arithmetic properties of their Fourier coefficients at cusps and Taylor coefficients at elliptic fixed-points --- in both cases integrality is related to the arithmeticity of the triangle group. As an application of our formulas, we provide an explicit modular interpretation of periods of 14 families of Calabi-Yau threefolds over the thrice-punctured sphere.
Gross and Zagier conjectured that the CM values (of certain Hecke translates) of the automorphic Green function $G_s(z_1,z_2)$ for the elliptic modular group at positive integral spectral parameter $s$ are given by logarithms of algebraic numbers in suitable class fields. We prove a partial average version of this conjecture, where we sum in the first variable $z_1$ over all CM points of a fixed discriminant $d_1$ (twisted by a genus character), and allow in the second variable the evaluation at individual CM points of discriminant $d_2$. This result is deduced from more general statements for…
Machine learning automorphic forms for black holes
本文通过在前馈神经网络上训练自守形式傅里叶系数，证明机器学习能从截断展开中准确预测模权，尤其在负权黑洞计数公式中表现突出，为量子引力对称性的自动检测提供了概念验证。
In this article, using methods from geometric analysis and theory of heat kernels, we derive qualitative estimates of automorphic cusp forms defined over quaternion algebras. Using which, we prove an average version of the holomorphic QUE conjecture. We then derive quantitative estimates of classical Hilbert modular cusp forms. This is a generalization of the results from [3] and [9] to higher dimensions.
We present an algorithm to determine if the $L$-series associated to an automorphic representation and the one associated to an elliptic curve over an imaginary quadratic field agree. By the work of Harris-Soudry-Taylor, Taylor and Berger-Harcos (cf. \cite{harris-taylor}, \cite{taylorII} and \cite{berger-harcos}) we can associate to an automorphic representation a family of compatible $p$-adic representations. Our algorithm is based on Faltings-Serre's method to prove that $p$-adic Galois representations are isomorphic.
Let E/Q be a real quadratic field and pi_0 a cuspidal, irreducible, automorphic representation of GL(2,A_E) with trivial central character and infinity type (2,2n+2) for some non-negative integer n. We show that there exists a non-zero Siegel paramodular newform F with weight, level, Hecke eigenvalues, epsilon factor and L-function determined explicitly by pi_0. We tabulate these invariants in terms of those of pi_0 for every rational prime p.
We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor--Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology. In practice, this imposes several restrictions -- one must be in a Shimura variety setting and the automorphic forms must be of regular weight at infinity. In this paper, we essentially show how to remove these restrictions. Our most general result is a modularity lifting theorem which, on…
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。
We study the behaviour of automorphic L-Invariants associated to cuspidal representations of GL(2) of cohomological weight 0 under abelian base change and Jacquet-Langlands lifts to totally definite quaternion algebras. Under a standard non-vanishing hypothesis on automorphic L-functions and some technical restrictions on the automorphic representation and the base field we get a simple proof of the equality of automorphic and arithmetic L-invariants. This together with Spiess' results on p-adic L-functions yields a new proof of the exceptional zero conjecture for modular elliptic curves - at …
We prove for L-function attached to an automorphic cusp form for the Hecke congruence group $Γ_0(D)$, which is also an eigenfunction of all the Hecke operators, that a positive proportion of its non-trivial zeros lie on the critical line. This result extends the work of J.L. Hafner of 1983 where the case of the full modular group is considered.
We classify the $(\mathfrak{g},K)$-modules generated by nearly holomorphic Hilbert-Siegel modular forms by the global method. As an application, we study the image of projection operators on the space of nearly holomorphic Hilbert-Siegel modular forms with respect to infinitesimal characters in terms of $(\mathfrak{g},K)$-modules.
Many results are known regarding how much local information is required to determine a global object, such as a modular form, or a Galois or automorphic representation. We survey some things that are known and expected, and then explain recent joint work with Dinakar Ramakrishnan about comparing degree 2 Artin and automorphic representations which a priori may not correspond at certain infinite sets of places.
This article is a research exposition based on the author's talk at the International Colloquium on Automorphic Representations and L-Functions, 2012, held at TIFR, Mumbai. We consider some special cases of the following question: when is a natural subset of the Fourier coefficients sufficient to uniquely determine a "modular form"? Two kinds of modular forms are considered in this article: a) classical modular forms of half-integral weight, and b) Siegel modular forms of genus 2 and integral weight. These two apparently different scenarios turn out to be closely related. Our results were moti…
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