YoungJu Choie
Generalizing a result of \cite{Z1991, CPZ} about elliptic modular forms, we give a closed formula for the sum of all Hilbert Hecke eigenforms over a totally real number field with strict class number $1$, multiplied by their period polynomials, as a single product of the Kronecker series.
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This is a sequel to the paper [F. Breuer, H.-G. Rück, Drinfeld modular polynomials in higher rank, J. Number Theory 129 (2009), 59-83.], in which we introduced Drinfeld modular polynomials of higher rank, using an analytic construction. These polynomials relate the isomorphism invariants of Drinfeld F_q[T]-modules of rank r\geq 2 linked by isogenies of a specified type. In the current paper, we give an algebraic construction of greater generality, and prove a generalization of the Kronecker congruences relations, which describe what happens when modular polynomials associated to P-isogenies ar…
Twisted Kronecker series and periods of modular forms on $Γ_0(N)$
本文构造特征扭曲的 Kronecker 级数族，并在 N 无平方因子时用其乘积给出 $Γ_0(N)$ 上 Hecke 特征形式及其扭曲周期多项式之和的闭式公式，推广了 Zagier 恒等式。
An Eisenstein--Kronecker Modular Symbol and Beilinson--Kato Distributions
本文构造了一个Eisenstein-Kronecker模符号，其特化恢复Borisov-Gunnells符号，并通过归一化将其分布等同于Colmez的Beilinson-Kato分布实现，从而建立了循环Manin关系并生成相关线性关系。
We present algorithms to perform modular polynomial multiplication or modular dot product efficiently in a single machine word. We pack polynomials into integers and perform several modular operations with machine integer or floating point arithmetic. The modular polynomials are converted into integers using Kronecker substitution (evaluation at a sufficiently large integer). With some control on the sizes and degrees, arithmetic operations on the polynomials can be performed directly with machine integers or floating point numbers and the number of conversions can be reduced. We also present …
On the integrality of modular functions over $\mathbb{Z}[j]$ and Kronecker-type congruences
本文研究了模函数在$\mathbb{Z}[j]$上的整性，并证明了在特定条件下，由模函数定义的函数满足克罗内克型同余关系。
Euler-Kronecker constants of modular forms: beyond Dirichlet $L$-series
本文构造涉及 ${\mathfrak S}_3$、${\mathfrak S}_4$ 扩张所对应 Artin $L$-级数的 Euler–Kronecker 常数例子，并据此解决 2023 年关于模形式同余的问题。
The classical Kronecker limit formula gives the constant term of the non-holomorphic Eisenstein series E(z,s) for SL(2,Z) at s=1 in terms of the Dedekind eta function. Here we compute the analagous formula for an Eisenstein series twisted by modular symbols (periods of weight two holomorphic cusp forms) for general Fuchsian groups of the first kind.
We develop two applications of the Kronecker's limit formula associated to elliptic Eisenstein series: A factorization theorem for holomorphic modular forms, and a proof of Weil's reciprocity law. Several examples of the general factorization results are computed, specifically for certain moonshine groups, congruence subgroups, and, more generally, non-compact subgroups with one cusp. In particular, we explicitly compute the Kronecker limit function associated to certain elliptic points for a few small level moonshine groups.
Let $Γ\subset\mathrm{PSL}_{2}(\mathbb{R})$ be a Fuchsian subgroup of the first kind acting by fractional linear transformations on the upper half-plane $\mathbb{H}$, and let $M=Γ\backslash\mathbb{H}$ be the associated finite volume hyperbolic Riemann surface. Associated to any cusp of $M$, there is the classically studied non-holomorphic (parabolic) Eisenstein series. In 1979, Kudla and Millson studied non-holomorphic (hyperbolic) Eisenstein series associated to any closed geodesic on $Γ\backslash\mathbb{H}$. In 2004, Jorgenson and Kramer introduced so-called elliptic Eisenstein series associa…
On the locally analytic completed cohomology of modular curves
本文综述了关于模曲线完成同调局部解析向量的研究，探讨了相关方法及贡献。
In this paper, we prove the Eichler cohomology theorem of weakly parabolic generalized modular forms of real weights on subgroups of finite index in the full modular group. We explicitly establish the isomorphism for large weights by constructing the map from the space of cusp forms to the cohomology group.
We construct bases for the spaces of higher order modular forms of all orders and weights. We also provide a cohomological interpretation of these forms.
Congruences for traces of singular moduli and Hurwitz - Kronecker class numbers
本文研究了模数迹与Hurwitz-Kronecker类数之间的同余关系，通过U-算子作用下的模形式同余，揭示了模11下的特殊同余条件。
We describe extension classes arising in the $\ell$-adic and Hodge cohomology of Hilbert modular varieties, generalising results of Caspar to arbitrary dimensions. We show that this description is consistent with the "plectic conjectures" of Nekovář and the second author.
The Mordell-Tornheim zeta function: Kronecker limit type formula, Series Evaluations and Applications
本文为 Mordell-Tornheim zeta 函数建立 Kronecker 极限型公式，统一多种经典模关系，并由此得到级数求值与新的混合函数方程族。
The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalised from modular curves to Hecke triangle surfaces. A sketch is included of how the modular symbols formalism gives rise to the standard algorithms for the computation of holomorphic modular forms. Precise and explicit connections are established to the cohomology of Hecke triangle surfaces and group cohomology. In all the note a general commutative ring is used as coefficient ring in view of applications to the computation of modular forms over rings different from the complex numbers. T…
Let Gamma be a congruence subgroup of the Picard modular group of an imaginary number field k, and let D be the associated symmetric space. We describe a method to compute the integral cohomology of the locally symmetric space Gamma\D. The method is implemented for the case k=Q(i) and k=Q(sqrt(-3)), and the cohomology is computed for various Gamma.
This is a continuation of our previous work on the locally analytic vectors of the completed cohomology of modular curves. We construct differential operators on modular curves with infinite level at p in both "holomorphic" and "anti-holomorphic" directions. As applications, we reprove a classicality result of Emerton which says that every absolutely irreducible two dimensional Galois representation which is regular de Rham at p and appears in the completed cohomology of modular curves comes from an eigenform. Moreover we give a geometric description of the locally analytic representations of …
Let $F$ be a totally real number field and let $p$ be a prime unramified in $F$. We prove the existence of Galois pseudo-representations attached to mod $p^m$ Hecke eigenclasses of paritious weight occurring in the coherent cohomology of Hilbert modular varieties for $F$ of level prime to $p$.
We propose an action of a certain motivic cohomology group on the coherent cohomology of Hilbert modular varieties, extending conjectures of Venkatesh, Prasanna, and Harris. The action is described in two ways: on cohomology modulo $p$ and over $\mathbb C$, and we conjecture that they both lift to an action on cohomology with integral coefficients. The conjecture is supported by theoretical evidence based on Stark's conjecture on special values of Artin $L$-functions, and by numerical evidence in base change cases.
We compute the cohomology with compact supports of a Picard modular surface as a virtual module over the product of the appropriate Galois group and the appropriate Hecke algebra. We use the method developed by Ihara, Langlands, and Kottwitz: comparison of the Grothendieck-Lefschetz formula and the Arthur-Selberg trace formula. Our implementation of this method takes as its starting point the works of Laumon and Morel.
The classical Kronecker limit formula describes the constant term in the Laurent expansion at the first order pole of the non-holomorphic Eisenstein series associated to the cusp at infinity of the modular group. Recently, the meromorphic continuation and Kronecker limit type formulas were investigated for non-holomorphic Eisenstein series associated to hyperbolic and elliptic elements of a Fuchsian group of the first kind by Jorgenson, Kramer and the first named author. In the present work, we realize averaged versions of all three types of Eisenstein series for $Γ_0(N)$ as regularized theta …
In this paper, we compare two different constructions of $p$-adic $L$-functions for modular forms and their relationship to Galois cohomology: one using Kato's Euler system and the other using Emerton's $p$-adically completed cohomology of modular curves. At a more technical level, we prove the equality of two elements of a local Iwasawa cohomology group, one arising from Kato's Euler system, and the other from the theory of modular symbols and $p$-adic local Langlands correspondence for $GL_2(\mathbb{Q}_p)$. We show that this equality holds even in the cases when the construction of $p$-adic …
Completed cohomology of Hilbert modular varieties below middle degree
研究希尔伯特模簇低度完备上同调的局部解析向量，证明通用包络代数作用需通过商分解，并给出维数界与稠密性结果。
Building on ideas of Vatsal, Cornut proved a conjecture of Mazur asserting the generic nonvanishing of Heegner points on an elliptic curve E as one ascends the anticyclotomic Z_p-extension of a quadratic imaginary extension K/Q. In the present article Cornut's result is extended by replacing the elliptic curve E with the Galois cohomology of Deligne's 2-dimensional l-adic representation attached to a modular form of weight 2k>2, and replacing the family of Heegner points with an analogous family of special cohomology classes.
We describe torsion classes in the first cohomology group of $\text{SL}_2(\mathbb{Z})$. In particular, we obtain generalized Dickson's invariants for p-power polynomial rings. Secondly, we describe torsion classes in the zero-th homology group of $\text{SL}_2(\mathbb{Z})$ as a module over the torsion invariants. As application, we obtain various congruences between cuspidal forms of level one and Eisenstein series.
On the étale cohomology of Hilbert modular varieties with torsion coefficients
本文研究带挠系数的希尔伯特模簇平展上同调，借鉴酉志村簇方法得到一般情形下的中间次数集中零化定理与非一般情形的次数界，借助几何Jacquet--Langlands函子性证明并应用于p进局部朗兰兹对应。
For any square-free integer $N$ such that the "moonshine group" $Γ_0(N)^+$ has genus zero, the Monstrous Moonshine Conjectures relate the Hauptmoduli of $Γ_0(N)^+$ to certain McKay-Thompson series associated to the representation theory of the Fischer-Griess monster group. In particular, the Hauptmoduli admits a $q$-expansion which has integer coefficients. In this article, we study the holomorphic function theory associated to higher genus moonshine groups $Γ_0(N)^+$. For all moonshine groups of genus up to and including three, we prove that the corresponding function field admits two generat…
Applications of patching the coherent cohomology of modular curves
本文应用Taylor-Wiles-Kisin补丁方法研究模曲线的相干上同调最小级，通过q展开原理建立补丁模块的乘法一性，并证明晶格变形环的部分规范化的Cohen-Macaulay性，进一步得出晶格点在特征p下的Zariski稠密性及Serre模p四元数模形式的乘法一性结果。
We construct a $(\mathfrak{gl}_2, B(\mathbb{Q}_p))$ and Hecke-equivariant cup product pairing between overconvergent modular forms and the local cohomology at $0$ of a sheaf on $\mathbb{P}^1$, landing in the compactly supported completed $\mathbb{C}_p$-cohomology of the modular curve. The local cohomology group is a highest-weight Verma module, and the cup product is non-trivial on a highest weight vector for any overconvergent modular form of infinitesimal weight not equal to $1$. For classical weight $k\geq 2$, the Verma has an algebraic quotient $H^1(\mathbb{P}^1, \mathcal{O}(-k))$, and on …
We give a new proof of the slope classicality theorem in classical and higher Coleman theory for modular curves at arbitrary level using the completed cohomology classes attached to overconvergent modular forms. The latter give an embedding of the quotient of overconvergent modular forms by classical modular forms, which is the obstruction space for classicality in either cohomological degree, into a unitary representation of $\mathrm{GL}_2(\mathbb{Q}_p)$. The $U_p$ operator becomes a double-coset, and unitarity yields the slope vanishing.
Let $E(z,s)$ be the non-holomorphic Eisenstein series for the modular group $SL(2,{\mathbb Z})$. The classical Kronecker limit formula shows that the second term in the Laurent expansion at $s=1$ of $E(z,s)$ is essentially the logarithm of the Dedekind eta function. This eta function is a weight $1/2$ modular form and Dedekind expressed its multiplier system in terms of Dedekind sums. Building on work of Goldstein, we extend these results from the modular group to more general Fuchsian groups $Γ$. The analogue of the eta function has a multiplier system that may be expressed in terms of a map …
Mean squares of quadratic twists of the Fourier coefficients of modular forms
本文渐近求值了给定模形式傅里叶系数在二次扭曲下的平滑求和版本，涉及Kronecker符号及正奇无平方整数求和。
For a totally real number field $F$ and a nonarchimedean prime $\mathfrak{p}$ of $F$ lying above a prime number $p$ we introduce certain sheaf cohomology groups that intertwine the $\mathfrak{p}^{\infty}$-tower of a quaternionic Hilbert modular variety associated to a quaternion algebra $D$ over $F$ that is split at $\mathfrak{p}$ and a $p$-adically admissible representation of $\mbox{PGL}_2(F_{\mathfrak{p}})$. Applied to infinitesimal $p$-adic deformations of the local factor at $\mathfrak{p}$ of a cuspidal automorphic representation $π$ of $D^*(\mathbb{A})$ this yields a natural construction…
Using holomorphic projection, we work out a parametrization for all relations of products (resp. Rankin-Cohen brackets) of weight $\tfrac 32$ mock modular forms with holomorphic shadow and weight $\tfrac 12$ modular forms in the spirit of the Kronecker-Hurwitz class number relations. In particular we obtain new proofs for several class number relations among which some are classical, others are relatively new. We also obtain similar results for the mock theta functions.
Let F be a real quadratic field with ring of integers O and with class number 1. Let Gamma be a congruence subgroup of GL_2 (O). We describe a technique to compute the action of the Hecke operators on the cohomology H^3 (Gamma; C). For F real quadratic this cohomology group contains the cuspidal cohomology corresponding to cuspidal Hilbert modular forms of parallel weight 2. Hence this technique gives a way to compute the Hecke action on these Hilbert modular forms.
Let p be a prime number. The Hasse invariant is a modular form modulo p that is often used to produce congruences between modular forms of different weights. We show how to produce such congruences between forms of weights 2 and p+1, in terms of group cohomology. We also show how our method works in the contexts of quadratic imaginary fields (where there is no Hasse invariant available) and Hilbert modular forms over totally real fields of even degree.
We construct an integral model of the perfectoid modular curve. Studying this object, we prove some vanishing results for the coherent cohomology at perfectoid level. We use a local duality theorem at finite level to compute duals for the coherent cohomology of the integral perfectoid curve. Specializing to the structural sheaf, we can describe the dual of the completed cohomology as the inverse limit of the integral cusp forms of weight $2$ and trace maps.
Two results on Eisenstein part of homology and cohomology groups of Bianchi modular groups
本文明确构造了 Bianchi 模群同调群的 Eisenstein 环，并研究上同调部分，进而得到尖点上同调群的渐近维数公式。
In a paper published in 1959, Shimura presented an elegant calculation of the critical values of L-functions attached to elliptic modular forms using the first cohomology group. We will show that a similar calculation is possible for Hilbert modular forms over real quadratic fields using the second cohomology group. We present explicit numerical examples calculated by this method.
Several-variable Kronecker limit formula over global function fields
本文为全局函数域上多变量 Eisenstein 级数建立了 Kronecker 极限公式，利用 Berkovich 解析结构对 Heegner 闭链进行参数化，并应用于解释 Dedekind-Weil zeta 函数和 Dirichlet $L$-函数的 Kronecker 项。
Elliptic modular graph functions and forms (eMGFs) are defined for arbitrary graphs as natural generalizations of modular graph functions and forms obtained by including the character of an Abelian group in their Kronecker--Eisenstein series. The simplest examples of eMGFs are given by the Green function for a massless scalar field on the torus and the Zagier single-valued elliptic polylogarithms. More complicated eMGFs are produced by the non-separating degeneration of a higher genus surface to a genus one surface with punctures. eMGFs may equivalently be represented by multiple integrals ove…
We provide a new and simple automorphic method using Eisenstein series to study the equidistribution of modular symbols modulo primes, which we apply to prove an average version of a conjecture of Mazur and Rubin. More precisely, we prove that modular symbols corresponding to a Hecke basis of weight 2 cusp forms are asymptotically jointly equidistributed mod $p$ while we allow restrictions on the location of the cusps. As an application, we obtain a residual equidistribution result for Dedekind sums. Furthermore, we calculate the variance of the distribution and show a surprising bias with con…
We study the $Λ$-module structure of the ordinary parts of the arithmetic cohomology groups of modular Jacobians made out of various towers of modular curves. We prove that the ordinary parts of $Λ$-adic Selmer groups coming from two different towers have "almost same" $Λ$-module structures. We also prove the cotorsionness of $Λ$-adic Tate-Shafarevich group under mild assumptions.
Eisenstein-Kronecker classes, integrality of critical values of Hecke $L$-functions and $p$-adic interpolation
本文证明任意全虚数域上代数Hecke特征标的临界$L$-值除以周期为代数整数，并构造$p$-进测度插值普通情形下的临界值，推广了Damerell、Shimura和Katz的结果。
For N=5, 6 and 7, using the classification of perfect quadratic forms, we compute the homology of the Voronoi cell complexes attached to the modular groups SL_N(\Z) and GL_N(\Z). From this we deduce the rational cohomology of those groups.
In the literature, the standard approach to finding bases of spaces of modular forms is via modular symbols and the homology of modular curves. By using the Eichler-Shimura isomorphism, a work by Wang shows how one can use a cohomological viewpoint to determine bases of spaces of cusp forms on $Γ_0(N)$ of weight $k \geq 2$ and character $χ$. It is interesting to look at the complexity of this alternative approach, and we do this for an explicit implementation of the algorithm suggested by Wang.
We formulate a detailed conjectural Eichler-Shimura type formula for the cohomology of local systems on a Picard modular surface associated to the group of unitary similitudes $\mathrm{GU}(2,1,\mathbb{Q}(\sqrt{-3}))$. The formula is based on counting points over finite fields on curves of genus three which are cyclic triple covers of the projective line. Assuming the conjecture we are able to calculate traces of Hecke operators on spaces of Picard modular forms. We provide ample evidence for the conjectural formula. Along the way we prove new results on characteristic polynomials of Frobenius …
Plectic Lie Algebra Action on Hilbert modular varieties
本文构造了Hilbert模簇局部解析完备上同调上plectic Galois作用的一部分，证明了中维度以下完备上同调消失，并使用了部分Sen算子。
This is a report on a joint project in experimental mathematics with Jonas Bergström and Carel Faber where we obtain information about modular forms by counting curves over finite fields.
Iterated integrals and cohomology
本研究受Manin非交换模符号与伪theta函数研究启发，提出标准上同调的扩展形式，既为相关迭代积分提供上同调解释，也给出多重zeta值关系的模对应物。
Eisenstein-Kronecker classes on Hilbert moduli spaces and a new construction of Katz's $p$-adic measure
将Kings-Sprang的Eisenstein-Kronecker类推广到Hilbert模空间上的万有交换概形，并与Katz的Eisenstein级数比较，构造插值这些类的$p$-进测度，恢复Katz的$p$-进Eisenstein测度。
Let K be an imaginary quadratic field with class number one and ring of integers O. We prove that mod l, a system of Hecke eigenvalues occurring in the first cohomology group of some congruence subgroup Gamma of SL(2,O) can be realized in the first cohomology with trivial coefficients after increasing the level of Gamma by l.
A $p$-adic cohomological approach to congruences of meromorphic modular forms
本文提出了一种基于p进上同调的方法，用于研究解析模形式的同余关系，通过刚性上同调与晶格结构的相互作用，建立了弗罗贝尼乌斯作用与U_p算子之间的联系。
The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let's mention : (1) the control of the image of the Galois representation modulo $p$, (2) Hida's congruence criterion outside an explicit set of primes $p$, and (3) the freeness of the integral cohomology of the Hilbert modular variety over certain local components of the Hecke algebra and the Gorenstein property of these local algebras. We study the arithmetic of the Hilbert modular forms by studying their modulo $p$ Galois representations and our main…
In this paper, we construct a pairing between modular forms of positive real weight and elements of certain Eichler cohomology groups that were introduced by Knopp in 1974. We use spectral theory of automorphic forms to show that this pairing is perfect for all positive weights except 1. The approach in this paper gives a new proof of a theorem by Knopp and Mawi from 2010 for all real weights excluding 1 and also a version of this theorem for vector-valued modular forms.
In a previous paper [CG], we showed how one could generalize Taylor-Wiles modularity lifting theorems [Wil95, TW95] to contexts beyond those in which the automorphic forms in question arose from the middle degree cohomology of Shimura varieties, in particular, to contexts in which the relevant automorphic forms contributed to cohomology in exactly two degrees. In this sequel, we extend our method to the general case in which Galois representations are expected to occur in cohomology, contingent on the (as yet unproven) existence of certain Galois representations with the expected properties. A…
Let $F$ be a totally real field in which $p$ is unramified. We prove that, if a cuspidal overconvergent Hilbert cuspidal form has small slopes under $U_p$-operators, then it is classical. Our method follows the original cohomological approach of Coleman. The key ingredient of the proof is giving an explicit description of the Goren-Oort stratification of the special fiber of the Hilbert modular variety. A byproduct of the proof is to show that, at least when $p$ is inert, of the rigid cohomology of the ordinary locus has the same image as the classical forms in the Grothendieck group of Hecke …
We interpolate cohomology classes attached to families of Hilbert modular forms. Using this we construct a two variable $p$-adic L-function which interpolates one variable $p$-adic L-functions.
We develop a regularization for Petersson inner products of arbitrary weakly holomorphic modular forms, generalizing several known regularizations. As one application, we extend work of Duke, Imamoglu, and Toth on regularized inner products of weakly holomorphic modular forms of weights $0$ and $3/2$. These regularized inner products can be evaluated in terms of the coefficients of holomorphic parts of harmonic Maass forms of dual weights. Moreover, we study the errors of modularity of the holomorphic parts of such a harmonic Maass forms and show that they induce cocyles in the first parabolic…
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