Tiago J. Fonseca
We survey some key developments in the theory of transcendental numbers, paying special attention to Nesterenko's theorem on values of Eisenstein series and emphasizing its underlying geometric aspects. We finish with a brief discussion on periods and related open problems.
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A note on transcendence of special values of functions related to modularity
本文研究具有代数傅里叶系数的模函数及亚纯（拟）模形式在任意同余子群下特殊值的超越性，统一并推广了 Schneider、Nesterenko 等人的经典定理。
We study the values of the recently introduced involution J (jimm) of the real line, which is equivariant with the action of the group PGL(2,Z). We test our conjecture that this involution sends algebraic numbers of degree at least three to transcendental values. We also deduce some theoretical results concerning the continued fraction statistics of the generic values of this involution and compare them with the experimental results.
Real geometric transcendence for the Gamma function
研究伽马函数映射下实代数曲线的性质，证明x轴是唯一满足条件的曲线，并应用相关结果探讨曼因-穆姆福德猜想的推广。
It is a classical fact that the elliptic modular functions satisfies an algebraic differential equation of order 3, and none of lower order. We show how this generalizes to Siegel modular functions of arbitrary degree. The key idea is that the partial differential equations they satisfy are governed by Gauss--Manin connections, whose monodromy groups are well-known. Modular theta functions provide a concrete interpretation of our result, and we study their differential properties in detail in the case of degree 2.
We extend some results of Gun, Murty, and Rath on elliptic modular forms. We take ANY Fuchsian triangle group with a cusp and look at power series expansions in a natural parameter around that cusp. Consider the automorphic forms for such a triangle group whose power series expansions in the natural parameter have algebraic coefficients. We show that the zeros of such forms are either transcendental or are "CM." By "CM," we mean they correspond to abelian varieties with complex multiplication. This result is the first of its kind in the case of non-arithmetic groups.
Real geometric transcendence for uniformization maps of algebraic Riemann surfaces
研究非单连通光滑连通复代数曲线\(X\)中包含其万有覆盖\(\tilde{X}\)中实代数曲线弧像的不可约实代数曲线集，给出该集合非空或无限时\(X\)的条件，还明确描述特定\(X\)情形下的该集合。
Functional Transcendence of Periods and the Geometric André--Grothendieck Period Conjecture
本文通过证明代数簇族中代数形式积分的函数超越性，给出混合 Hodge 结构变分的 Ax–Schanuel 结果，并据此建立几何形式的 André–Grothendieck 周期猜想。
In the present paper, we introduce a special function on the Drinfeld period domain $Ω^{r}$ for $r\geq 2$ which gives the false Eisenstein series of Gekeler when $r=2$. We also study its functional equation and relation with quasi-periodic functions of a Drinfeld module as well as transcendence of its values at CM points.
We present a general result of simultaneous approximation to several transcendental real, complex or p-adic numbers xi_1,...,xi_t by conjugate algebraic numbers of bounded degree over Q, provided that the given transcendental numbers xi_1,...,xi_t generate over Q a field of transcendence degree one. We provide sharper estimates for example when xi_1,...,xi_t form an arithmetic progression with non-zero algebraic difference, or a geometric progression with non-zero algebraic ratio different from a root of unity. In this case, we also obtain by duality a version of Gel'fond's transcendence crite…
Using geometric methods we prove the standard period-index conjecture for the Brauer group of a field of transcendence degree 2 over a finite field.
Transcendence of simple geodesics on finite modular covers
研究有限指数子群对应的模覆盖上简单测地线的端点性质，证明若测地线简单则端点要么有理数、二次无理数或超越数，并验证了最小测地线叶层情形。
Geometry and Transcendence of the Hexponential
本文研究模群第二导出子群对应的六穿孔平面覆盖及其 hexponential 单值化映射，构造基于 Sturmian 序列的截面，证明其取值要么是 Markov 二次无理数要么是超越数，并给出 hexp 的连分数展开。
On nearly holomorphic Drinfeld modular forms for admissible coefficient rings
本文推广了先前工作，针对$\mathrm{GL}_2(A)$的同余子群引入了近亚纯Drinfeld模形式的概念，通过扩展德拉姆层给出了其代数与解析描述的比较同构，并证明了其在CM点处特殊值的超越性及与CM Drinfeld $A$-模周期的关联。
We address the problem of the determination of the images of three-dimensional geometric and modular Galois representations. In the modular case the existence of these representations is only conjectural. We give conditions to ensure that the images are as large as possible for almost every prime. We compute some examples and stress the applications to inverse Galois theory.
We give a lower bound for the number of ordinary lines spanned by a modular hyperbola when the modulus is a prime power. We also give a partial answer to a question of Shparlinski.
Bringmann, Guerzhoy and Kane have shown how to correct mock modular forms by a certain linear combination of the Eichler integral of their shadows in order to obtain p-adic modular forms in the sense of Serre. In this paper, we give a new proof of their results (for good primes p) by employing the geometric theory of harmonic Maass forms developed by the first author and the theory of overconvergent modular forms due to Katz and Coleman.
We introduce a geometric formalism for studying modular forms of half-integral weight and explore some of its basic properties. Geometric Hecke operators are constructed and some basic spaces of $p$-adic forms are introduced. The $p$-adic theory is greatly expanded in subsequent papers, making that part of this paper largely obsolete.
We carry out a thorough study of weight-shifting operators on Hilbert modular forms in characteristic $p$, generalizing the author's prior work with Sasaki to the case where $p$ is ramified in the totally real field $F$. In particular we use the partial Hasse invariants and Kodaira-Spencer filtrations defined by Reduzzi and Xiao to improve on Andreatta and Goren's construction of partial $Θ$-operators, obtaining ones whose effect on weights is optimal from the point of view of geometric Serre weight conjectures. Furthermore we describe the kernels of partial $Θ$-operators in terms of images of…
In a series of papers we have been studying the geometric theta correspondence for non-compact arithmetic quotients of symmetric spaces associated to orthogonal groups. It is our overall goal to develop a general theory of geometric theta liftings in the context of the real differential geometry/topology of non-compact locally symmetric spaces of orthogonal and unitary groups which generalizes the theory of Kudla-Millson in the compact case. In this paper we study in detail the geometric theta lift for Hilbert modular surfaces. In particular, we will give a new proof and an extension (to all f…
Rational points on modular curves: parameterization and geometric explanations
该研究在Zywina的Serre均匀性猜想条件下，通过参数化方法揭示了模曲线上的有理点与椭圆曲线j不变量之间的几何联系。
Let p be a prime and F a totally real field in which p is unramified. We consider mod p Hilbert modular forms for F, defined as sections of automorphic line bundles on Hilbert modular varieties of level prime to p in characteristic p. For a mod p Hilbert modular Hecke eigenform of arbitrary weight (without parity hypotheses), we associate a two-dimensional representation of the absolute Galois group of F, and we give a conjectural description of the set of weights of all eigenforms from which it arises. This conjecture can be viewed as a "geometric" variant of the "algebraic" Serre weight conj…
Geometric construction of modular polynomials with level structures
本文提出了一种纯代数方法，用于构造具有更高层次结构的模多项式，展示了与椭圆曲线模型直接相关的不变量的模多项式存在性，并给出了计算这些多项式的算法。
We prove a geometric formula for the cycle integrals of Parson's weight 2k modular integrals in terms of the intersection angles of geodesics on modular curves. Our result is an analog for modular integrals of a classical formula for the cycle integrals of certain hyperbolic Poincaré series, due to Katok. On the other hand, it extends a recent geometric formula of Matsusaka and Duke, Imamoglu, and Tóth for the cycle integrals of weight 2 modular integrals.
On the geometric Serre weight conjecture for Hilbert modular forms
本文证明了 Diamond 和 Sasaki 提出的几何 Serre 权重猜想，在无奇偶性假设下适用于 $p \geq 5$ 惰性的实二次域和 $p \geq \min\{5, [F:\mathbb{Q}]\}$ 完全分裂的全实域。
We give a new construction of $p$-adic overconvergent Hilbert modular forms by using Scholze's perfectoid Shimura varieties at infinite level and the Hodge--Tate period map. The definition is analytic, closely resembling that of complex Hilbert modular forms as holomorphic functions satisfying a transformation property under congruence subgroups. As a special case, we first revisit the case of elliptic modular forms, extending recent work of Chojecki, Hansen and Johansson. We then construct sheaves of geometric Hilbert modular forms, as well as subsheaves of integral modular forms, and vary ou…
In this work, we give a purely analytic introduction to the phenomenon of mirror symmetry for quintic threefolds via classical hypergeometric functions and differential equations for them. Starting with a modular map and recent transcendence results for its values, we regard a mirror map $z(q)$ as a concept generalizing the modular one. We give an alternative approach demonstrating the existence of non-linear differential equations for the mirror map, and exploit both an elegant construction of Klemm-Lian-Roan-Yau and the Ax theorem to prove that the Yukawa coupling $K(q)$ does not satisfy any…
The Geometric Unitary Kudla Conjecture
本文基于几何单位元Kudla猜想，证明对称形式傅里叶-雅科比级数收敛并等于Hermitian Hilbert模形式的展开，验证了猜想并在任意余维数上去除Li-Liu的算术内积公式的模性假设。
We give a survey of a variety of recent results about the distribution and some geometric properties of points $(x,y)$ on modular hyperbolas $xy \equiv a \pmod m$. We also outline a very diverse range of applications of such results, discuss multivariate generalisations and suggest a number of open problems of different levels of difficulty.
We give a geometrical construction of the canonical automorphic factor for the Jacobi group and construct new vector valued modular forms from Jacobi forms by differentiating them with respect to toroidal variables and then evaluating at zero.
Formal deformations of modular forms and multiple L-values
本文利用形变理论的上同调与李代数方法证明模形式形变的存在性和本质唯一性，并构造涵盖所有权重的泛族，从而赋予非临界多重 L 值新的几何联系。
Let K be a number field with euclidean ring of integers O. Let G be a finite-index torsion-free subgroup of Sp(2n, O). We exhibit a finite, geometrically defined spanning set of the top dimensional integral cohomology of G by generalizing the modular symbol algorithm of Ash and Rudolph.
We prove a gluing lemma for sections of line bundles on a rigid analytic variety. We apply the lemma, in conjunction with a result of Buzzard's, to give a proof of (a generalization) of Coleman's theorem which states that overconvergent modular forms of small slope are classical. The proof is "geometric" in nature, and is suitable for generalization to other PEL Shimura varieties.
The Linear Slicing Method for Equal Sums of Like Powers: Modular and Geometric Constraints
本文提出了一种等幂和的线性切片方法，通过模数约束和几何排除区原理，证明了在特定条件下，对于足够大的k，不存在解。
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。
A modular ramified geometric Satake equivalence
将分叉几何Satake等价从有理系数推广到模与整系数，涵盖Zhu及第三作者的结果。
Geometric interpretation of Hermitian modular forms via Burkhardt invariants
本文利用Burkhardt不变量，通过格极化K3曲面的逆周期映射给出Hermite模形式的完整theta表达式，建立了K3曲面模空间、theta函数与秩5有限复反射群之间的非平凡联系。
Theta correspondence and the Borisov-Gunnells relations
本文通过几何θ对应研究模形式之间的关系，揭示了埃西恩系列之间联系的几何证明，并与李定理建立联系。
This paper gives an expository account of our experiments concerning relations between modular forms for congruence subgroups of SL(3,Z) and three dimensional Galois representations. The main new result presented here is a calculation of the variations of the Hodge structure corresponding to the motives we consider in realizing the Galois representations. It turns out that the period spaces for the Hodge structures are four dimensional, while the geometric realizations of such Hodge structures can appear in subspaces of dimension at most one.
Pseudodifferential Jacobi forms and Geometric Rankin-Cohen Brackets
通过Jacobi群在伪微分算子上的作用，构造了Jacobi形式空间与不变伪微分算子空间的同构，并导出了带复参数的几何Rankin-Cohen括号族。
Let $p$ be a prime number and let $n$ be a positive integer prime to $p$. By an Ihara result we mean existence of an injection with torsion-free cokernel from a full lattice in the space of $p$-old modular forms, into a full lattice in the space of all modular forms of level $np$. In this paper, we prove Ihara results for genus two Siegel modular forms, Siegel-Jacobi forms and Hilbert modular forms. We also propose a geometric formulation for the notion of $p$-old Siegel modular forms of genus two.
We provide a geometric Hodge-Tate map giving generic description of the overconvergent modular symbols of some p-adic (accessible) weight k, base-changed to C_p, in terms of overconvergent modular forms of weight k+2.
The modular $j$-function is a bijective map from $X_0(1) \setminus \{\infty\}$ to $\mathbb{C}$. A natural question is to describe the inverse map. Gauss offered a solution to the inverse problem in terms of the arithmetic-geometric mean. This method relies on an elliptic curve model and the Gaussian hypergeometric series. Here we use the theory of polar harmonic Maass forms to solve the inverse problem by directly examining the Fourier expansion of the weight $2$ polar harmonic Maass form obtained by specializing the logarithmic derivative of the denominator formula for the Monster Lie algebra…
We generalize some of the results of Andreatta, Iovita, and Pilloni and the author to Hodge type Shimura varieties having non-empty ordinary locus. For any $p$-adic weight $κ$, we give a geometric definition of the space of overconvergent modular forms of weight $κ$ in terms of sections of a sheaf. We show that our sheaves live in analytic families, interpolating the classical sheaves for integral weights. We define an action of the Hecke algebra, including a completely continuous operator at $p$. In some simple cases, we also build the eigenvariety.
We interpolate the Gauss-Manin connection in p-adic families of nearly overconvergent modular forms. This gives a family of Maass-Shimura type differential operators from the space of nearly overconvergent modular forms of type r to the space of nearly overconvergent modular forms of type r + 1 with p-adic weight shifted by 2. Our construction is purely geometric, using Andreatta-Iovita-Stevens and Pilloni's geometric construction of eigencurves, and should thus generalize to higher rank groups.
This article lays the foundations for the study of modular forms transforming with respect to representations of Fuchsian groups of genus zero. More precisely, we define geometrically weighted graded modules of such modular forms, where the graded structure comes from twisting with all isomorphism classes of line bundles on the corresponding compactified modular curve, and we study their structure by relating it to the structure of vector bundles over orbifold curves of genus zero. We prove that these modules are free whenever the Fuchsian group has at most two elliptic points. For three or mo…
Rings of Hilbert modular forms, computations on Hilbert modular surfaces, and the Oda-Hamahata conjecture
针对全实数域上椭圆曲线的模性解析与几何概念差异问题，本文通过显式计算对实二次域上椭圆曲线且希尔伯特模簇几何亏格为1的情形，证明了浜畑猜想中希尔伯特模簇与椭圆曲线共轭乘积的对应关系。
The Geometry of Drinfeld Modular Forms
本文为同余子群的Drinfeld模形式代数提供几何视角，通过建立线丛截面环与模形式代数的同构实现几何计算，并推广了代数分解结果。
A database of basic numerical invariants of Hilbert modular surfaces
本文介绍用于计算Hilbert模曲面几何不变量的算法，并报告相应实现，以构建基本数值不变量数据库。
Geometric realizations of representations for $\text{PSL}(2, \mathbb{F}_p)$ and Galois representations arising from defining ideals of modular curves
该研究构造了PSL(2, F_p)表示的几何实现，建立了模曲线定义理想、PSL(2, F_p)表示与伽罗瓦表示间的对应，是多个数论几何核心理论的阿贝尔几何对应物。
This paper presents the theory of holomorphic vector valued modular forms from a geometric perspective. More precisely, we define certain holomorphic vector bundles on the modular orbifold of generalized elliptic curves whose sections are vector valued modular forms. This perspective simplifies the theory, and it clarifies the role that exponents of representations of SL_2(Z) play in the holomorphic theory of vector valued modular forms. Further, it allows one to use standard techniques in algebraic geometry to deduce free-module theorems and dimension formulae (deduced previously by other aut…
In this paper, we discuss the theory of the Siegel modular variety in the aspects of arithmetic and geometry. This article covers the theory of Siegel modular forms, the Hecke theory, a lifting of elliptic cusp forms, geometric properties of the Siegel modular variety, (hypothetical) motives attached to Siegel modular forms and a cohomology of the Siegel modular variety.
Hecke operators on topological modular forms
本文将经典模形式的Adams运算、Hecke算子和Atkin-Lehner对合精化为拓扑模形式TMF上的稳定算子，利用代数几何与同伦论技术证明其性质，推导Ramanujan经典同余式并给出满足Maeda猜想的新Hecke算子族。
Let $p$ be an unramified prime in a totally real field $L$ such that $h^+(L)=1$. Our main result shows that Hilbert modular newforms of parallel weight two for $Γ_0(p)$ can be constructed naturally, via classical theta series, from modules of isogenies of superspecial abelian varieties with real multiplication on a Hilbert moduli space. This can be viewed as a geometric reinterpretation of the Eichler Basis Problem for Hilbert modular forms.
The theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type. We generalize the Kudla-Millson relation between intersection numbers of cycles and Fourier coefficients of Siegel modular forms to the case where the cycles have local coefficients. Now the generating series of the cycles give rise to vector-valued Siegel modular forms. The underlying correspondence between the highest weights of the orthogonal and the symplectic group coincides with the one obtained by Adams for which we provide a geometric interpretation.
We give a comprehensive treatment of the transformation laws of theta functions from an algebro-geometric perspective, that is, in terms of moduli of abelian schemes. This is accomplished by introducing geometric notions of theta-descent structures, metaplectic stacks, and bundles of half-forms, whose analytic incarnations underlie different aspects of the classical transformation laws. As an application, we lay the foundations for a geometric theory of modular forms of half-integral weight and, more generally, for modular forms taking values in the Weil representation. We discuss further appl…
In this paper we give an explicit formula for the number of subgroups of the modular group of a given index that are genus zero and torsion-free and a formula for their conjugacy classes. We do so by exhibiting a correspondence between these groups and the trivalent maps on a sphere. We focus on the particular case of the subgroups of index 18 which have some interesting geometric properties.
In this article we introduce Hecke operators on the differential algebra of geometric quasi-modular forms. As an application for each natural number $d$ we construct a vector field in six dimensions which determines uniquely the polynomial relations between the Eisenstein series of weight 2,4 and 6 and their transformation under multiplication of the argument by $d$, and in particular, it determines uniquely the modular curve of degree $d$ isogenies between elliptic curves.
Regularized Integrals on Riemann Surfaces and Modular Forms
本文提出一种基于共形几何的内禀正则化方法，用于在Riemann曲面上对具有任意全纯极点的微分形式进行积分，并应用于建立位形空间（含Feynman图积分）的解析理论，证明了椭圆曲线上正则化图积分的殆全纯模性质，给出了有序A-圈积分拟模性的几何证明与组合公式。
We investigate geometric and combinatorial aspects of the mysterious relationship between the action of the motivic Galois group on the motivic fundamental group of the projective line punctured at zero, infinity, and N-th roots of unity, and the geometry of modular manifolds for the congruence subgroup Y(m;N) of GL(m,Z) fixing (0, ... , 0, 1) modulo N. To achieve this, we consider a canonical collection of elements in the image of the motivic Galois Lie algebra, spanning it over Q. These elements are the motivic correlators at N-th roots of unity. We assign to them chains in the complex compu…
We give a general recipe for explicitly constructing asymptotically optimal towers of modular curves such as {X_0(l^n): n=1,2,3,...}. We illustrate the method by giving equations for eight towers with various geometric features. We conclude by observing that such towers are all of a specific recursive form, and speculate that perhaps every tower of this form that attains the Drinfeld-Vladut bound is modular.
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