Eyal Z. Goren, Payman L Kassaei
We obtain new results on the geometry of Hilbert modular varieties in positive characteristic and morphisms between them. Using these results and methods of rigid geometry, we develop a theory of canonical subgroups for abelian varieties with real multiplication.
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These are the notes from a survey talk given at Arbeitstagung 2001 covering the author's work with Lev Borisov and Sorin Popescu on toric varieties, modular forms, and equations of modular curves.
This is a survey article about Siegel modular varieties over the complex numbers. It is written mostly from the point of view of moduli of abelian varieties, especially surfaces. We cover compactification of Siegel modular varieties; classification of the compactified varieties by Kodaira dimension, etc.; moduli of abelian surfaces and especially applications of the lifting of Jacobi forms to modular forms; projective models of some special Siegel modular 3-folds; non-principally polarized abelian surfaces; and constructing degenerating families of abelian varieties.
We compute the tautological ring for a Hilbert modular variety at an unramified prime. The method generalises that of van der Geer from the Siegel case.
We consider several maps that occur naturally between modular Shimura varieties, Hilbert-Blumenthal varieties and the moduli spaces of polarized abelian varieties when forgetting certain endomorphism structures. We prove that, up to birational equivalences, these forgetful maps coincide with the natural projection by suitable abelian groups of Atkin-Lehner involutions.
We give an abstract characterization of the Satake compactification of a general Drinfeld modular variety. We prove that it exists and is unique up to unique isomorphism, though we do not give an explicit stratification by Drinfeld modular varieties of smaller rank which is also expected. We construct a natural ample invertible sheaf on it, such that the global sections of its $k$-th power form the space of (algebraic) Drinfeld modular forms of weight $k$. We show how the Satake compactification and modular forms behave under all natural morphisms between Drinfeld modular varieties; in particu…
We use vanishing results for sheaf cohomology on Siegel modular varieties to study two lifting problems: (a) When can Siegel modular forms (mod p) be lifted to characteristic zero? This uses and extends previous results for cusp forms by Stroh and Lan-Suh. (b) When is the restriction of Siegel modular forms to the boundary of the moduli space a surjective map? We investigate this question in arbitrary characteristic, generalising analytic results of Weissauer and Arakawa.
We define modular equations in the setting of PEL Shimura varieties as equations describing Hecke correspondences, and prove upper bounds on their degrees and heights. This extends known results about elliptic modular polynomials, and implies complexity bounds for number-theoretic algorithms using these modular equations. In particular, we obtain tight degree bounds for modular equations of Siegel and Hilbert type for abelian surfaces.
We construct local charts for the ramified cusps of the Hilbert modular variety of level $Γ_1(\mathfrak{c},\mathfrak{n})$. This allows us, following closely M. Rapoport and C.-L. Chai, to construct arithmetic toroidal and minimal compactifications of these modular varieties. Some applications to modular forms are given. See our joint work with J. Tilouine for other applications.
Embeddings of Certain Exceptional Shimura Varieties into Siegel Modular Varieties
本文定义了一类包含某些例外群局部Shimura变种的局部Shimura变种，并为该类构建了从(G, μ)-显示到p-可除群的函子。作为应用，证明了对于该类，局部Shimura变种在无穷级可代表且完美oid。考虑该类的全局对应物，将某些例外Shimura变种嵌入到Siegel模变种中。特别是，证明它们在无穷级是完美oid。
Let $\cal A$ be a maximal (or more generally a hereditary) order in a central simple algebra over a global field $F$ of positive characteristic. We study the reduction of the modular scheme of $\cal A$-elliptic sheaves at all places of $F$. We show that some of these varieties - for different $\cal A$ - are twists of each other. This result can be viewed as a global analogue of the Cherednik-Drinfeld theorem for these varieties.
Explicit Kodaira-Spencer map over Hilbert modular varieties
本文显式计算 Hilbert-Siegel 模簇和扭曲 Hilbert 模簇上的 Kodaira-Spencer 映射及其对 Hodge 丛度量的影响，推广了 arXiv:2205.11334 的结果。
We compare the asymptotic grows of the number of rational points on modular varieties of D-elliptic sheaves over finite fields to the grows of their Betti numbers as the degree of the level tends to infinity. This is a generalization to higher dimensions of a well-known result for modular curves. As a consequence of the main result, we also produce a new asymptotically optimal sequence of curves.
Non-critical equivariant L-values of modular abelian varieties
本文证明了数域上强模阿贝尔簇非临界L值的Beilinson猜想等变版本，并应用于非CM强模Q曲线，得到Zagier和Deninger猜想的弱形式。
Let $G$ be a semiabelian variety and $C$ a curve in $G$ that is not contained in a proper algebraic subgroup of $G$. In this situation, conjectures of Pink and Zilber imply that there are at most finitely many points contained in the so-called unlikely intersections of $C$ with subgroups of codimension at least $2$. In this note, we establish this assertion for general semiabelian varieties over $\bar{\mathbb{Q}}$. This extends results of Maurin and Bombieri, Habegger, Masser, and Zannier in the toric case as well as Habegger and Pila in the abelian case.
Varieties with prescribed finite unramified Brauer groups and subgroups precisely obstructing the Hasse principle
针对数域上簇的Hasse原理阻碍问题，证明对任意非零有限阿贝尔群对$B_0\subset B$，存在射影线上的范丛簇$X$，其非分歧Brauer群同构于$B$且$B_0$是$B$中阻碍Hasse原理的最小子群。
There is a mysterious connection between the multiple polylogarithms at N-th roots of unity and modular varieties. In this paper we "explain" it in the simplest case of the double logarithm. We introduce an Euler complex data on modular curves. It includes a length two complex on every modular curve. Their second cohomology groups recover the Beilinson-Kato Euler system in K_2 of modular curves. We show that the above connection in the double logarithm case is provided by the specialization at a cusp of the Euler complex data on the modular curve Y_1(N). Furthermore, specializing the Euler com…
We recall first the analytic theory of the Hilbert modular varieties of level $Γ_1(\mathfrak{c},\mathfrak{n})$ and their compactifications. We construct arithmetic toroidal compactifications of the universal Hilbert-Blumenthal abelian variety (and of its fiber products), over the compactifications of the Hilbert modular variety constructed by the first author. We give several applications related to Hodge theory, Hilbert-Jacobi modular forms and Hilbert modular forms of half-integer weight.
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.
By combining theorems of Drinfeld and Strauch, we show that the monodromy representation on the special fibre of a Drinfeld modular variety, with level not divisible by the characteristic, is surjective. We illustrate this result in the special case of Drinfeld $\mathbb{F}_q[t]$-modules in level $t$, and apply this to show that the Kronecker factors of a Drinfeld modular polynomial in rank $r$ are irreducible.
We give new examples of weight three cusp forms on noncongruence subgroups of SL(2, Z) whose Scholl representation is modular and which satisfy three term Atkin-Swinnerton-Dyer relations.
We construct $16$ reflection groups $Γ$ acting on symmetric domains $\mathcal{D}$ of Cartan type IV, for which the graded algebras of modular forms are freely generated by forms of the same weight, and in particular the Satake--Baily--Borel compactification of $\mathcal{D} / Γ$ is isomorphic to a projective space. Four of these are previously known results of Freitag--Salvati Manni, Matsumoto, Perna and Runge. In addition we find several new modular groups of orthogonal type whose algebras of modular forms are freely generated.
We prove that for any even lattice L of signature (2,n), the modular variety defined by the orthogonal group of the lattice L+mE_8 is of general type when m is sufficiently large.
Partial Hasse invariants for genus zero curves in Hilbert modular varieties
本文研究了Hilbert模形式中零 genus 曲线的部分Hassé不变量的特征零提升，并通过计算Teichmüller曲线得到类似Deuring的公式。
On the cohomological dimension of Siegel modular varieties and the modularity of formal Siegel modular forms
本文证明亏格g≥2时Siegel模簇的凝聚上同调维数至多为g(g+1)/2-2，由此推出其紧化边界满足Grothendieck-Lefschetz条件，从而形式Siegel模形式自动成为经典模形式，推广了Bruinier和Raum的工作。
We discuss the following two problems: 1) The properties of the multiple zeta-values and their generalizations, multiple polylogarithms at N-th roots of unity; 2) The action of the absolute Galois group on the pro-l-completion of the fundamental group of the projective line without zero, infinity, and all N-th roots of unity; and a surprising connection of these problems with the geometry and topology of modular varieties for GL_m.
On abelian points of varieties intersecting subgroups in a torus
在自然条件下，研究了闭不可约子簇X与托尔中连通代数子群的并集的非异常子集上的阿贝尔点集是有限的，并推广了Ostafe、Sha、Shparlinski和Zannier(2017)的结果，同时在非连通代数子群情况下推广了其结构定理，并在曲线和算术动力系统中得到了相关结果。
This article has three goals. First, we generalize the result of Deuring and Serre on the characterization of supersingular locus of modular curves to all Shimura varieties given by totally indefinite quaternion algebras over totally real number fields. Second, we generalize the result of Ribet on arithmetic level raising to such Shimura varieties in the inert case. Third, as an application to number theory, we use the previous results to study the Selmer group of certain triple product motive of an elliptic curve, in the context of the Bloch--Kato conjecture.
We explore an analogue of the André-Oort conjecture for subvarieties of Drinfeld modular varieties. The conjecture states that a subvariety $X$ of a Drinfeld modular variety contains a Zariski-dense set of complex multiplication (CM) points if and only if $X$ is a "special" subvariety (i.e. $X$ is defined by requiring additional endomorphisms). We prove this conjecture in two cases. Firstly when $X$ contains a Zariski-dense set of CM points with a certain behaviour above a fixed prime (which is the case if these CM points lie in one Hecke orbit), and secondly when $X$ is a curve containing inf…
We expand on Nekovář's construction of the plectic half transfer to define a plectic Galois action on Hilbert modular varieties. More precisely, we study in a unifying fashion Shimura varieties associated to groups that differ only in the centre from $R_{F/\mathbb{Q}}{\rm GL}_2$. We define plectic Galois actions on the CM points and on the set of connected components of these Shimura varieties, and show that these two actions are compatible. This extends the plectic conjecture of Nekovář--Scholl.
In this paper, we introduce an algebro-geometric formulation for Faltings' theorem on diophantine approximation on abelian varieties using an improvement of Faltings-Wustholz observation over number fields. In fact, we prove that, for any geometrically irreducible sub-variety E of an abelian variety A and any finitely generated subgroup F of A(C) we have an estimate of the form d_v(E;x) >cH(x)^d for for some constant c where d_v(E;x) denotes the distance of a point x in F outside E and v is a place of K. This was proved before, only for F being the set of rational points of A over a number fie…
We study moduli spaces of abelian varieties in positive characteristic, more specifically the moduli space of principally polarized abelian varieties on the one hand, and the analogous space with Iwahori type level structure, on the other hand. We investigate the Ekedahl-Oort stratification on the former, the Kottwitz-Rapoport stratification on the latter, and their relationship. In this way, we obtain structural results about the supersingular locus in the case of Iwahori level structure, for instance a formula for its dimension in case $g$ is even.
In their paper Livné and Yui (math.AG/0304497) discuss several examples of non-rigid Calabi-Yau varieties which admit semi-stable K3-fibrations with 6 singular fibres over a base which is a rational modular curve. They also establish the modularity of the L-function of these examples. The purpose of this note is to point out that the examples which were listed in their paper, but which do not lead to semi-stable fibrations, are still modular in the sense that their L-function is associated to modular forms. We treat the case associated to the group Gamma_1(7) in detail, but our technique also …
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…
Mazur, Tate, and Teitelbaum gave a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for elliptic curves. We provide a generalization of their conjecture in the good ordinary case to higher dimensional modular abelian varieties over the rationals by constructing the p-adic L-function of a modular abelian variety and showing that it satisfies the appropriate interpolation property. This relies on a careful normalization of the p-adic L-function, which we achieve by a comparison of periods. Our generalization agrees with the conjecture of Mazur, Tate, Teitelbaum in dimension 1 and the …
Let $N\subset \RR^{r}$ be a lattice, and let $°\colon N \to \CC$ be a piecewise-linear function that is linear on the cones of a complete rational polyhedral fan. Under certain conditions on $°$, the data $(N,°)$ determines a function $f\colon {\HHH}\to \CC$ that is a holomorphic modular form of weight $r$ for the congruence subgroup $Γ_{1} (l) $. Moreover, by considering all possible pairs $(N ,°)$, we obtain a natural subring ${\TTT} (l)$ of modular forms with respect to $Γ_{1} (l) $. We construct an explicit set of generators for $\TTT (l)$, and show that ${\TTT} (l)$ is stable under the ac…
We show that certain abelian varieties over $\Q$ with bad reduction at one prime only are modular by using methods based on the tables of Odlyzko and class field theory.
Irregular cusps of orthogonal modular varieties
本文研究了正交模簇非正则尖点的环面紧化，澄清了 Kodaira 维数的尖点形式判据，并证明了大多数偶格的稳定正交群无非正则尖点。
Let f be a weight two newform for Gamma_1(N) without complex multiplication. In this article we study the conductor of the absolutely simple factors B of the variety A_f over certain number fields L. The strategy we follow is to compute the restriction of scalars Res_{L/\Q}(B), and then to apply Milne's formula for the conductor of the restriction of scalars. In this way we obtain an expression for the local exponents of the conductor N_L(B). Under some hypothesis it is possible to give global formulas relating this conductor with N. For instance, if N is squarefree we find that N_L(B) belongs…
Formal Siegel modular forms for arithmetic subgroups
本文研究算术子群的形式西格尔模形式，证明在仿射覆盖数有上界时其均可由经典全纯西格尔模形式的傅里叶-雅可比展开给出，并验证属2至4时该条件恒成立，进而应用于属2无平方水平参数模群。
On the p-adic integration over Igusa towers of Siegel modular varieties
本文提出了一种针对Igusa塔的显式p进积分理论，用于研究模Siegel流形的互反律。
In this paper, we study the torsion subgroup of $J_0(N)$ over the field generated by those points in the cuspidal group, where $N$ is an odd positive integer. We prove that, considered as Hecke modules, this group and the cuspidal subgroup are both supported at the maximal Eisenstein ideals.
Classically, congruence subgroups of the modular group, which can be described by congruence relations, play important roles in group theory and modular forms. In reality, the majority of finite index subgroups of the modular group are noncongruence. These groups as well as their modular forms are central players of this survey article. Differences between congruence and noncongruence subgroups and modular forms will be discussed. We will mainly focus on three interesting aspects of modular forms for noncongruence subgroups: the unbounded denominator property, modularity of the Galois represen…
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.
We formulate a question regarding uniform versions of "large Galois image properties" for modular abelian varieties of higher dimension, generalizing the well-known case of elliptic curves. We then answer our question affirmatively in the exceptional image case, and provide lower estimates for uniform bounds in the remaining cases.
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.
In this paper, we determine all modular Jacobian varieties $J_1(M,MN)$ over the number field $\mathbb{Q}(ζ_M)$ with the Mordell-Weil rank zero following the method of Derickx, Etropolski, van Hoeij, Morrow, and Zureick-Brown.
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。
In this paper, we will study modular Abelian varieties with odd congruence numbers by examining the cuspidal subgroup of $J_0(N)$. We will show that the conductor of such Abelian varieties must be of a special type. For example, if $N$ is the conductor of an absolutely simple modular Abelian variety with an odd congruence number, then $N$ has at most two prime divisors, and if $N$ is odd, then $N=p^α$ or $N=pq$ for some prime $p$ and $q$. In the second half of this paper, we will focus on modular elliptic curves with odd modular degree. Our results, combined with the work of Agashe, Ribet, and…
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 $p$.
In this paper we will use experimental and computational methods to find modular forms for non-congruence subgroups, and the modular forms for congruence subgroups that they are associated with via the Atkin--Swinnerton-Dyer correspondence. We also prove a generalization of a criterion due to Ligozat for an eta-quotient to be a modular function.
We prove that up to scaling there are only finitely many integral lattices L of signature (2,n) with n>20 or n=17 such that the modular variety defined by the orthogonal group of L is not of general type. In particular, when n>107, every modular variety defined by an arithmetic group for a rational quadratic form of signature (2,n) is of general type. We also obtain similar finiteness in n>8 for the stable orthogonal groups. As a byproduct we derive finiteness of lattices admitting reflective modular form of bounded vanishing order, which proves a conjecture of Gritsenko and Nikulin.
Modular Zilber-Pink for geometrically generic varieties
该研究证明，对满足特定投影条件的$\mathrm{Y}(1)^n$的子簇，模块化齐尔伯-平克猜想成立。
Let $D$ be an odd square-free positive integer and $C$ a divisor of $D$. For any quadratic character $χ$ modulo $C$, we prove that the $χ$-part of the group $J_0(DC)_\text{tor}$ of torsion points of $J_0(DC)$ coincides with the $χ$-part of its cuspidal subgroup, away from those primes of bad reduction or where possible congruences between oldforms and newforms occur.
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。
In this article we study a differential algebra of modular-type functions attached to the periods of a one parameter family of Calabi-Yau varieties which is mirror dual to the universal family of quintic threefolds. Such an algebra is generated by seven functions satisfying functional and differential equations in parallel to the modular functional equations of classical Eisenstein series and the Ramanujan differential equation. Our result is the first example of automorphic-type functions attached to varieties whose period domain is not Hermitian symmetric. It is a reformulation and realizati…
We give a computationally effective criterion for determining whether a finite-index subgroup of SL(2, Z) is a congruence subgroup, extending earlier work of Hsu for subgroups of PSL(2, Z).
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…
Runge's method is a tool to figure out integral points on curves effectively in terms of height. This method has been generalised to varieties of any dimension, unfortunately its conditions of application are often too restrictive. In this paper, we provide a further generalisation intended to be more flexible while still effective, and exemplify its applicability by giving finiteness results for integral points on some Siegel modular varieties. As a special case, we obtain a totally explicit finiteness result for integral points on the Siegel modular variety $A_2(2)$.
We generalize to Hilbert modular varieties of arbitrary dimension the work of W. Duke (Inventiones 1988) on the equidistribution of Heegner points and of primitive positively oriented closed geodesics in the Poincare upper half plane, subject to certain subconvexity results. We also prove vanishing results for limits of cuspidal Weyl sums associated with analogous problems for the Siegel upper half space of degree 2. In particular, these Weyl sums are associated with families of Humbert surfaces in Siegel 3-folds and of modular curves in these Humbert surfaces.
We present a database of several hundred modular forms up to and including weight six on noncongruence subgroups of index $\leq 17$. In addition, our database contains expressions for the Belyi map for genus zero subgroups and equations of the corresponding elliptic curves for genus one subgroups and numerical approximations of noncongruence Eisenstein series to 1500 digits precision.
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