Representation of modular invariant function by generators of a modular function field
Noburo Ishii
给出了用模函数域的生成元表示模不变函数的方法。
We give a method of representing the modular invariant function by generators of a modular function field.
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We define a modular function which is a generalization of the elliptic modular lambda function. We show this function and the modular invariant function generate the modular function field with respect to the principal congruence subgroup. Further we study its values at imaginary quadratic points.
For a positive integer $N$ divisible by $4,5,6,7$ or $9$, let $\mathcal{O}_{1,N}(\mathbb{Q})$ be the ring of weakly holomorphic modular functions for the congruence subgroup $Γ_1(N)$ with rational Fourier coefficients. We present explicit generators of the ring $\mathcal{O}_{1,N}(\mathbb{Q})$ over $\mathbb{Q}$ by making use of modular units which have infinite product expansions.
For positive integers $g$ and $N$, let $\mathcal{F}_N$ be the field of meromorphic Siegel modular functions of genus $g$ and level $N$ whose Fourier coefficients belong to the $N$th cyclotomic field. We present explicit generators of $\mathcal{F}_N$ over $\mathcal{F}_1$ in terms of quotient of theta constants, when $g\geq2$ and $N\geq 3$.
We give bounds on the degree of generators for the ideal of relations of the graded algebras of modular forms with coefficients in $\mathbb{Q}$ over congruence subgroups $Γ_0(N)$ for $N$ satisfying some congruence conditions and for $Γ_1(N)$. We give similar bounds for the graded $\mathbb{Z}[\frac{1}{N}]$-algebra of modular forms on $Γ_1(N)$ with coefficients in $\mathbb{Z}[\frac{1}{N}]$. For a prime $p \geq 5$, we give a lower bound on the highest weight appearing in a minimal list of generators for $Γ_0(p)$, and we identify a set of generators for the graded algebra $M(Γ_0(p),\mathbb{Z})$ of…
We study graded rings of modular forms over congruence subgroups, with coefficients in a subring $A$ of $\mathbb{C}$, and specifically the highest weight needed to generate these rings as $A$-algebras. In particular, we determine upper bounds, independent of $N$, for the highest needed weight that generates the $\mathbb{C}$-algebras of modular forms over $Γ(N)$, $Γ_1(N)$ and $Γ_0(N)$ with some conditions on $N$. For $N \geq 5$, we prove that the $\mathbb{Z}[1/N]$-algebra of modular forms over $Γ_1(N)$ with coefficients in $\mathbb{Z}[1/N]$ is generated in weight at most 3. We give an algorithm…
Explicit generators of the space of modular forms
本文通过Rankin-Cohen括号构造了权为κ、水平为1的尖点形式空间的显式生成集，并给出了其对偶空间的显式周期生成集。
We construct generators for modules of vector-valued Picard modular forms on a unitary group of type (2,1) over the Eisenstein integers. We also calculate eigenvalues of Hecke operators acting on cusp forms.
In this paper we will describe all vector-valued Siegel modular forms of degree 2 and weight ${\rm Sym}^6({\rm St}) \otimes \det^{k}({\rm St})$ with $k$ odd. These vector-valued forms constitute a module over the ring of classical Siegel modular forms of degree 2 and even weight and this module turns out to be free. In order to find generators, we generalize certain Rankin-Cohen differential operators on triples of classical Siegel modular forms that were first considered by Ibukiyama and we find a Rankin-Cohen bracket on vector-valued Siegel modular forms.
We give two explicit sets of generators of the group of invertible regular functions over QQ on the modular curve Y1(N). The first set of generators is very surprising. It is essentially the set of defining equations of Y1(k) for k <= N/2 when all these modular curves are simultaneously embedded into the affine plane, and this proves a conjecture of Derickx and Van Hoeij. This set of generators is an elliptic divisibility sequence in the sense that it satisfies the same recurrence relation as the elliptic division polynomials. The second set of generators is explicit in terms of classical anal…
Hashimoto and Ueda determined the weights of generators of the graded ring of modular forms on the Cayley half-space of degree two. In this paper we describe explicit generators. We show that the graded ring can be generated by Eisenstein series by calculating the dimensions of the homogeneous components. Moreover we give two distinct constructions for the skew-symmetric form of weight 252.
In our first article in this series ("Modular Invariant of Quantum Tori I: Definitions Nonstandard and Standard" arXiv:0909.0143) a modular invariant of quantum tori was defined. In this paper, we consider the case of the quantum torus associated to the golden mean. We show that the modular invariant is approximately 9538.249655644 by producing an explicit formula for it involving weighted versions of the Rogers-Ramanujan functions.
In this paper, we obtain two effective bounds for the $j$-invariant of integral points on certain modular curves which has positive genus and less than three cusps.
In this paper, we give some effective bounds for the $j$-invariant of integral points on arbitrary modular curves over arbitrary number fields assuming that the number of cusps is not less than 3.
For numerical semigroups with a specified list of (not necessarily minimal) generators, we describe the asymptotic distribution of factorization lengths with respect to an arbitrary modulus. In particular, we prove that the factorization lengths are equidistributed across all congruence classes that are not trivially ruled out by modular considerations.
In this paper we show that under certain condition the Fontaine--Mazur $L$-invariant for a Hilbert eigenform coincides with its Teitelbaum type $L$-invariant, and thus prove a conjecture of Chida, Mok and Park.
Inspired by prior work of Bruinier and Ono and Mertens and Rolen, we study class polynomials for non-holomorphic modular functions arising from modular forms of negative weight. In particular, we give general conditions for the irreducibility of class polynomials. This allows us to easily generate infintely many new class invariants.
Explicit height bounds on modular polynomials for the elliptic $j$-invariant, cube root of $j$, and Weber modular function $\mathfrak f$
研究改进椭圆\(j\)不变量等相关模多项式显式上界，通过改进Breuer等人给出的界，还证明了\(j\)立方根及韦伯模函数\(\mathfrak f\)的模多项式高度新显式上界。
In this paper we describe a method for computing a basis for the space of weight $2$ cusp forms invariant under a non-split Cartan subgroup of prime level $p$. As an application we compute, for certain small values of $p$, explicit equations over $\bf Q$ for the canonical embeddings of the associated modular curves.
Let $K$ be an imaginary quadratic field where $p$ splits. We study signed Selmer groups for non-ordinary modular forms over the anticyclotomic $\mathbf{Z}_p$-extension of $K$, showing that their $μ$-invariants vanish. This generalizes and gives a new proof of a recent result of Matar on the vanishing of the $μ$-invariants of plus and minus signed Selmer groups for elliptic curves.
Given a cuspidal Hilbert modular eigenform $π$ of parallel weight 2 and a nonarchimedian place $\mathfrak p$ of the underlying totally real field such that the local component of $π$ at $\mathfrak p$ is the Steinberg representation, one can associate two types of $\mathscr L$-invariants, one defined in terms of the cohomology of arithmetic groups and the other in terms of the Galois representation associated to $π$. We show that the $\mathscr L$-invariants are the same.
The modular invariant of rank 1 Drinfeld modules is introduced and used to formulate and prove an exact analog of the Weber-Fueter theorem for global function fields. The main ingredient in the proof is a version of Shimura's Main Theorem of Complex Multiplication for global function fields, which is also proved here.
On $λ$-invariants of congruent modular forms in the anticyclotomic, indefinite setting
本文在Selmer群正秩的反cyclotomic不定情形下，提出了比较模形式$\lambda$-不变量的精确可计算公式，改进了Hatley-Lei的不可计算误差项结果。
We provide explicit bounds on the difference of heights of the $j$-invariants of isogenous elliptic curves defined over $\overline{\mathbb{Q}}$. The first one is reminiscent of a classical estimate for the Faltings height of isogenous abelian varieties, which is indeed used in the proof. We also use an explicit version of Silverman's inequality and isogeny estimates by Gaudron and Rémond. We give applications to the study of Vélu's formulas and of modular polynomials.
The purpose of this paper is to prove the equality between the algebraic Iwasawa $λ$-invariant and the analytic Iwasawa $λ$-invariant for a Hilbert cusp form of parallel weight $2$ at an ordinary prime $p$ when the associated residual Galois representation is reducible. This is a generalization of a result of R. Greenberg and V. Vatsal.
On anticyclotomic mu-invariants of modular forms
设f是权为2的平凡特征标的模形式。还固定一个虚二次域K。利用Bertolini-Darmon和Vatsal的工作，研究了K的反分圆Zp-扩张上f的p-adic Selmer群的μ-不变量.特别是，我们验证了在这种情况下的主要猜想的mu部分。这一结果的证明是基于对模形式的同余的分析，从而导致了一个理论上的定量版本的水平降低（我们验证的情况下，马祖尔的原则适用）。
$λ$-invariant stability in families of modular Galois representations
本文从定量角度研究权为 2 的模形式族中 Iwasawa $λ$-不变量的变化，提供了 $λ$-不变量增长或保持稳定的频率下界。
Let $F$ be a totally real field of degree $g$, and let $p$ be a prime number. We construct $g$ partial Hasse invariants on the characteristic $p$ fiber of the Pappas-Rapoport splitting model of the Hilbert modular variety for $F$ with level prime to $p$, extending the usual partial Hasse invariants defined over the Rapoport locus. In particular, when $p$ ramifies in $F$, we solve the problem of lack of partial Hasse invariants. Using the stratification induced by these generalized partial Hasse invariants on the splitting model, we prove in complete generality the existence of Galois pseudo-re…
Invariants for the Weil representation and Modular Units for Orthogonal Groups of Signature (2,2)
该研究证明了含自迷向子群的判别群的Weil表示不变量空间由自迷向子群特征函数张成，并应用该结果通过Borcherds乘积构造了符号(2,2)正交群的模单位。
We compute the arithmetic L-invariants (of Greenberg-Benois) of twists of symmetric powers of p-adic Galois representations attached to Iwahori level Hilbert modular forms (under some technical conditions). Our method uses the automorphy of symmetric powers and the study of analytic Galois representations on p-adic families of automorphic forms over symplectic and unitary groups. Combining these families with some explicit plethysm in the representation theory of GL(2), we construct global Galois cohomology classes with coefficients in the symmetric powers and provide formulae for the L-invari…
This is the first of a series of two papers in which we present a solution to Manin's Real Multiplication program -- an approach to Hilbert's 12th problem for real quadratic extensions of $\mathbb{Q}$ -- in positive characteristic, using quantum analogs of the exponential function and the modular invariant. In this first paper, we treat the problem of Hilbert class field generation. If $k=\mathbb{F}_{q}(T)$ and $k_{\infty}$ is the analytic completion of $k$, we introduce the quantum modular invariant \[ j^{\rm qt}: k_{\infty}\multimap k_{\infty}\] as a multivalued, modular invariant function. …
Number of $K$-rational points with given $j$-invariant on modular curves
本文研究了模曲线中具有给定j不变量的K有理点数目，并提出算法确定CM点数目，同时探讨了椭圆曲线的循环同源性质。
We give generators and relations for the graded rings of Hermitian modular forms of degree two over the rings of integers in $\mathbb{Q}(\sqrt{-7})$ and $\mathbb{Q}(\sqrt{-11})$. In both cases we prove that the subrings of symmetric modular forms are generated by Maass lifts. The computation uses a reduction process against Borcherds products which also leads to a dimension formula for the spaces of modular forms.
The genus $4$ modular curve $X_{ns}(11)$ attached to a non-split Cartan group of level $11$ admits a model defined over $\mathbb{Q}$. We compute generators for its function field in terms of Siegel modular functions. We also show that its Jacobian is isomorphic over $\mathbb{Q}$ to the new part of the Jacobian of the classical modular curve $X_0(121)$.
Bianchi's elliptic quintic curves and several modular function fields of level ten
该研究利用拉马努金函数显式描述经典Bianchi椭圆五次曲线上的两个挠点，并以此为基础给出若干10级模函数域的生成元和定义方程。
Explicit bounds on the coefficients of modular polynomials for the elliptic $j$-invariant
本文研究椭圆 $j$-不变量模多项式 $Φ_N$ 的系数规模，给出适用于所有 $N\geq1$ 的显式上界，并证明其主项随 $N$ 趋于无穷时渐近最优。
In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary oc-tics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume values whose denominators are products of powers of primes of bad reduction for the associated hyperelliptic curves. We illustrate our theorem with explicit computations. This work is motivated by the study of the value of these modular functions at CM points of the Siegel upper-half space, which, if their…
N-systems, class polynomials for double eta-quotients and singular values of J-invariant function
Enge和Schertz给出了利用双η-商构造有限域上椭圆曲线的方法。在他们的方法中，需要计算出有限域上的模方程的解所对应的椭圆曲线上的有理点，因为我们事先无法知道模方程的哪个解是模不变量所对应的解。给出了模不变量是模多项式的重根的条件。在此基础上，提出了一种减少计算有理点个数的方法。
Recently, Bruinier and Ono found an algebraic formula for the partition function in terms of traces of singular moduli of a certain non-holomorphic modular function. In this paper we prove that the rational polynomial having these singuar moduli as zeros is (essentially) irreducible, settling a question of Bruinier and Ono. The proof uses careful analytic estimates together with some related work of Dewar and Murty, as well as extensive numerical calculations of Sutherland.
K3 Surfaces and Orthogonal Modular Forms
本文研究了具有特定自同构群和Picard秩的K3曲面的模形式生成元，并通过Weierstrass型方程揭示其结构。
The quantum modular invariant of a real number is defined as a discontinuous, PGL(2,Z)-invariant multi-valued map using the distance-to-the-nearest-integer function. On the rationals, the quantum modular invariant is shown to be infinity and for quadratic irrationalities PARI/GP experiments suggest it is a finite set. In the case of the golden mean, we produce explicit formulas involving weighted versions of the Rogers-Ramanujan functions for the experimental supremum and infimum of its quantum modular invariant. We then define a universal modular invariant as a continuous and single valued ma…
Non-holomorphic modular forms from zeta generators
本文利用 zeta 生成元分析等变迭代 Eisenstein 积分，从奇 Riemann zeta 值推导出多重 zeta 值与模图形式的出现，并构造三重 Eisenstein 积分的模补全，在 Brown 框架下显式实现模图形式。
Modular forms via invariant theory
本文提出利用不变量理论从已知模形式构造新模形式的两种方法，分别推广了Rankin-Cohen括号并处理了亏格大于1的向量值模形式。
Modular invariance, modular identities and supersingular j-invariants
对于每一个$k$维模不变向量空间，我们将权为$2k$的$SL(2,\mathbb{Z})$上的模形式联系起来.我们探讨了这种形式的数论性质，并找到了它为零的一个充分条件，从而产生模恒等式（例如，Ramanujan-Watson的模恒等式。此外，我们集中在一个家庭的模不变空间来自适当的二维空间，通过对称的权力建设。特别地，我们考虑了一类$D_4^{(1)}$型仿射Kac-Moody李代数的由某些一级模的分次维数所张成的二维空间。在这种情况下，与$k$次对称幂相关联的模形式的约简模素数$p=2k+3 \geq 5$将超奇椭圆曲线分类为特征$p$。这种构造也对金子和扎吉尔研究的某些模形式给出了新的解释。
This is a survey based on the construction of Siegel modular forms of degree 2 and 3 using invariant theory in joint work with Fabien Cléry and Carel Faber.
Modular Symbols and Equivariant Birational Invariants
本文研究经典模符号与有限阿贝尔群相关的Kontsevich-Pestun-Tschinkel群之间的关系，探讨其等变双有理不变量。
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。
Consider the full modular group $\sf{PSL}_{2}(\mathbb{Z})$ with presentation $\langle U,S|U^3,S^2\rangle$. Motivated by our investigations on quasi-modular forms and the Minkowski question mark function (so that this paper might be considered as a necessary appendix), we are lead to the following natural question. Some words in the alphabet $\{U,S\}$ are equal to the unity; for example, $USU^3SU^2$ is such a word of length $8$, and $USU^3SUSU^3S^3U$ is such a word of length $15$. Given $n\in\mathbb{N}_{0}$. Find the number of words of length $n$ which are equal to the unity. This is the new en…
Siegel modular forms of degree three and invariants of ternary quartics
本研究确定了三次Siegel模形式分次环的结构，明确其由19个模形式生成并找到7个权的齐次参数系，同时建立了三元四次式的Dixmier-Ohno不变量与这些生成元的完整对应。
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。
The $E_8$ root lattice can be constructed from the modular curve $X(13)$ by the invariant theory for the simple group $\text{PSL}(2, 13)$. This gives a different construction of the $E_8$ root lattice. It also gives an explicit construction of the modular curve $X(13)$.
An invariant of a model of genus one curve is a polynomial in the coefficients of the model that is stable under certain linear transformations. The classical example of an invariant is the discriminant, which characterizes the singularity of models. The ring of invariants of genus one models over a field is generated by two elements. Fisher normalized these invariants for models of degree n=2,3,4 in such a way that these invariants are moreover defined over the integers. We provide an alternative way to express these normalized invariants using modular forms. This method relies on a direct co…
Infinite Families of Quantum Modular 3-Manifold Invariants
本文利用AJK理论扩展了3-流形的格上同调与BPS q-级数，首次为无穷多族负定plumbed 3-流形计算了AJK级数，并构造了可视为WRT不变量形变且具量子模性质的无穷多族不变量。
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。
In this paper, we study quantum modular forms in connection to quantum invariants of plumbed 3-manifolds introduced recently by Gukov, Pei, Putrov, and Vafa. We explicitly compute these invariants for any $3$-leg star plumbing graphs whose associated matrix is unimodular and positive definite. For these graphs we confirm a quantum modularity conjecture of Gukov. We also analyze the invariants for general $n$-leg star graphs with unimodular plumbing matrices, and prove that they can be expressed as linear combinations of quantum modular forms.
Gukov-Pei-Putrov-Vafa constructed $q$-series invariants called homological blocks in a physical way in order to categorify Witten-Reshetikhin-Turaev (WRT) invariants and conjectured that radial limits of homological blocks are WRT invariants. In this paper, we prove their conjecture for unimodular H-graphs. As a consequence, it turns out that the WRT invariants of H-graphs yield quantum modular forms of depth two and of weight one with the quantum set $\mathbb{Q}$. In the course of the proof of our main theorem, we first write the invariants as finite sums of rational functions. We second carr…
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