L. J. P. Kilford
In this note we consider a question of Ono, concerning which spaces of classical modular forms can be generated by sums of $η$-quotients. We give some new examples of spaces of modular forms which can be generated as sums of $η$-quotients, and show that we can write all modular forms of level $Γ_0(N)$ as rational functions of $η$-products.
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We describe an algorithm for computing a $\Q$-rational model for the quotient of a modular curve by an automorphism group, under mild assumptions on the curve and the automorphisms, by determining $q$-expansions for a basis of the corresponding space of cusp forms. We also give a moduli interpretation for general morphisms between modular curves.
An eta-quotient of level $N$ is a modular form of the shape $f(z) = \prod_{δ| N} η(δz)^{r_δ}$. We study the problem of determining levels $N$ for which the graded ring of holomorphic modular forms for $Γ_{0}(N)$ is generated by (holomorphic, respectively weakly holomorphic) eta-quotients of level $N$. In addition, we prove that if $f(z)$ is a holomorphic modular form that is non-vanishing on the upper half plane and has integer Fourier coefficients at infinity, then $f(z)$ is an integer multiple of an eta-quotient. Finally, we use our results to determine the structure of the cuspidal subgroup…
Unexpected primes of good reduction in quotients of modular and Shimura curves
研究权重2、平方自由层数N且有固定阿特金 - 勒纳符号模式的新形式零维空间，借此对平方自由层数的模曲线和志村曲线阿特金 - 勒纳商中良好约化的意外素数进行分类。
Tetragonal modular quotients $X_0^+(N)$
本研究确定了所有$\bb Q$或$\bb C$-角率为4的模商曲线$X_0^+(N)$，并由此得到若干$X_0(N)$的$\bb Q$-角率为8的新案例，丰富了模曲线角率的相关结果。
From the Modularity Theorem proven by Wiles, Taylor, et al, we know that all elliptic curves are modular. It has been shown by Martin and Ono exactly which are represented by eta-quotients, and some examples of elliptic curves represented by modular forms that are linear combinations of eta-quotients have been given by Pathakjee, RosnBrick, and Yoong. In this paper, we first show that eta-quotients which are modular for any congruence subgroup of level $N$ coprime to $6$ can be viewed as modular for $Γ_0(N)$. We then categorize when even weight eta-quotients can exist in $M_k (Γ_1(p))$ and $M_…
Graded Algebras of Modular Forms and Sign Changes of Fourier Coefficients for Eta Quotients
本文确定了N=8,12,16,20,32时模形式分次代数的显式结构，给出eta商生成元与关系，并利用eta商基得到三个特定空间中傅里叶系数符号周期性的充要条件，最终找出所有符号周期性的eta商。
In this paper, we will study the arithmetic of the Eisenstein part of the modular Jacobians. In the first section, we introduce some general preliminaries of the arithmetic theory of modular curves that we will need later. In the second section, we give an example of modular abelian varieties due to Gross and study its properties in some details. In the third section, we define Eisenstein quotients of the modular Jacobians in general and give a criterion of the non-triviality of Heegner points on such Eisenstein quotients. The last two sections return to the concrete examples when the level of…
We prove a sub-convex estimate for the sup-norm of $L^2$-normalized holomorphic modular forms of weight $k$ on the upper half plane, with respect to the unit group of a quaternion division algebra over $\mf Q$. More precisely we show that when the $L^2$ norm of an eigenfunction $f$ is one, | f |_\infty \ll k^{1/2 - 12/131 + \varepsilon}, for any $\varepsilon>0$ and for all $k$ sufficiently large.
In [4] and [5], Folsom presents a family of modular units as higher-level analogues of the Rogers-Ramanujan $q$-continued fraction. These units are constructed from analytic solutions to the higher-order $q$-recurrence equations of Selberg. Here, we consider another family of modular units, which are quotients of Hall-Littlewood $q$-series that appear in the generalized Rogers-Ramanujan type identities of [6]. In analogy with the results of Folsom, we provide a formula for the rank of the subgroup these units generate and show that their specializations at the cusp $0$ generate a subgroup of t…
The well-known fact that all elliptic curves are modular, proven by Wiles, Taylor, Breuil, Conrad and Diamond, leaves open the question whether there exists a 'nice' representation of the modular form associated to each elliptic curve. Here we provide explicit representations of the modular forms associated to certain Legendre form elliptic curves 2_E_1(λ) as linear combinations of quotients of Dedekind's eta-function. We also give congruences for some of the modular forms' coefficients in terms of Gaussian hypergeometric functions.
Modular Units on $X_{1}( p)$ and Quotients of the Cuspidal Group
本文研究素数级 p≥5 时 X_1(p) 上的模单元群，给出其显式基，并利用该基数值研究尖点群结构及确定其一个特定的大商群。
Enge and Schertz gave the method of using the double eta-quotient for the construction of elliptic curves over finite fields. In their method, it is necessary to count the number of rational points of elliptic curves corresponding to solutions of the modular equation over a finite field, because in advance we can not know which solution of the modular equation is that corresponding to the modular invariant. We give a condition that the modular invariant is a multiple root of the modular polynomial. Consequently, we give a method to reduce the amount of computation in the process of counting th…
We use the method of quadratic Chabauty on the quotients $X_0^+(N)$ of modular curves $X_0(N)$ by their Fricke involutions to provably compute all the rational points of these curves for prime levels $N$ of genus four, five, and six. We find that the only such curves with exceptional rational points are of levels $137$ and $311$. In particular there are no exceptional rational points on those curves of genus five and six. More precisely, we determine the rational points on the curves $X_0^+(N)$ for $N=137,173,199,251,311,157,181,227,263,163,197,211,223,269,271,359$.
Tetragonal modular quotients $X_0^{+d}(N)$
本文研究了模曲线 $X_0(N)$ 在 Atkin-Lehner 对合 $w_d$ 下的商曲线 $X_0(N)/w_d$，确定了所有 $\mathbb{Q}$-gonality 和 $\mathbb{C}$-gonality 等于 4 的商曲线。
Tetragonal modular quotients $X_0^*(N)$
本文确定了所有复数亏格和有理数亏格为4的模商曲线 $X_0^*(N)$，其中有理数情形除级别378外全部给出。
Quadratic Chabauty for Atkin-Lehner quotients of modular curves via weakly holomorphic modular forms: Hodge Filtrations
本研究针对素级亏格≥2的Atkin-Lehner商模曲线，提出用弱全纯模形式计算二次Chabauty方法中关键的Hodge滤过的算法，提升了计算效率，并在$X_0^+(193)$上发现了权2尖点形式积分的同余关系。
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…
Tetragonal modular quotients of $X_0(N)$
针对模曲线$X_0(N)$的Atkin-Lehner对合商问题，本文确定了满足阶数条件的子群对应的$\mathbb{C}$与$\mathbb{Q}$-四边形商曲线，完成了$X_0(N)$所有$\mathbb{C}$-四边形Atkin-Lehner商的分类。
Dimension formulas for modular form spaces of rational weights, the classification of eta-quotient characters and an extension of Martin's theorem
本文给出由分数指数eta-商诱导乘子系统的有理权模形式空间的显式维数公式，推广了Yves Martin的乘法性全纯eta-商列表，并完全分类了整数指数eta-商诱导的乘子系统。
The modular automorphisms of quotient modular curves
本文确定了 4、9 不整除层数 N 时任意商模曲线的模自同构群，发现涉及 w_25 时的非预期 3 阶自同构，并推出充分大 N 下 X_0^*(N^2) 的完整自同构群。
The classical modular equations involve bivariate polynomials that can be seen to be univariate with coefficients in the modular invariant $j$. Kiepert found modular equations relating some $η$-quotients and the Weber functions $γ_2$ and $γ_3$. In the present work, we extend this idea to double $η$-quotients and characterize all the parameters leading to this kind of equation. We give some properties of these equations, explain how to compute them and give numerical examples.
We study the homology and the Borel-Moore homology with coefficients in $\mathbb{Q}$ of a quotient (called arithmetic quotient) of the Bruhat-Tits building of $\mathrm{PGL}$ of a nonarchimedean local field of positive characteristic by an arithmetic subgroup (a special case of the general definition in Harder's article (Invent.\ Math.\ 42, 135-175 (1977)). We define an analogue of modular symbols in this context and show that the image of the canonical map from homology to Borel-Moore homology is contained in the sub $\mathbb{Q}$-vector space generated by the modular symbols. By definition, th…
Let $A$ be the coordinate ring of a projective smooth curve over a finite field minus a closed point. For a nontrivial ideal $I \subset A$, Drinfeld defined the notion of structure of level $I$ on a Drinfeld module. We extend this to that of level $N$, where $N$ is a finitely generated torsion $A$-module. The case where $N=(I^{-1}/A)^d$, where $d$ is the rank of the Drinfeld module,coincides with the structure of level $I$. The moduli functor is representable by a regular affine scheme. The automorphism group $\mathrm{Aut}_{A}(N)$ acts on the moduli space. Our theorem gives a class of subgroup…
Maximal curves over finite fields and a modular isogeny
本文证明了在含$11^5$个元素的域上存在达到Hasse-Weil-Serre上界的亏格$7$和$12$曲线，通过推广Chen同源结果计算商模曲线点数并发现多项已知最大点数新纪录。
The topic of this paper concerns a certain relation between the jacobians of various quotients of the modular curve $X(p)$, which relates the jacobian of the quotient of $X(p)$ by the normaliser of a non-split Cartan subgroup of $GL_2(F_p)$ to the jacobians of more standard modular curves. In this paper, we confirm a conjecture of Merel found in a paper of Darmon-Merel, "Winding quotients and some variants of Fermat's Last Theorem", Crelle, v. 490, p. 81-100, 1997, which describes this relation in terms of explicit correspondences. The method used is to reduce the conjecture to showing a certa…
Eta-Quotient Representations for a Three Parameter Family of Modular Functions Associated with the Rogers-Ramanujan Continued Fraction
本文构造了与Rogers-Ramanujan连分数相关的三参数模函数族，推导其递归eta商表示，并应用于剖分公式，特别是用于获得受限奇数差分拆的5-剖分公式。
We complete the computation of all $\mathbb{Q}$-rational points on all the $64$ maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty--Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method combined with the Mordell-Weil sieve. Additionally, for square-free levels $N$, we classify all $\mathbb{Q}$-rational points as cusps, CM points (including their CM field and $j$-invariants) and exceptional ones. We further indicate how to use this …
Euler characteristic of crepant resolutions of specific modular quotient singularities
本文研究正特征下特殊线性群有限子群商奇点双解析解欧拉示性数的McKay对应推广，利用有限域上wild McKay对应证明其对特定半直积群成立，并提出模情形下的猜想。
In this paper we study algebras of modular forms on unitary groups of signature $(n,1)$. We give a necessary and sufficient condition for an algebra of unitary modular forms to be free in terms of the modular Jacobian. As a corollary we obtain a criterion that guarantees in many cases that, if $L$ is an even lattice with complex multiplication and the ring of modular forms for its orthogonal group is a polynomial algebra, then the ring of modular forms for its unitary group is also a polynomial algebra. We prove that a number of rings of unitary modular forms are freely generated by applying t…
In this paper, we provide refined sufficient conditions for the quadratic Chabauty method to produce a finite set of points, with the conditions on the rank of the Jacobian replaced by conditions on the rank of a quotient of the Jacobian plus an associated space of Chow-Heegner points. We then apply this condition to prove the finiteness of this set for any modular curves $X_{\mathrm{ns} }^+ (N)$ and $X_0 ^+ (N)$ of genus at least 2 with N prime. The proof relies on the existence of a quotient of their Jacobians whose Mordell-Weil rank is equal to its dimension (and at least 2), which is prove…
We complete several generating functions to non-holomorphic modular forms in two variables. For instance, we consider the generating function of a natural family of meromorphic modular forms of weight two. We then show that this generating series can be completed to a smooth, non-holomorphic modular form of weights 3/2 and two. Moreover, it turns out that the same function is also a modular completion of the generating function of weakly holomorphic modular forms of weight 3/2, which prominently appear in work of Zagier on traces of singular moduli.
In connection with our previous investigation about Siegel threefolds which admit a Calabi--Yau model, we consider ball quotients which belong to the unitary group $\U(1,3)$. In this paper we determine a very particular example of a Picard modular variety of general type. Really we determine the ring of modular forms. This algebra has 25 generators, 15 modular forms $B_i$ of weight one and ten modular forms $C_j$ of weight 2. Both will appear as Borcherds products. We determine the ideal of relations. The forms $C_i$ are cuspidal. Their squares define holomorphic differential forms on the non-…
Atkin--Lehner Quotients of Modular Curves at Primes of Bad Reduction
本文构造了模曲线 $X_0(N)$ 的 Atkin--Lehner 商在坏约化素数处的半稳定模型，并给出特殊纤维的显式描述与亏格公式。
Periodic sign changes for weakly holomorphic $η$-quotients
该研究聚焦弱全纯η-商模形式的符号变化问题，通过给出负权、零权、正权三类形式的代表性示例，为相关数论研究提供了基础案例参考。
Let $Γ\subset \mathbf{PU}(2,1)$ be a lattice which is not co-compact, of finite Bergman-covolume and acting freely on the open unit ball $\mathbf{B} \subset \mathbb{C}^2$. Then the compactification $X = \bar{Γ\setminus \mathbf{B}}$ is a smooth projective surface with an elliptic compactification divisor $D = X \setminus (Γ\setminus \mathbf{B})$. In this short note we discover a new class of unramified ball-quotients $X$. We consider ball-quotients $X$ with $kod(X) = h^1(X, \mathcal{O}_X) = 1$. We prove that all minimal surfaces with finite Mordell-Weil group in the class described become after…
Let $N\geq 1$ be a non-square free integer and let $W_N$ be a non-trivial subgroup of the group of the Atkin-Lehner involutions of $X_0(N)$ such that the modular curve $X_0(N)/W_N$ has genus at least two. We determine all pairs $(N,W_N)$ such that $X_0(N)/W_N$ is a bielliptic curve and the pairs $(N,W_N)$ such that $X_0(N)/W_N$ has an infinite number of quadratic points over $\mathbb{Q}$.
We show that every modular form on $Γ_0(2^n)$ ($n\geq2$) can be expressed as a sum of eta-quotients. Furthermore, we construct a primitive generator of the ring class field of the order of conductor $4N$ ($N\geq1$) in an imaginary quadratic field in view of the special value of certain eta-quotient.
The Shimura correspondence connects modular forms of integral weights and half-integral weights. One of the directions is realized by the Shintani lift, where the inputs are holomorphic differentials and the outputs are holomorphic modular forms of half-integral weight. In this article, we generalize this lift to differentials of the third kind. As an application we obtain a modularity result concerning the generating series of winding numbers of closed geodesics on the modular curve.
Exceptional Congruences for Eta-quotient newforms
本文研究了eta-quotient新形式系数的I型和II型同余，利用模形式理论证明了这些同余及对素数幂的扩展。
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.
An elementary proof of Newman's eta-quotient theorem
本文给出纽曼 eta-商定理的初等证明，关键观察是用上三角、下三角子群加任意同余子群生成 Gamma_1(N)，并借助避免 Dedekind 和的简单乘子系恒等式建立模性。
In this paper, we investigate which modular form spaces can contain $η$-quotients, functions of the form $f(τ) = \prod_{δ\mid N} η(δτ)^{r_δ}$. For $k\geq 2$ even and $N$ coprime to 6, we give necessary conditions for the space $M_k(Γ_1(N))$ to contain $η$-quotients. We then show that these conditions are sufficient as well for a large family of squarefree levels $N$ coprime to 6.
We prove hypergeometric type identities for a function defined in terms of quotients of the $p$-adic gamma function. We use these identities to prove a supercongruence conjecture of Rodriguez-Villegas between a truncated $_4F_3$ hypergeometric series and the Fourier coefficients of certain weight four modular form.
An integral power series is called lacunary modulo $M$ if almost all of its coefficients are divisible by $M$. Motivated by the parity problem for the partition function, $p(n)$, Gordon and Ono studied the generating functions for $t$-regular partitions, and determined conditions for when these functions are lacunary modulo powers of primes. We generalize their results in a number of ways by studying infinite products called Dedekind eta-quotients and generalized Dedekind eta-quotients. We then apply our results to the generating functions for the partition functions considered by Nekrasov, Ok…
Eta quotients on $Γ_0(6)$ yield evaluations of sunrise integrals at 2, 3, 4 and 6 loops. At 2 and 3 loops, they provide modular parametrizations of inhomogeneous differential equations whose solutions are readily obtained by expanding in the nome $q$. Atkin-Lehner transformations that permute cusps ensure fast convergence for all external momenta. At 4 and 6 loops, on-shell integrals are periods of modular forms of weights 4 and 6 given by Eichler integrals of eta quotients. Weakly holomorphic eta quotients determine quasi-periods. A Rademacher sum formula is given for Fourier coefficients of …
We study the meromorphic modular forms defined as sums of -k (k>1) powers of integral quadratic polynomials with negative discriminant. These functions can be viewed as meromorphic analogues of the holomorphic modular forms defined in the same way with positive discriminant, first investigated by Zagier in connection with the Doi-Naganuma map and then by Kohnen and Zagier in connection with the Shimura-Shintani lifts. We compute the Fourier coefficients of these meromorphic modular forms and we show that they split into the sum of a meromorphic modular form with computable algebraic Fourier co…
Throughout the 1980's, Kudla and the second named author studied integral transforms from rapidly decreasing closed differential forms on arithmetic quotients of the symmetric spaces of orthogonal and unitary groups to spaces of classical Siegel and Hermitian modular forms. These transforms came from the theory of dual reductive pairs and the theta correspondence. They computed the Fourier expansion of these transforms in terms of periods over certain totally geodesic cycles . This also gave rise to the realization of intersection numbers of these `special' cycles with cycles with compact supp…
Representability of systems of proportionally modular numerical semigroups
研究比例模数值半群系统的可表示性问题，核心方法是从其商描述出发，通过双生成元数值半群实现每个商，再用双分支典范等变分解图及合适重数胶合，主要贡献是证明其可由特定加权齐次曲面奇点的分解表示。
It is known that all modular forms on SL_2(Z) can be expressed as a rational function in eta(z), eta(2z) and eta(4z). By utilizing known theorems, and calculating the order of vanishing, we can compute the eta-quotients for a given level. Using this count, knowing how many eta-quotients are linearly independent and using the dimension formula, we can figure out a subspace spanned by the eta-quotients. In this paper, we primarily focus on the case where N=p a prime.
Ramanujan type congruences for quotients of Klein forms
本文基于Klein形式商构造level p模形式乘积，导出了素数p≥5幂次的Ramanujan型同余，完整列举了特定权与级数下的同余乘积，并构建了模形式分次代数的生成集与级数分解方法。
Congruences of modular forms and modularity of Tate-Shafarevich classes
本文在适当假设下证明了有理数上椭圆曲线的$p$-挠Tate-Shafarevich类在模曲线Jacobi商群中的可见性，验证了Jetchev-Stein猜想，其核心方法是利用Bertolini-Darmon二部Kolyvagin系统的非平凡性来推导Selmer群模$p$的基。
In this paper, we study the explicit structure of the graded ring of elliptic modular forms for the congruence subgroup $Γ_0(N)$ with N=1,2,3,4,5,6,7,8,9,10,12,16,18,25. More precisely, making use of some relations between Fourier expansions, we determine the ring by a quotient of the polynomial ring in several variables.
On the modularity of the odd rank generating functions
本文研究奇秩生成函数的变换公式，构造弱Maass形式和弱模形式族，并给出模5的奇秩恒等式，类似于Ramanujan经典分拆恒等式。
In 1919, P. A. MacMahon studied generating functions for generalized divisor sums. In this paper, we provide a framework in which to view these generating functions in terms of Jacobi forms, and prove that they are quasi-modular forms.
By work of Bringmann, Ono, and Rhoades it is known that the generating function of the $M_2$-rank of partitions without repeated odd parts is the so-called holomorphic part of a certain harmonic Maass form. Here we improve the standing of this function as a harmonic Maass form and show more can be done with this function. In particular we show the related harmonic Maass form transforms like the generating function for partitions without repeated odd parts (which is a modular form). We then use these improvements to determine formulas for the rank differences modulo $7$. Additionally we give id…
The Explicit Hypergeometric-Modularity Method I
本文提出显式超几何-模性方法，将给定超几何数据关联到模形式，利用拉马努金椭圆函数理论、形式群律和超同余，构造eta商并给出应用。
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.
Let $k\geq 2$ be an integer and $j$ an integer satisfying $1\leq j \leq 4k-5$. We define a family $\{ C_{j,k}(z) \}_{1\leq j \leq 4k-5} $ of eta quotients, and prove that this family constitute a basis for the space $S_{2k} (Γ_0 (12))$ of cusp forms of weight $2k$ and level $12$. We then use this basis together with certain properties of modular forms at their cusps to prove an extension of the Ramanujan-Mordell formula.
Double coset operators and eta-quotients
本文研究可改变模形式特征的广义双陪集算子，明确刻画其作用，并应用于 eta-商，给出算子构造、推广 Newman 定理的系数递推公式以及 eta-幂表示为 eta-商线性组合的判据与算法。
In this paper, we derive eight basic identities of symmetry in three variables related to Bernoulli polynomials and power sums. These and most of their corollaries are new, since there have been results only about identities of symmetry in two variables. These abundance of symmetries shed new light even on the existing identities so as to yield some further interesting ones. The derivations of identities are based on the $p$-adic integral expression of the generating function for the Bernoulli polynomials and the quotient of integrals that can be expressed as the exponential generating functio…
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