Congruence relations satisfied by quaternionic modular forms

Sur le spectre et la topologie des variétés hyperboliques de congruence : les cas complexe et quaternionien

Building on results of Arthur and Mok, we extend to (finite volume) complex and quaternionic hyperbolic manifolds the results of arXiv:1004.1085. For the spherical spectrum our results are optimal. Finally, as an application we prove a Lefschetz property for the restriction map between arithmetic quotients of complex balls. This generalizes a recent theorem of Arvind Nair and gives an optimal version of it.

Supercongruences satisfied by coefficients of 2F1 hypergeometric series

Recently, Chan, Cooper and Sica conjectured two congruences for coefficients of classical 2F1 hypergeometric series which also arise from power series expansions of modular forms in terms of modular functions. We prove these two congruences using combinatorial properties of the coefficients.

Congruences between modular forms and related modules

We fix $\ell$ a prime and let $M$ be an integer such that $\ell\not|M$; let $f\in S_2(Γ_1(M\ell^2))$ be a newform supercuspidal of fixed type related to the nebentypus, at $\ell$ and special at a finite set of primes. Let $\TT^ψ$ be the local quaternionic Hecke algebra associated to $f$. The algebra $\TT^ψ$ acts on a module $\mathcal M^ψ_f$ coming from the cohomology of a Shimura curve. Applying the Taylor-Wiles criterion and a recent Savitt's theorem, $\TT^ψ$ is the universal deformation ring of a global Galois deformation problem associated to $\orho_f$. Moreover $\mathcal M^ψ_f$ is free of …

广义立方和超立方划分的算术性质

Arithmetic Properties of Generalized Cubic and Overcubic Partitions

本文研究了广义立方和超立方划分函数的若干同余关系，利用q级数技术和模形式理论证明了模2和12的幂次同余关系及密度结果。

On the arithmetic and geometry of binary Hamiltonian forms

Given an indefinite binary quaternionic Hermitian form $f$ with coefficients in a maximal order of a definite quaternion algebra over $\mathbb Q$, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most $s$ by $f$, as $s$ tends to $+\infty$. We compute the volumes of hyperbolic 5-manifolds constructed by quaternions using Eisenstein series. In the Appendix, V. Emery computes these volumes using Prasad's general formula. We use hyperbolic geometry in dimension 5 to describ…

Note on Igusa's cusp form of weight 35

A congruence relation satisfied by Igusa's cusp form of weight 35 is presented. As a tool to confirm the congruence relation, a Sturm-type theorem for the case of odd-weight Siegel modular forms of degree 2 is included.

源自局部的希尔伯特模形式欧拉特征 congruences

Hilbert modular Eisenstein congruences of local origin

本文研究了希尔伯特模形式与欧拉特征级数之间的欧拉特征 congruences，探讨了其模数来源及新形式满足的条件。

On theta series attached to the Leech lattice

Some congruence relations satisfied by the theta series associated with the Leech lattice are given.

Recurrence relations satisfied by the traces of singular moduli for $Γ_0(N)$

We compute the divisor of the modular equation on the modular curve $Γ_0(N) \backslash \mathbb H^*$ and then find recurrence relations satisfied by the modular traces of the Hauptmodul for any congruence subgroup $Γ_0(N)$ of genus zero. We also introduce the notions and properties of $Γ$-equivalence and $Γ$-reduced forms about binary quadratic forms. Using these, we can explicitly compute the recurrence relations for $N = 2, 3, 4, 5$.

Congruence properties of Taylor coefficients of modular forms

In their work, Serre and Swinnerton-Dyer study the congruence properties of the Fourier coefficients of modular forms. We examine similar congruence properties, but for the coefficients of a modified Taylor expansion about a CM point $τ$. These coefficients can be shown to be the product of a power of a constant transcendental factor and an algebraic integer. In our work, we give conditions on $τ$ and a prime number $p$ that, if satisfied, imply that $p^m$ divides the algebraic part of all the Taylor coefficients of $f$ of sufficiently high degree. We also give effective bounds on the largest …

On Atkin and Swinnerton-Dyer Congruence Relations (2)

In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier coefficients at infinity satisfy a three-term Atkin and Swinnerton-Dyer congruence relation, which is the $p$-adic analogue of the three-term recursion satisfied by the coefficients of classical Hecke eigen forms. We also show that there is an automorphic $L$-function over $\mathbb Q$ whose local factors agree with those of the $l$-adic Scholl representations …

经典模多项式系数的同余性质

Congruence properties of the coefficients of the classical modular polynomials

本文针对素数 $\ell$，以模不变函数 $j(z)$ 的傅里叶系数形式给出了经典模多项式 $\ell$ 个非平凡系数的闭式公式，并据此推导出同余式，支持了关于这些系数满足素数 $2,3$ 和 $5$ 的幂的同余猜想。

Three-dimensional imprimitive representations of the modular group and their associated modular forms

This paper uses previous results of the authors on vector-valued modular forms to study certain non-congruence modular forms. We prove that these forms have unbounded denominators, and in certain cases we verify congruences of Atkin--Swinnerton-Dyer type satisfied by the Fourier coefficients of these forms. Our results rest on group-theoretic facts about the modular group, a detailed study of its imprimitive three-dimensional representations, and the theory of their associated vector-valued modular forms. For the proof of the congruences we also make essential use of a result of Katz.

Thejitha与Fathima近期同余式的初等证明及推广

Elementary Proofs and Generalizations of Recent Congruences of Thejitha and Fathima

本文针对Thejitha和Fathima提出的、计数特定着色分拆的函数$a_5(n)$的类Ramanujan同余式，用生成函数运算与初等$q$-级数分拆的纯初等方法给出证明，并将结果推广为无穷族可除性性质。

涉及Rogers-Ramanujan连分数的某些无穷乘积的符号模式与同余式

Sign Patterns and Congruences of certain infinite products involving the Rogers-Ramanujan continued fraction

本文研究涉及Rogers-Ramanujan连分数的无穷乘积系数符号规律，证明特定算术级数中系数的正负性，并发现若干模3、模4和模15的同余式。

The Ramanujan-Serre differential operators and certain elliptic curves

For several congruence subgroups of low levels and their conjugates, we derive differential equations satisfied by the Eisenstein series of weight 4 and relate them to elliptic curves, whose associated new forms of weight 2 constitute the list of Martin and Ono of new forms given by eta-products/quotients.

椭圆曲线与模形式的傅里叶系数

Elliptic curves and Fourier coefficients of meromorphic modular forms

本文研究了模形式系数的同余关系，揭示其与椭圆曲线对称幂及$p$-adic Chowla--Selberg周期的联系，并通过超几何函数和Borcherds--Shimura提升进行证明。

$k$-伸长分拆钻石的无穷多同余性质的初等证明

Elementary Proofs of Infinitely Many Congruences for $k$-Elongated Partition Diamonds

本文用初等生成函数与经典q级数方法，为k-伸长分拆钻石数d_k在无穷多个k值下证明无穷多同余性质，扩展了Andrews和Paule的模形式结果。

Schur型过划分模2的小幂次的算术性质的初等证明

Elementary Proofs of Arithmetic Properties for Schur-Type Overpartitions Modulo Small Powers of 2

本文针对Schur型过划分计数函数$S(n)$，用初等方法证明了其模16的一族无限Ramanujan型同余式，扩展了Broudy和Lovejoy关于模2的小幂次的结果。

广义三次分拆模5幂次同余族：基于局部化方法

A congruence family modulo powers of 5 for generalized cubic partitions via the localization method

本文利用 Banerjee-Smoot 局部化方法，在模曲线 X_0(10) 上证明了广义三次分拆函数 a_3(n) 模 5 幂的同余族，推广了 Watson-Atkin 方法。

Mex-related partitions and relations to ordinary partition and singular overpartitions

In a recent paper, Andrews and Newman introduced certain families of partition functions using the minimal excludant or "mex" function. In this article, we study two of the families of functions Andrews and Newman introduced, namely $p_{t,t}(n)$ and $p_{2t,t}(n)$. We establish identities connecting the ordinary partition function $p(n)$ to $p_{t,t}(n)$ and $p_{2t,t}(n)$ for all $t\geq 1$. Using these identities, we prove that the Ramanujan's famous congruences for $p(n)$ are also satisfied by $p_{t,t}(n)$ and $p_{2t,t}(n)$ for infinitely many values of $t$. Very recently, da Silva and Sellers …

有理数型模形式模ℓ

Modular forms of CM type mod $\ell$

本文研究了模ℓ下的有理数型模形式的同余问题，证明了在特定条件下此类同余总是成立，并探讨了三种情况下的模形式性质。

定义模形式和 Jacobi 形式的微分方程

Defining differential equations for modular forms and Jacobi forms

本文定义了模形式和 Jacobi 形式满足的非线性微分方程，利用李群作用下的微分不变量紧凑表达，并将结果推广至 Jacobi 形式的不变偏微分方程系统。