Andrew Fiori, Cameron Franc
We study modular forms for the minimal index noncongruence subgroups of the modular group. Our main theorem is a proof of the unbounded denominator conjecture for these groups, and we also provide a study of the Fourier coefficients of Eisenstein series for one of these minimal groups.
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In this paper we show two classes of noncongruence subgroups satisfy the so-called unbounded denominator property. In particular, we establish our conjecture in [KL08] which says that every type II noncongruence character group of Gamma^0(11) satisfies the unbounded denominator property.
In this paper, we consider modular forms for finite index subgroups of the modular group whose Fourier coefficients are algebraic. It is well-known that the Fourier coefficients of any holomorphic modular form for a congruence subgroup (with algebraic coefficients) have bounded denominators. It was observed by Atkin and Swinnerton-Dyer that this is no longer true for modular forms for noncongruence subgroups and they pointed out that unbounded denominator property is a clear distinction between modular forms for noncongruence and congruence modular forms. It is an open question whether genuine…
Given a finite index subgroup of $SL_2(\mathbb Z)$ with modular curve defined over $\mathbb Q$, under the assumption that the space of weight $k$ ($ \ge 2$) cusp forms is $1$-dimensional, we show that a form in this space with Fourier coefficients in $\mathbb Q$ has bounded denominators if and only if it is a congruence modular form.
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 define the notion of a $G$-structure for elliptic curves, where $G$ is a finite 2-generated group. When $G$ is abelian, a $G$-structure is the same as a classical congruence level structure. There is a natural action of $\text{SL}_2(\mathbb{Z})$ on these level structures. If $Γ$ is a stabilizer of this action, then the quotient of the upper half plane by $Γ$ parametrizes isomorphism classes of elliptic curves equipped with $G$-structures. When $G$ is "sufficiently" nonabelian, the stabilizers $Γ$ are noncongruence. As a result we realize noncongruence modular curves as moduli spaces of elli…
The Unbounded Denominators Conjecture
该研究证明了SL₂(Z)有限指数子群非同余模形式的无界分母猜想及其Mason向量值模形式推广，结合新算术全纯性界等工具，为有理共形场论同余性质提供新路径。
An Explicit Belyi Map for the Wiman Sextic and Cusp Forms for a Noncongruence Subgroup
该研究针对维曼六次曲线的光滑射影模型确定显式Belyi函数，结合非同余子群$\boldsymbol{\tilde{W}}$的模函数描述其复单值化，通过$\boldsymbol{F}_5$上的退化与Puiseux级数展开，给出权2下非同余子群无界分母猜想的不同证明。
We study the question of when the coefficients of a hypergeometric series are p-adically unbounded for a given rational prime p. Our first main result is a necessary and sufficient criterion (applicable to all but finitely many primes) for determining when the coefficients of a hypergeometric series with rational parameters are p-adically unbounded. This criterion is then used to show that the set of unbounded primes for a given series is, up to a finite discrepancy, a finite union of primes in arithmetic progressions. This set can be computed explicitly. We characterize when the density of th…
A thorough analysis is made of the Fourier coefficients for vector-valued modular forms associated to three-dimensional irreducible representations of the modular group. In particular, the following statement is verified for all but a finite number of equivalence classes: if a vector-valued modular form associated to such a representation has rational Fourier coefficients, then these coefficients have "unbounded denominators", i.e. there is a prime number p, depending on the representation, which occurs to an arbitrarily high power in the denominators of the coefficients. This provides a verif…
Let $ρ: SL(2,\mathbb{Z})\to GL(2,\mathbb{C})$ be an irreducible representation of the modular group such that $ρ(T)$ has finite order $N$. We study holomorphic vector-valued modular forms $F(τ)$ of integral weight associated to $ρ$ which have \emph{rational} Fourier coefficients. (These span the complex space of all integral weight vector-valued modular forms associated to $ρ$.) As a special case of the main Theorem, we prove that if $N$ does \emph{not} divide 120 then every nonzero $F(τ)$ has Fourier coefficients with \emph{unbounded denominators}.
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.
Let $ρ$ denote an irreducible two-dimensional representation of $Γ_{0}(2)$. The collection of vector-valued modular forms for $ρ$, which we denote by $M(ρ)$, form a graded and free module of rank two over the ring of modular forms on $Γ_{0}(2)$, which we denote by $M(Γ_{0}(2))$. For a certain class of $ρ$, we prove that if Z is any vector-valued modular form for $ρ$ whose component functions have algebraic Fourier coefficients then the sequence of the denominators of the Fourier coefficients of both component functions of Z is unbounded. Our methods involve computing an explicit basis for $M(ρ…
Signed $p$-adic $L$-functions of Bianchi modular forms
本文研究了Bianchi模形式的有符号p-adic L-函数，通过p-稳定化形式和局部解析分布，将双变量幂级数分解为有界系数的线性组合。
A conjecture of Zagier and the value distribution of quantum modular forms
本文证明八字结彩色 Jones 多项式相关函数在部分商无界的无理数处连续，并给出一致光滑逼近及有理参数下对数不变量的极限分布。
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