arXiv subjects

Ling Long

Publications and source records attributed to Ling Long.

34 records · Page 2Linked to original sources

arXiv2014

Atkin and Swinnerton-Dyer congruences and noncongruence modular forms

Atkin and Swinnerton-Dyer congruences are special congruence recursions satisfied by coefficients of noncongruence modular forms. These are in some sense $p$-adic analogues of Hecke recursion satisfied by classic Hecke eigenforms. They actually appeared in different context and sometimes can be obtained using the theory of formal groups. In this survey paper, we introduce the Atkin and Swinnerton-Dyer congruences, and discuss some recent progress on this topic.

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arXiv2014

Some supercongruences occurring in truncated hypergeometric series

For the purposes of this paper supercongruences are congruences between terminating hypergeometric series and quotients of $p$-adic Gamma functions that are stronger than those one can expect to prove using commutative formal group laws. We prove a number of such supercongruences by using classical hypergeometric transformation formulae. These formulae (see the appendix), most of which are decades or centuries old, allow us to write the terminating series as the ratio of products of of $Γ$-values. At this point sums have become quotients. Writing these $Γ$-quotients as $Γ_p$-quotients, we are in a situation that is well-suited for proving $p$-adic congruences. These $Γ_p$-functions can be $p$-adically approximated by their Taylor series expansions. Sometimes there is cancelation of the lower order terms, leading to stronger congruences. Using this technique we prove, among other things, a conjecture of Kibelbek and a strengthened version of a conjecture of van Hamme.

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arXiv2012

Supercongruences and Complex Multiplication

We study congruences involving truncated hypergeometric series of the form_rF_{r-1}(1/2,...,1/2;1,...,1;λ)_{(mp^s-1)/2} = \sum_{k=0}^{(mp^s-1)/2} ((1/2)_k/k!)^r λ^k where p is a prime and m, s, r are positive integers. These truncated hypergeometric series are related to the arithmetic of a family of algebraic varieties and exhibit Atkin and Swinnerton-Dyer type congruences. In particular, when r=3, they are related to K3 surfaces. For special values of λ, with s=1 and r=3, our congruences are stronger than what can be predicted by the theory of formal groups because of the presence of elliptic curves with complex multiplications. They generalize a conjecture made by Rodriguez-Villegas for the λ=1 case and confirm some other supercongruence conjectures at special values of λ.

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arXiv2011

Jacobsthal identity for Q(sqrt(-2))

Let $p$ be a prime congruent to 1 or 3 modulo 8 so that the equation $p=a^2+2b^2$ is solvable in integers. In this paper, we obtain closed-form expressions for $a$ and $b$ in terms of Jacobsthal sums. This is analogous to a classical identity of Jacobsthal.

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arXiv2011

Galois Representations with Quaternion Multiplications Associated to Noncongruence Modular Forms

In this paper we study the compatible family of degree-4 Scholl representations $ρ_{\ell}$ associated with a space $S$ of weight $κ> 2$ noncongruence cusp forms satisfying Quaternion Multiplications over a biquadratic field $K$. It is shown that when either $K$ is totally real or $κ$ is odd, $ρ_\ell$ is automorphic, that is, its associated L-function has the same Euler factors as the L-function of an automorphic form for $GL_4(\mathbb Q)$. Further, it yields a relation between the Fourier coefficients of noncongruence cusp forms in $S$ and those of certain automorphic forms via the three-term Atkin and Swinnerton-Dyer congruences.

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arXiv2011

On l-adic representations for a space of noncongruence cuspforms

This paper is concerned with a compatible family of 4-dimensional \ell-adic representations ρ_{\ell} of G_\Q:=\Gal(\bar \Q/\Q) attached to the space of weight 3 cuspforms S_3 (Γ) on a noncongruence subgroup Γ\subset \SL. For this representation we prove that: 1.)It is automorphic: the L-function L(s, ρ_{\ell}^{\vee}) agrees with the L-function for an automorphic form for \text{GL}_4(\mathbb A_{\Q}), where ρ_{\ell}^{\vee} is the dual of ρ_{\ell}. 2.) For each prime p \ge 5 there is a basis h_p = \{h_p ^+, h_p ^- \} of S_3 (Γ) whose expansion coefficients satisfy 3-term Atkin and Swinnerton-Dyer (ASD) relations, relative to the q-expansion coefficients of a newform f of level 432. The structure of this basis depends on the class of p modulo 12. The key point is that the representation $ρ_{\ell}$ admits a quaternion multiplication structure in the sense of a recent work of Atkin, Li, Liu and Long.

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arXiv2010

Hypergeometric evaluation identities and supercongruences

In this article, we provide an application of hypergeometric evaluation identities, including a strange valuation of Gosper, to prove several supercongruences related to special valuations of truncated hypergeometric series. In particular, we prove a conjecture of van Hamme.

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arXiv2010

Fourier coefficients of noncongruence cuspforms

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.

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arXiv2009

Zeros of some level 2 Eisenstein series

The zeros of classical Eisenstein series satisfy many intriguing properties. Work of F. Rankin and Swinnerton-Dyer pinpoints their location to a certain arc of the fundamental domain, and recent work by Nozaki explores their interlacing property. In this paper we extend these distribution properties to a particular family of Eisenstein series on Gamma(2) because of its elegant connection to a classical Jacobi elliptic function cn(u) which satisfies a differential equation. As part of this study we recursively define a sequence of polynomials from the differential equation mentioned above that allow us to calculate zeros of these Eisenstein series. We end with a result linking the zeros of these Eisenstein series to an L-series.

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arXiv2008

On modular forms for some noncongruence subgroups of SL_2(Z) II

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.

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arXiv2007

Computations with finite index subgroups of $PSL_2(\mathbb Z)$ using Farey Symbols

Finite index subgroups of the modular group are of great arithmetic importance. Farey symbols, introduced by Ravi Kulkarni in 1991, are a tool for working with these groups. Given such a group $Γ$, a Farey symbol for $Γ$ is a certain finite sequence of rational numbers (representing vertices of a fundamental domain of $Γ$) together with pairing information for the edges between the vertices. They are a compact way of encoding the information about the group and they provide a simple way to do calculations with the group. For example: calculating an independent set of generators and decomposing group elements into a word in these generators, finding coset representatives, elliptic points, and genus of the group, testing if the group is congruence, etc. In this expository article, we will discuss Farey Symbols and explicit algorithms for working with them.

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arXiv2007

On modular forms for some noncongruence arithmetic subgroups

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 noncongruence modular forms (with algebraic coefficients) always satisfy the unbounded denominator property. Here, we give a partial positive answer to the above open question by constructing special finite index subgroups of SL_2(Z) called character groups and discuss the properties of modular forms for some groups of this kind.

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arXiv2007

Finite index subgroups of the modular group and their modular forms

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 representation arising from noncongruence cuspforms, and Atkin and Swinnerton-Dyer congruences.

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arXiv2007

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 attached to the space of noncongruence cusp forms.

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arXiv2007

On Atkin and Swinnerton-Dyer congruence relations (3)

In the previous two papers with the same title ([LLY05] by W.C. Li, L. Long, Z. Yang and [ALL05] by A.O.L. Atkin, W.C. Li, L. Long), the authors have studied special families of cuspforms for noncongruence arithmetic subgroups. It was found that the Fourier coefficients of these modular forms at infinity satisfy three-term Atkin and Swinnerton-Dyer congruence relations which are the $p$-adic analogue of the three-term recursions satisfied by the coefficients of classical Hecke eigenforms. In this paper, we first consider Atkin and Swinnerton-Dyer type congruences which generalize the three-term congruences above. These weaker congruences are satisfied by cuspforms for special noncongruence arithmetic subgroups. Then we will exhibit an infinite family of noncongruence cuspforms, each of which satisfies three-term Atkin and Swinnerton-Dyer type congruences for almost every prime $p$. Finally, we will study a particular space of noncongruence cuspforms. We will show that the attached $l$-adic Scholl representation is isomorphic to the $l$-adic representation attached to a classical automorphic form. Moreover, for each of the four residue classes of odd primes modulo 12 there is a basis so that the Fourier coefficients of each basis element satisfy three-term Atkin and Swinnerton-Dyer congruences in the stronger original sense.

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arXiv2004

On Atkin-Swinnerton-Dyer congruence relations

In this paper we exhibit a noncongruence subgroup $\G$ whose space of weight 3 cusp forms $S_3(\G)$ admits a basis satisfying the Atkin-Swinnerton-Dyer congruence relations with two weight 3 newforms for certain congruence subgroups. This gives a modularity interpretation of the motive attached to $S_3(\G)$ by A. Scholl and also verifies the Atkin-Swinnerton-Dyer congruence conjecture for this space.

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