Publications (34)
Fourier coefficients of noncongruence cuspforms
Wen-Ching Winnie Li, Ling Long
Given a finite index subgroup of with modular curve defined over , under the assumption that the space of weight () cusp forms is -dimen…
Atkin and Swinnerton-Dyer congruences for meromorphic modular forms
Michael Allen, Ling Long, Hasan Saad
In the 1970's, Atkin and Swinnerton-Dyer conjectured that Fourier coefficients of holomorphic modular cusp forms on noncongruence subgroups of satisfy cer…
Supercongruences and Complex Multiplication
Jonas Kibelbek, Ling Long, Kevin Moss +2
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…
Zeros of some level 2 Eisenstein series
Sharon Garthwaite, Ling Long, Holly Swisher +1
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 doma…
ADAgent: LLM Agent for Alzheimer's Disease Analysis with Collaborative Coordinator
Wenlong Hou, Guangqian Yang, Ye Du +5
Alzheimer's disease (AD) is a progressive and irreversible neurodegenerative disease. Early and precise diagnosis of AD is crucial for timely intervention and treatment planning to…
On Atkin and Swinnerton-Dyer congruence relations (3)
Ling Long
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 cu…
Hypergeometric Functions over Finite Fields
Jenny Fuselier, Ling Long, Ravi Ramakrishna +2
Building on the developments of many people including Evans, Greene, Katz, McCarthy, Ono, Roberts, and Rodriguez-Villegas, we consider period functions for hypergeometric type alge…
Some Numeric Hypergeometric Supercongruences
Ling Long
In this article, we list a few hypergeometric supercongruence conjectures based on two evaluation formulas of Whipple and numeric data computed using Magma and Sagemath.
Computations with finite index subgroups of using Farey Symbols
Chris A. Kurth, Ling Long
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…
Supercongruences for rigid hypergeometric Calabi--Yau threefolds
Ling Long, Fang-Ting Tu, Noriko Yui +1
We establish the supercongruences for the fourteen rigid hypergeometric Calabi--Yau threefolds over conjectured by Rodriguez-Villegas in 2003. Our first method is based…
On l-adic representations for a space of noncongruence cuspforms
Jerome W. Hoffman, Ling Long, Helena Verrill
This paper is concerned with a compatible family of 4-dimensional \ell-adic representations Ï_{\ell} of G_\Q:=\Gal(\bar \Q/\Q) attached to the space of weight 3 cuspforms S_3 (Î)…
Potentially -type Galois representations associated to noncongruence modular forms
Wen-Ching Winnie Li, Tong Liu, Ling Long
In this paper, we consider Galois representations of the absolute Galois group attached to modular forms for noncongruence subgroups o…
Hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions
Alyson Deines, Jenny G. Fuselier, Ling Long +2
In this paper, we investigate the relationships among hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions through some families of `hyperg…
Traces of Hecke Operators via Hypergeometric Character Sums
Jerome W. Hoffman, Wen-Ching Winnie Li, Ling Long +1
In this paper we obtain explicit formulas for the traces of Hecke operators on spaces of cusp forms in certain instances related to arithmetic triangle groups. These expressions ar…
Some supercongruences occurring in truncated hypergeometric series
Ling Long, Ravi Ramakrishna
For the purposes of this paper supercongruences are congruences between terminating hypergeometric series and quotients of -adic Gamma functions that are stronger than those one…
The Explicit Hypergeometric-Modularity Method II
Michael Allen, Brian Grove, Ling Long +1
In the first paper of this sequence, we provided an explicit hypergeometric modularity method by combining different techniques from the classical, -adic, and finite field setti…
On a conjecture of Kimoto and Wakayama
Ling Long, Robert Osburn, Holly Swisher
We prove a conjecture due to Kimoto and Wakayama from 2006 concerning Apery-like numbers associated to a special value of a spectral zeta function. Our proof uses hypergeometric se…
Hodge numbers of hypergeometric data
Ling Long, Yifan Yang
In this paper, based on the toric hypergeometric model given in a paper by Beukers--Cohen--Mellit, we provide two other ways to explain why the zig-zag diagram method can be used t…
A Whipple formula revisited
Wen-Ching Winnie Li, Ling Long, Fang-Ting Tu
A well-known formula of Whipple relates certain hypergeometric values and . In this paper we revisit this relation from the viewpoint of the underlying hyperge…
Atkin and Swinnerton-Dyer congruences and noncongruence modular forms
Wen-Ching Winnie Li, Ling Long
Atkin and Swinnerton-Dyer congruences are special congruence recursions satisfied by coefficients of noncongruence modular forms. These are in some sense -adic analogues of Heck…
On Atkin and Swinnerton-Dyer Congruence Relations (2)
A. O. L. Atkin, Wen-Ching Winnie Li, Ling Long
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 clas…
On modular forms for some noncongruence subgroups of SL_2(Z) II
Ling Long, Chris Kurth
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 t…
On modular forms for some noncongruence arithmetic subgroups
Chris Kurth, Ling Long
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…
The Explicit Hypergeometric-Modularity Method I
Michael Allen, Brian Grove, Ling Long +1
The theories of hypergeometric functions and modular forms are highly intertwined. For example, particular values of truncated hypergeometric functions and hypergeometric character…
Hypergeometric evaluation identities and supercongruences
Ling Long
In this article, we provide an application of hypergeometric evaluation identities, including a strange valuation of Gosper, to prove several supercongruences related to special va…
Jacobsthal identity for Q(sqrt(-2))
Ki-Ichiro Hashimoto, Ling Long, Yifan Yang
Let be a prime congruent to 1 or 3 modulo 8 so that the equation is solvable in integers. In this paper, we obtain closed-form expressions for and in terms…
Characterization of intersecting families of maximum size in
Ling Long, Rafael Plaza, Peter Sin +1
We consider the action of the -dimensional projective special linear group on the projective line over the finite field $\F_q$, where is an odd prime po…
AD-CARE: A Guideline-grounded, Modality-agnostic LLM Agent for Real-world Alzheimer's Disease Diagnosis with Multi-cohort Assessment, Fairness Analysis, and Reader Study
Wenlong Hou, Sheng Bi, Guangqian Yang +16
Alzheimer's disease (AD) is a growing global health challenge as populations age, and timely, accurate diagnosis is essential to reduce individual and societal burden. However, rea…
Finite index subgroups of the modular group and their modular forms
Ling Long
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 majo…
The Explicit Hypergeometric Modularity Method III
Brian Grove, Ling Long, Esme Rosen +1
We refine the Explicit Hypergeometric Modularity Method (EHMM) and develop a variant that applies to a broader class of hypergeometric data. As an application, we establish the mod…
Generalized Legendre curves and Quaternionic Multiplication
Alyson Deines, Jenny G. Fuselier, Ling Long +2
This paper is devoted to abelian varieties arising from generalized Legendre curves. In particular, we consider their corresponding Galois representations, periods, and endomorphis…
Galois Representations with Quaternion Multiplications Associated to Noncongruence Modular Forms
A. O. L. Atkin, Wen-Ching Winnie Li, Tong Liu +1
In this paper we study the compatible family of degree-4 Scholl representations associated with a space of weight noncongruence cusp forms satisfying Quater…
Computing Special -Values of Certain Modular Forms with Complex Multiplication
Wen-Ching Winnie Li, Ling Long, Fang-Ting Tu
In this expository paper, we illustrate two explicit methods which lead to special -values of certain modular forms admitting complex multiplication (CM), motivated in part by p…
On Atkin-Swinnerton-Dyer congruence relations
Wen-Ching Winnie Li, Ling Long, Zifeng Yang
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 tw…