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Ling Long

Publications and source records attributed to Ling Long.

At least 19 recordsLinked to original sources
arXiv2026

The Explicit Hypergeometric Modularity Method III

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 modularity of hypergeometric Galois representations arising from length-$4$ data that are not necessarily defined over $\mathbb{Q}$. We also use this method to give explicit constructions of nine modular forms associated with hypergeometric rigid Calabi-Yau threefolds conjectured to be modular by Rodriguez-Villegas. The modularity of these threefolds was first proved by Long-Tu-Yui-Zudilin using a different approach based on the Faltings-Serre method. Moreover, the explicit nature of the method makes it well suited to the computation of special $L$-values of the associated modular forms.

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arXiv2026

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

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, real-world AD assessment is hampered by incomplete, heterogeneous multimodal data and variability across sites and patient demographics. Although large language models (LLMs) have shown promise in biomedicine, their use in AD has largely been confined to answering narrow, disease-specific questions rather than generating comprehensive diagnostic reports that support clinical decision-making. Here we expand LLM capabilities for clinical decision support by introducing AD-CARE, a modality-agnostic agent that performs guideline-grounded diagnostic assessment from incomplete, heterogeneous inputs without imputing missing modalities. By dynamically orchestrating specialized diagnostic tools and embedding clinical guidelines into LLM-driven reasoning, AD-CARE generates transparent, report-style outputs aligned with real-world clinical workflows. Across six cohorts comprising 10,303 cases, AD-CARE achieved 84.9% diagnostic accuracy, delivering 4.2%-13.7% relative improvements over baseline methods. Despite cohort-level differences, dataset-specific accuracies remain robust (80.4%-98.8%), and the agent consistently outperforms all baselines. AD-CARE reduced performance disparities across racial and age subgroups, decreasing the average dispersion of four metrics by 21%-68% and 28%-51%, respectively. In a controlled reader study, the agent improved neurologist and radiologist accuracy by 6%-11% and more than halved decision time. The framework yielded 2.29%-10.66% absolute gains over eight backbone LLMs and converges their performance. These results show that AD-CARE is a scalable, practically deployable framework that can be integrated into routine clinical workflows for multimodal decision support in AD.

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arXiv2025

Atkin and Swinnerton-Dyer congruences for meromorphic modular forms

In the 1970's, Atkin and Swinnerton-Dyer conjectured that Fourier coefficients of holomorphic modular cusp forms on noncongruence subgroups of $\text{SL}_2(\mathbb{Z})$ satisfy certain $p$-adic recurrence relations which are analogous to Hecke's recurrence relations for congrunece subgroups. In 1985, this was proven in seminal work of Scholl and it was recently extended to weakly holomorphic modular forms by Kazalicki and Scholl. We show that Atkin and Swinnerton-Dyer type congruences extend to the setting of meromorphic modular forms and that the $p$-adic recurrence relations arise from Scholl's congruences in addition to a contribution of fibers of universal elliptic curves at the poles. Moreover, when the poles are located at CM points, we exploit the CM structure to reduce these $p$-adic recurrence relations to $2$-term relations and we give explicit examples. Using this framework, we partially prove conjectures that certain meromorphic modular forms are magnetic.

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arXiv2025

ADAgent: LLM Agent for Alzheimer's Disease Analysis with Collaborative Coordinator

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 alleviate the progressive neurodegeneration. However, most existing methods rely on single-modality data, which contrasts with the multifaceted approach used by medical experts. While some deep learning approaches process multi-modal data, they are limited to specific tasks with a small set of input modalities and cannot handle arbitrary combinations. This highlights the need for a system that can address diverse AD-related tasks, process multi-modal or missing input, and integrate multiple advanced methods for improved performance. In this paper, we propose ADAgent, the first specialized AI agent for AD analysis, built on a large language model (LLM) to address user queries and support decision-making. ADAgent integrates a reasoning engine, specialized medical tools, and a collaborative outcome coordinator to facilitate multi-modal diagnosis and prognosis tasks in AD. Extensive experiments demonstrate that ADAgent outperforms SOTA methods, achieving significant improvements in accuracy, including a 2.7% increase in multi-modal diagnosis, a 0.7% improvement in multi-modal prognosis, and enhancements in MRI and PET diagnosis tasks.

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arXiv2024

The Explicit Hypergeometric-Modularity Method II

In the first paper of this sequence, we provided an explicit hypergeometric modularity method by combining different techniques from the classical, $p$-adic, and finite field settings. In this article, we explore an application of this method from a motivic viewpoint through some known hypergeometric well-poised formulae of Whipple and McCarthy. We first use the method to derive a class of special weight three modular forms, labeled as $\mathbb{K}_2$-functions. Then using well-poised hypergeometric formulae we further construct a class of degree four Galois representations of the absolute Galois groups of the corresponding cyclotomic fields. These representations are then shown to be extendable to $G_{\mathbb{Q}}$ and the $L$-function of each extension coincides with the $L$-function of an automorphic form.

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arXiv2024

Traces of Hecke Operators via Hypergeometric Character Sums

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 are in terms of hypergeometric character sums over finite fields, a theory developed largely by Greene, Katz, Beukers-Cohen-Mellit, and Fuselier-Long-Ramakrishna-Swisher-Tu. Our approach, in contrast to the previous works, is uniform and more geometric, and it works equally well for forms on elliptic modular curves and Shimura curves. The same method can be applied to obtain eigenvalues of Hecke operators as well.

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arXiv2024

Hodge numbers of hypergeometric data

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 to compute Hodge numbers for hypergeometric data defined over $\mathbb Q$.

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arXiv2024

The Explicit Hypergeometric-Modularity Method I

The theories of hypergeometric functions and modular forms are highly intertwined. For example, particular values of truncated hypergeometric functions and hypergeometric character sums are often congruent or equal to Fourier coefficients of modular forms. In this series of papers, we develop and explore an explicit "Hypergeometric-Modularity" method for associating a modular form to a given hypergeometric datum. In particular, for certain length three and four hypergeometric data we give an explicit method for finding a modular form $f$ such that the corresponding hypergeometric Galois representation has a subrepresentation isomorphic to the Deligne representation of $f$. Our method utilizes Ramanujan's theory of elliptic functions to alternative bases, commutative formal group laws, and supercongruences. As a byproduct, we give a collection of eta quotients with multiplicative coefficients constructed from hypergeometric functions. In the second paper, we discuss a number of applications, including explicit connections between hypergeometric values and periods of these explicit eta quotients as well as evaluation formulae for certain special $L$-values.

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arXiv2021

A Whipple $_7F_6$ formula revisited

A well-known formula of Whipple relates certain hypergeometric values $_7F_6(1)$ and $_4F_3(1)$. In this paper we revisit this relation from the viewpoint of the underlying hypergeometric data $HD$, to which there are also associated hypergeometric character sums and Galois representations. We explain a special structure behind Whipple's formula when the hypergeometric data $HD$ are primitive and self-dual. If the data are also defined over $\mathbb Q$, by the work of Katz, Beukers, Cohen, and Mellit, there are compatible families of $\ell$-adic representations of the absolute Galois group of $\mathbb Q$ attached to $HD$. For specialized choices of $HD$, these Galois representations are shown to be decomposable and automorphic. As a consequence, the values of the corresponding hypergeometric character sums can be explicitly expressed in terms of Fourier coefficients of certain modular forms. We further relate the hypergeometric values $_7F_6(1)$ in Whipple's formula to the periods of these modular forms.

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arXiv2017

Supercongruences for rigid hypergeometric Calabi--Yau threefolds

We establish the supercongruences for the fourteen rigid hypergeometric Calabi--Yau threefolds over $\mathbb Q$ conjectured by Rodriguez-Villegas in 2003. Our first method is based on Dwork's theory of $p$-adic unit roots and it allows us to establish the supercongruences between the truncated hypergeometric series and the corresponding unit roots for ordinary primes. The other method makes use of the theory of hypergeometric motives, in particular, adapts the techniques from the recent work of Beukers, Cohen and Mellit on finite hypergeometric sums over $\mathbb Q$. Essential ingredients in executing the both approaches are the modularity of the underlying Calabi--Yau threefolds and a $p$-adic perturbation method applied to hypergeometric functions.

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arXiv2016

Potentially $\text{GL}_2$-type Galois representations associated to noncongruence modular forms

In this paper, we consider Galois representations of the absolute Galois group $\text{Gal}(\overline {\mathbb Q}/\mathbb Q)$ attached to modular forms for noncongruence subgroups of $\text{SL}_2(\mathbb Z)$. When the underlying modular curves have a model over $\mathbb Q$, these representations are constructed by Scholl and are referred to as Scholl representations, which form a large class of motivic Galois representations. In particular, by a result of Belyi, Scholl representations include the Galois actions on the Jacobian varieties of algebraic curves defined over $\mathbb Q$. As Scholl representations are motivic, they are expected to correspond to automorphic representations according to the Langlands philosophy. Using recent developments in the automorphy lifting theorem, we obtain various automphy and potential automorphy results for potentially $\text{GL}_2$-type Galois representations associated to noncongruence modular forms. Our results are applied to various kinds of examples. Especially, we obtain potential automorphy results for Galois representations attached to an infinite family of spaces of weight 3 noncongruence cusp forms of arbitrarily large dimensions.

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arXiv2016

Characterization of intersecting families of maximum size in $PSL(2,q)$

We consider the action of the $2$-dimensional projective special linear group $PSL(2,q)$ on the projective line $PG(1,q)$ over the finite field $\F_q$, where $q$ is an odd prime power. A subset $S$ of $PSL(2,q)$ is said to be an intersecting family if for any $g_1,g_2 \in S$, there exists an element $x\in PG(1,q)$ such that $x^{g_1}= x^{g_2}$. It is known that the maximum size of an intersecting family in $PSL(2,q)$ is $q(q-1)/2$. We prove that all intersecting families of maximum size are cosets of point stabilizers for all odd prime powers $q>3$.

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arXiv2015

Hypergeometric Functions over Finite Fields

Building on the developments of many people including Evans, Greene, Katz, McCarthy, Ono, Roberts, and Rodriguez-Villegas, we consider period functions for hypergeometric type algebraic varieties over finite fields and consequently study hypergeometric functions over finite fields in a manner that is parallel to that of the classical hypergeometric functions. Using a comparison between the classical gamma function and its finite field analogue the Gauss sum, we give a systematic way to obtain certain types of hypergeometric transformation and evaluation formulas over finite fields and interpret them geometrically using a Galois representation perspective. As an application, we obtain a few finite field analogues of algebraic hypergeometric identities, quadratic and higher transformation formulas, and evaluation formulas. We further apply these finite field formulas to compute the number of rational points of certain hypergeometric varieties.

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arXiv2014

Generalized Legendre curves and Quaternionic Multiplication

This paper is devoted to abelian varieties arising from generalized Legendre curves. In particular, we consider their corresponding Galois representations, periods, and endomorphism algebras. For certain one parameter families of 2-dimensional abelian varieties of this kind, we determine when the endomorphism algebra of each fiber defined over the algebraic closure of $\Bbb Q$ contains a quaternion algebra.

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arXiv2014

On a conjecture of Kimoto and Wakayama

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 series and p-adic analysis.

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