Subtopic Deep Dive
Automorphic Forms on GL(n)
Research Guide
What is Automorphic Forms on GL(n)?
Automorphic forms on GL(n) are automorphic representations and cusp forms associated with the general linear group GL(n) over local and global fields, studied for their construction, Fourier coefficients, and L-functions.
This subtopic covers cuspidal automorphic forms on GL(n) and their functorial properties, central to the Langlands program. Key works include Gelbart and Jacquet (1978, 453 citations) relating representations of GL(2) and GL(3), and Kim (2002, 663 citations) proving functoriality for the exterior square of GL(4). Over 10 highly cited papers from 1978-2016 address these forms.
Why It Matters
Automorphic forms on GL(n) underpin the Langlands program by connecting Galois representations to automorphic ones, enabling proofs of functoriality (Kim, 2002) and potential automorphy (Barnet-Lamb et al., 2011). They yield explicit L-function constructions, impacting number theory and arithmetic geometry. Applications include Shimura varieties (Kottwitz, 1992) and geometric Langlands via gauge theory (Kapustin and Witten, 2007).
Key Research Challenges
Functoriality Conjectures
Proving Langlands functoriality transfers automorphic forms between GL(n) and other groups. Kim (2002) established it for GL(4) exterior square, but general cases remain open. Arthur and Clozel (2016) advanced trace formula applications.
Explicit Constructions
Constructing cuspidal forms with prescribed properties on higher GL(n). Gelbart and Jacquet (1978) linked GL(2) and GL(3), yet explicit families for n≥4 challenge researchers. Barnet-Lamb et al. (2011) used potential automorphy for Calabi-Yau varieties.
L-Indistinguishability
Defining and verifying L-indistinguishability for automorphic forms on GL(n). Labesse and Langlands (1979) introduced it for SL(2), with extensions to GL(n) unresolved. Ties to geometric Langlands (Kapustin and Witten, 2007).
Essential Papers
Electric-magnetic duality and the geometric Langlands program
Anton Kapustin, Edward Witten · 2007 · Communications in Number Theory and Physics · 920 citations
The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N = 4 super Yang-Mills theory in four dimensions.The key ingredients a...
Functoriality for the exterior square of 𝐺𝐿₄ and the symmetric fourth of 𝐺𝐿₂
Henry Kim · 2002 · Journal of the American Mathematical Society · 663 citations
In this paper we prove the functoriality of the exterior square of cusp forms on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper ...
A relation between automorphic representations of ${\rm GL}(2)$ and ${\rm GL}(3)$
Stephen Gelbart, Hervé Jacquet · 1978 · Annales Scientifiques de l École Normale Supérieure · 453 citations
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Higher regulators and values of L-functions
Alexander Beilinson · 1985 · Journal of Mathematical Sciences · 451 citations
A Family of Calabi–Yau Varieties and Potential Automorphy II
Tom Barnet-Lamb, David Geraghty, Michael Harris et al. · 2011 · Publications of the Research Institute for Mathematical Sciences · 387 citations
We prove new potential modularity theorems for n-dimensional essentially self-dual l -adic representations of the absolute Galois group of a totally real field. Most notably, in the ordinary case w...
Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula. (AM-120), Volume 120
James Arthur, Laurent Clozel · 2016 · 371 citations
A general discovered by Robert Langlands and named by him the principle, predicts relations between forms on arithmetic subgroups of different reductive groups. Langlands functoriality relates t...
Gauge theory, ramification, and the geometric Langlands program
Sergei Gukov, Edward Witten · 2006 · Current Developments in Mathematics · 310 citations
In the gauge theory approach to the geometric Langlands program, ramification can be described in terms of "surface operators," which are supported on two-dimensional surfaces somewhat as Wilson or...
Reading Guide
Foundational Papers
Start with Gelbart and Jacquet (1978) for GL(2)-GL(3) relations, then Kim (2002) for functoriality proof, and Arthur-Clozel (2016) for trace formulas; these establish core constructions and methods.
Recent Advances
Kapustin-Witten (2007) for geometric Langlands duality; Barnet-Lamb et al. (2011) for potential automorphy; Gukov-Witten (2006) for gauge theory ramification.
Core Methods
Hecke operators, Fourier coefficients, exterior square lifts (Kim, 2002), base change, L-functions, trace formulas (Arthur-Clozel, 2016).
How PapersFlow Helps You Research Automorphic Forms on GL(n)
Discover & Search
Research Agent uses searchPapers and citationGraph to map foundational works like Gelbart and Jacquet (1978) from Kim (2002) citations, then findSimilarPapers for GL(n) functoriality extensions. exaSearch uncovers recent GL(n) L-function papers beyond top lists.
Analyze & Verify
Analysis Agent applies readPaperContent to extract functoriality proofs from Kim (2002), verifies claims via verifyResponse (CoVe) against Arthur and Clozel (2016), and runs PythonAnalysis for L-function coefficient statistics with NumPy. GRADE grading scores evidence strength in Langlands conjectures.
Synthesize & Write
Synthesis Agent detects gaps in GL(n) constructions post-Gelbart-Jacquet (1978), flags contradictions in automorphy lifts; Writing Agent uses latexEditText, latexSyncCitations for theorems, latexCompile for manuscripts, and exportMermaid for Hecke algebra diagrams.
Use Cases
"Compute Fourier coefficients from Kim (2002) exterior square functoriality using code examples."
Research Agent → searchPapers('Kim 2002 functoriality code') → Code Discovery (paperExtractUrls → paperFindGithubRepo → githubRepoInspect) → runPythonAnalysis (NumPy simulation of GL(4) coefficients) → matplotlib plots of eigenvalues.
"Draft LaTeX proof outline for GL(3) automorphic relation extending Gelbart-Jacquet."
Synthesis Agent → gap detection on GL(n>3) → Writing Agent → latexEditText (insert theorem) → latexSyncCitations (add 1978 paper) → latexCompile → PDF with Hecke operator diagram via latexGenerateFigure.
"Find GitHub repos implementing trace formulas for GL(n) automorphic forms."
Research Agent → citationGraph('Arthur Clozel 2016') → Code Discovery (paperExtractUrls → paperFindGithubRepo → githubRepoInspect) → runPythonAnalysis (verify repo trace formula code on sample L-functions) → exportCsv of results.
Automated Workflows
Deep Research workflow scans 50+ GL(n) papers via searchPapers → citationGraph, producing structured reports on functoriality progress from Gelbart-Jacquet (1978) to Kim (2002). DeepScan applies 7-step analysis with CoVe checkpoints to verify automorphy in Barnet-Lamb et al. (2011). Theorizer generates hypotheses for GL(n) L-indistinguishability from Labesse-Langlands (1979).
Frequently Asked Questions
What defines automorphic forms on GL(n)?
Cusp forms and representations on GL(n) over global fields with Hecke eigenvalues and Fourier expansions. Central to Langlands via L-functions (Gelbart and Jacquet, 1978).
What are key methods in this subtopic?
Functoriality lifts (Kim, 2002), base change (Arthur and Clozel, 2016), and potential automorphy (Barnet-Lamb et al., 2011) construct forms explicitly.
What are seminal papers?
Gelbart-Jacquet (1978, 453 citations) on GL(2)-GL(3); Kim (2002, 663 citations) on GL(4) exterior square; Kapustin-Witten (2007, 920 citations) on geometric Langlands.
What open problems exist?
General functoriality for GL(n), explicit constructions beyond n=4, full L-indistinguishability (Labesse-Langlands, 1979). Ramification in geometric settings (Gukov-Witten, 2006).
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Part of the Advanced Algebra and Geometry Research Guide