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Physical Sciences › Mathematics › Mathematical Physics
SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS
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Paper Summary
Paperzilla title
Sym-metry Wins! Unlocking the Secrets of Automorphic Forms
This paper proves the automorphy of symmetric power liftings for all n ≥ 1 for cuspidal Hecke eigenforms of level 1 and weight k > 2. The result also applies to a more general class of eigenforms, including those associated with semistable elliptic curves, significantly advancing Langlands's functoriality principle.
Possible Conflicts of Interest
None identified
Identified Weaknesses
Limited Practical Applications
The paper focuses on automorphy of symmetric powers of automorphic representations, a highly specialized area within number theory with limited direct real-world application.
High Barrier to Entry
The paper heavily relies on prior results and complex machinery, making it difficult for non-experts to grasp the core arguments and hindering broader accessibility.
Lack of Explicit Examples
While the results contribute to advancing theoretical understanding within the field, the lack of explicit examples and numerical computations limits the potential for immediate follow-up research and practical exploration of the findings.
Rating Explanation
The paper makes significant contributions to Langlands's functoriality principle by establishing the automorphy of symmetric power liftings for a wider class of modular forms, including those associated with semistable elliptic curves. The approach, combining Galois deformation theory, p-adic families, and eigenvarieties, is innovative and robust, thus earning a rating of 4 despite its technical complexity and specialized focus.
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Original Title:
SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS
File Name:
s10240-021-00127-3.pdf
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File Size:
1.68 MB
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July 14, 2025 at 11:12 AM
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