Superpowered Modular Forms: Level Up Your Symmetries!

Overview

Paper Summary › Explain Like I'm Five › Conflicts of Interest › Identified Limitations › Rating Explanation › Good to know › Topic Hierarchy › File Information ›

Paper Summary

Paperzilla title

This paper proves the automorphy of the symmetric power lifting Sym^n f for all n ≥ 1, where f is a cuspidal Hecke eigenform without complex multiplication. This removes a prior assumption requiring the absence of supercuspidal primes. The proof relies on a new automorphy lifting theorem and a refined "killing ramification" technique.

Explain Like I'm Five

Scientists found that a special way to change complicated math patterns always makes them into neat, repeating shapes. They even made it work for more kinds of patterns than ever before!

Possible Conflicts of Interest

None identified

Identified Limitations

Inaccessibility to a broader audience

The paper heavily relies on specialized mathematical concepts and notations, making it very difficult for anyone outside of the niche field of automorphic forms to understand. It assumes deep familiarity with Langlands functoriality, Galois representations, and advanced algebraic number theory.

Heavy dependence on prior work

The paper builds upon prior work by the authors and others, referencing it extensively. Without access to and understanding of these prerequisite papers (e.g. [NT21], [BLGGT14], [KW09a]), evaluating the current work's originality and significance is extremely challenging.

Technical complexity of the core proof idea

The "killing ramification" technique, while central to the proof, is presented quite technically. The introduction of "seasoned" automorphic representations adds another layer of complexity. A clearer, more intuitive explanation of this core idea would improve the paper's impact.

Rating Explanation

This paper makes a significant contribution to the field of automorphic forms by removing a key assumption from prior work on symmetric power functoriality for holomorphic modular forms. While highly technical and specialized, it presents a novel automorphy lifting theorem and an intricate proof. The impact is primarily within the specialized field, hence the rating of 4 rather than 5.

Good to know

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Topic Hierarchy

Domain: Physical Sciences

Field: Mathematics

Subfield: Mathematical Physics

File Information

Original Title: SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II

Uploaded: July 14, 2025 at 11:12 AM

Privacy: Public