Breakthrough Tracker record
Modularity theorems for a positive proportion of abelian surfaces
Under specified local, image and ordinary or distinguished hypotheses, the paper proves modularity for a positive proportion of abelian surfaces over the rational numbers.
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- Stable ID
math-modularity-abelian-surfaces-2025- Revision
math-modularity-abelian-surfaces-2025.v1- Field
- Mathematics · Arithmetic geometry
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Current · Expert-supported preprint and major advance
- Last checked
AI role
No generative-AI role was disclosed.
Record details
- Problem or result
- Modularity of abelian surfaces over the rational numbers
- Authors
- George Boxer, Frank Calegari, Toby Gee and Vincent Pilloni
- Institutions
- Imperial College London, University of Chicago and CNRS/Université Paris-Saclay affiliations are represented
- Result date
- Preprint submitted February 28, 2025
Why it matters
The work extends modularity machinery central to the proof of Fermat's Last Theorem from elliptic curves toward two-dimensional abelian varieties.
Limits
This is not a theorem that every abelian surface over the rational numbers is modular. The positive-proportion and hypothesis restrictions are essential.
Sources
- Primary: Boxer, Calegari, Gee and Pilloni, Modularity theorems for abelian surfaces
- Independent: Quanta, The Core of Fermat's Last Theorem Just Got Superpowered
Correction and revision history
- Added on 2026-07-22 with positive-proportion and technical hypotheses preserved against broader headline claims.
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