The Pythagoras number of fields of transcendence degree 1 over ℚ

2025/06/26 by Olivier Benoist, Benoist, Olivier
Mathematics · #11E12 #11E25 #14G12 #14G25 #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematical and Theoretical Analysis #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2506.21380

openalex publication_date 2025/06/26 · openalex created_date 2025/10/20 · openalex updated_date 2026/07/28

Abstract

We show that any sum of squares in a field of transcendence degree 1 over ℚ is a sum of 5 squares, answering a question of Pop and Pfister. We deduce this result from a representation theorem, in k(C), for quadratic forms of rank ≥ 5 with coefficients in k, where C is a curve over a number field k.

Citations

• Bloch's formula for 0-cycles with modulus and higher dimensional Class Field Theory
• Hasse principles for higher-dimensional fields

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