On elements of prime order in the plane Cremona group over a perfect field

2007/07/29 by Igor Dolgachev, Dolgachev, Igor V., V. A. Iskovskikh +1 · 3 citations
Mathematics · #12F12 #14E07 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.0707.4305

openalex publication_date 2007/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the plane Cremona group over a perfect field k of characteristic p ≥ 0 contains an element of prime order ℓ≥ 7 not equal to p if and only if there exists a 2-dimensional algebraic torus T over k such that T(k) contains an element of order ℓ. If p = 0 and k does not contain a primitive ℓ-th root of unity, we show that there are no elements of prime order ℓ > 7 in \Cr₂(k) and all elements of order 7 are conjugate.

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• Arithmetic toric varieties
• Representations of finite subgroups of Cremona groups
• Algebraic tori - thirty years after

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