Finite subgroups of Cremona group of rank 3 over the field of rational numbers

We give an explicit bound on orders of finite subgroups of Cremona group of rank three over $\mathbb{Q}$.

💡 Research Summary

The paper establishes an explicit universal bound on the order of any finite subgroup of the Cremona group of rank three over the rational numbers, Cr₃(ℚ). The author’s strategy is to reduce the problem to bounding finite subgroups of linear algebraic groups and of automorphism groups of certain three‑dimensional algebraic varieties that appear as minimal models of rational threefolds.

First, the author revisits classical results on finite subgroups of GLₙ and PGLₙ over number fields. Using Minkowski’s theorem, Schur’s theorem, and Serre’s refinements, the paper defines cyclotomic invariants mₚ(K) and tₚ(K) for a field K of characteristic zero, and from them constructs two families of p‑adic bounds νₛ𝚌ʰ(p,n) and νₛₑ(p,n). Theorem 2.5 (Serre–Schur) asserts that for any finite G⊂GLₙ(K) one has νₚ(|G|)≤νₛ𝚌ʰ(p,n). By passing to the reductive group PGLₙ, Corollary 2.14 gives νₚ(|G|)≤νₛₑ(p,n). The author then computes explicit numerical values of νₛ𝚌ʰ and νₛₑ for fields of degree ≤ 15 (Appendix A). As a consequence, concrete absolute bounds are obtained, e.g. |G|≤1 132 185 600 for any finite G⊂GL₄(K) with