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On a rationality problem for fields of cross-ratios II

  • Tran-Trung Nghiem (a1) and Zinovy Reichstein (a2)

Abstract

Let k be a field, $x_1, \dots , x_n$ be independent variables and let $L_n = k(x_1, \dots , x_n)$ . The symmetric group $\operatorname {\Sigma }_n$ acts on $L_n$ by permuting the variables, and the projective linear group $\operatorname {PGL}_2$ acts by

$$ \begin{align*} \begin{pmatrix} a & b \\ c & d \end{pmatrix}\, \colon x_i \longmapsto \frac{a x_i + b}{c x_i + d} \end{align*} $$
for each $i = 1, \dots , n$ . The fixed field $L_n^{\operatorname {PGL}_2}$ is called “the field of cross-ratios”. Given a subgroup $S \subset \operatorname {\Sigma }_n$ , H. Tsunogai asked whether $L_n^S$ rational over $K_n^S$ . When $n \geqslant 5,$ the second author has shown that $L_n^S$ is rational over $K_n^S$ if and only if S has an orbit of odd order in $\{ 1, \dots , n \}$ . In this paper, we answer Tsunogai’s question for $n \leqslant 4$ .

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Tran-Trung Nghiem was supported by a Fondation Hadamard Scholarship. Zinovy Reichstein was partially supported by National Sciences and Engineering Research Council of Canada Discovery grant 253424-2017.

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[GPW] Groupprops, The Group Properties Wiki, Subgroup structure of symmetric group ${S}_4$ . https://groupprops.subwiki.org/wiki/Subgroup_structure_of_symmetric_group:S4
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[RS20] Reichstein, Z. and Scavia, F., The Noether problem for spinor groups of small rank. J. Algebra 548(2020), 134152. MR4046156
[Se97] Serre, J.-P., Galois cohomology. Translated from the French by Patrick Ion and revised by the author. Springer-Verlag, Berlin, 1997. http://dx.doi.org/10.1007/978-3-642-59141-9
[Tsu17] Tsunogai, H., Toward Noether’s problem for the fields of cross-ratios . Tokyo J. Math. 39(2017), no. 3, 901922. http://dx.doi.org/10.3836/tjm/1491465735
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On a rationality problem for fields of cross-ratios II

  • Tran-Trung Nghiem (a1) and Zinovy Reichstein (a2)

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