On the simplest sextic fields and related Thue equations

We consider the parametric family of sextic Thue equations $$x^6-2mx^5y-5(m+3)x^4y^2-20x^3y^3+5mx^2y^4+2(m+3)xy^5+y^6=\lambda$$ where $m\in\mathbb{Z}$ is an integer and $\lambda$ is a divisor of $27(m^2+3m+9)$. We show that the only solutions to the equations are the trivial ones with $xy(x+y)(x-y)(x+2y)(2x+y)=0$.Comment: 12 pages, 2 table

Birational classification of fields of invariants for groups of order 128128

Let $G$ be a finite group acting on the rational function field $\mathbb{C}(x_g : g\in G)$ by $\mathbb{C}$-automorphisms $h(x_g)=x_{hg}$ for any $g,h\in G$. Noether's problem asks whether the invariant field $\mathbb{C}(G)=k(x_g : g\in G)^G$ is rational (i.e. purely transcendental) over $\mathbb{C}$. Saltman and Bogomolov, respectively, showed that for any prime $p$ there exist groups $G$ of order $p^9$ and of order $p^6$ such that $\mathbb{C}(G)$ is not rational over $\mathbb{C}$ by showing the non-vanishing of the unramified Brauer group: $Br_{nr}(\mathbb{C}(G))\neq 0$. For $p=2$, Chu, Hu, Kang and Prokhorov proved that if $G$ is a 2-group of order $\leq 32$, then $\mathbb{C}(G)$ is rational over $\mathbb{C}$. Chu, Hu, Kang and Kunyavskii showed that if $G$ is of order 64, then $\mathbb{C}(G)$ is rational over $\mathbb{C}$ except for the groups $G$ belonging to the two isoclinism families $\Phi_{13}$ and $\Phi_{16}$. Bogomolov and B\"ohning's theorem claims that if $G_1$ and $G_2$ belong to the same isoclinism family, then $\mathbb{C}(G_1)$ and $\mathbb{C}(G_2)$ are stably $\mathbb{C}$-isomorphic. We investigate the birational classification of $\mathbb{C}(G)$ for groups $G$ of order 128 with $Br_{nr}(\mathbb{C}(G))\neq 0$. Moravec showed that there exist exactly 220 groups $G$ of order 128 with $Br_{nr}(\mathbb{C}(G))\neq 0$ forming 11 isoclinism families $\Phi_j$. We show that if $G_1$ and $G_2$ belong to $\Phi_{16}, \Phi_{31}, \Phi_{37}, \Phi_{39}, \Phi_{43}, \Phi_{58}, \Phi_{60}$ or $\Phi_{80}$ (resp. $\Phi_{106}$ or $\Phi_{114}$), then $\mathbb{C}(G_1)$ and $\mathbb{C}(G_2)$ are stably $\mathbb{C}$-isomorphic with $Br_{nr}(\mathbb{C}(G_i))\simeq C_2$. Explicit structures of non-rational fields $\mathbb{C}(G)$ are given for each cases including also the case $\Phi_{30}$ with $Br_{nr}(\mathbb{C}(G))\simeq C_2\times C_2$.Comment: 31 page

Rationality problem for algebraic tori

We give the complete stably rational classification of algebraic tori of dimensions $4$ and $5$ over a field $k$. In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank $4$ and $5$ is given. We show that there exist exactly $487$ (resp. $7$, resp. $216$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $4$, and there exist exactly $3051$ (resp. $25$, resp. $3003$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $5$. We make a procedure to compute a flabby resolution of a $G$-lattice effectively by using the computer algebra system GAP. Some algorithms may determine whether the flabby class of a $G$-lattice is invertible (resp. zero) or not. Using the algorithms, we determine all the flabby and coflabby $G$-lattices of rank up to $6$ and verify that they are stably permutation. We also show that the Krull-Schmidt theorem for $G$-lattices holds when the rank $\leq 4$, and fails when the rank is $5$. Indeed, there exist exactly $11$ (resp. $131$) $G$-lattices of rank $5$ (resp. $6$) which are decomposable into two different ranks. Moreover, when the rank is $6$, there exist exactly $18$ $G$-lattices which are decomposable into the same ranks but the direct summands are not isomorphic. We confirm that $H^1(G,F)=0$ for any Bravais group $G$ of dimension $n\leq 6$ where $F$ is the flabby class of the corresponding $G$-lattice of rank $n$. In particular, $H^1(G,F)=0$ for any maximal finite subgroup $G\leq {\rm GL}(n,\mathbb{Z})$ where $n\leq 6$. As an application of the methods developed, some examples of not retract (stably) rational fields over $k$ are given.Comment: To appear in Mem. Amer. Math. Soc., 147 pages, minor typos are correcte