arXiv subjects

Ming-chang Kang

Publications and source records attributed to Ming-chang Kang.

At least 19 recordsLinked to original sources

arXiv2021

A two-dimensional rationality problem and intersections of two quadrics

Let $k$ be a field with char $k\neq 2$ and $k$ be not algebraically closed. Let $a\in k\setminus k^2$ and $L=k(\sqrt{a})(x,y)$ be a field extension of $k$ where $x,y$ are algebraically independent over $k$. Assume that $σ$ is a $k$-automorphism on $L$ defined by \[ σ: \sqrt{a}\mapsto -\sqrt{a},\ x\mapsto \frac{b}{x},\ y\mapsto \frac{c(x+\frac{b}{x})+d}{y} \] where $b,c,d \in k$, $b\neq 0$ and at least one of $c,d$ is non-zero. Let $L^{\langleσ\rangle}=\{u\in L:σ(u)=u\}$ be the fixed subfield of $L$. We show that $L^{\langleσ\rangle}$ is isomorphic to the function field of a certain surface in $P^4_k$ which is given as the intersection of two quadrics. We give criteria for the $k$-rationality of $L^{\langleσ\rangle}$ by using the Hilbert symbol. As an appendix of the paper, we also give an alternative geometric proof of a part of the result which is provided to the authors by J.-L. Colliot-Thélène.

math.AG↗

arXiv2020

Noether's problem for some subgroups of $S_{14}$: the modular case

Let $G$ be a subgroup of $S_{n}$, the symmetric group of degree $n$. For any field $k$, $G$ acts naturally on the rational function field $k(x_{1},\cdots,x_{n})$ via $k$-automorphisms defined by $σ\cdot x_{i}:=x_{σ\cdot i}$ for any $σ\in G$ and $1\leq i\leq n$. In this article, we will show that if $G$ is a solvable transitive subgroup of $S_{14}$ and $\text{char}(k)=7$, then the fixed subfield $k(x_{1},\cdots,x_{14})^{G}$ is rational (i.e., purely transcendental) over $k$. In proving the above theorem, we rely on the Kuniyoshi-Gaschütz Theorem or some ideas in its proof.

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arXiv2019

An application of cohomological invariants

Let $G$ be a finite group, $k$ be a field and $G\to GL(V_{\rm reg})$ be the regular representation of $G$ over $k$. Then $G$ acts naturally on the rational function field $k(V_{\rm reg})$ by $k$-automorphisms. Define $k(G)$ to be the fixed field $k(V_{\rm reg})^G$. Noether's problem asks whether $k(G)$ is rational (resp. stably rational) over $k$. When $k=\bQ$ and $G$ contains a normal subgroup $N$ with $G/H\simeq C_8$ (the cyclic group of order $8$), Jack Sonn proves that $\bQ(G)$ is not stably rational over $\bQ$, which is a non-abelian extension of a theorem of Endo-Miyata, Voskresenskii, Lenstra and Saltman for the abelian Noether's problem $\bQ(C_8)$. Using the method of cohomological invariants, we are able to generalize Sonn's theorem as follows. Theorem. Let $G$ be a finite group and $N$ $\lhd$ $G$ such that $G/N\simeq C_{2^n}$ with $n\geq 3$. If $k$ is a field satisfying that ${\rm char}\,k=0$ and $k(ζ_{2^n})/k$ is not a cyclic extension where $ζ_{2^n}$ is a primitive $2^n$-th root of unity, then $k(G)$ is not stably rational (resp. not retract rational) over $k$. \end{abstract}

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arXiv2019

Degree three unramified cohomology groups and Noether's problem for groups of order $243$

Let $k$ be a field and $G$ be a finite group acting on the rational function field $k(x_g : g\in G)$ by $k$-automorphisms defined as $h(x_g)=x_{hg}$ for any $g,h\in G$. We denote the fixed field $k(x_g : g\in G)^G$ by $k(G)$. Noether's problem asks whether $k(G)$ is rational (= purely transcendental) over $k$. It is well-known that if $C(G)$ is stably rational over $C$, then all the unramified cohomology groups $H_[nr}^i(C(G),Q/Z)=0$ for $i \ge 2$. Hoshi, Kang and Kunyavskii [HKK] showed that, for a $p$-group of order $p^5$ ($p$: an odd prime number), $H_[nr}^2(C(G),Q/Z)\neq 0$ if and only if $G$ belongs to the isoclinism family $Φ_{10}$. When $p$ is an odd prime number, Peyre [Pe3] and Hoshi, Kang and Yamasaki [HKY1] exhibit some $p$-groups $G$ which are of the form of a central extension of certain elementary abelian $p$-group by another one with $H_[nr}^2(C(G),Q/Z)=0$ and $H_[nr}^3(C(G),Q/Z)\neq 0$. However, it is difficult to tell whether $H_[nr}^3(C(G),Q/Z)$ is non-trivial if $G$ is an arbitrary finite group. In this paper, we are able to determine $H_[nr}^3(C(G),Q/Z)$ where $G$ is any group of order $p^5$ with $p=3, 5, 7$. Theorem 1. Let $G$ be a group of order $3^5$. Then $H_[nr}^3(C(G),Q/Z)\neq 0$ if and only if $G$ belongs to $Φ_7$. Theorem 2. If $G$ is a group of order $3^5$, then the fixed field $C(G)$ is rational if and only if $G$ does not belong to $Φ_{7}$ and $Φ_{10}$. Theorem 3. Let $G$ be a group of order $5^5$ or $7^5$. Then $H_[nr}^3(C(G),Q/Z)\neq 0$ if and only if $G$ belongs to $Φ_6$, $Φ_7$ or $Φ_{10}$. Theorem 4. If $G$ is the alternating group $A_n$, the Mathieu group $M_{11}$, $M_{12}$, the Janko group $J_1$ or the group $PSL_2(F_q)$, $SL_2(F_q)$, $PGL_2(F_q)$ (where $q$ is a prime power), then $H_[nr}^d(C(G),Q/Z)=0$ for any $d\ge 2$. Besides the degree three unramified cohomology groups, we compute also the stable cohomology groups.

math.AG↗

arXiv2018

Multiplicative Invariant Fields of Dimension \le 6

The finite subgroups of $GL_4(\bm{Z})$ are classified up to conjugation in \cite{BBNWZ}; in particular, there exist $710$ non-conjugate finite groups in $GL_4(\bm{Z})$. Each finite group $G$ of $GL_4(\bm{Z})$ acts naturally on $\bm{Z}^{\oplus 4}$; thus we get a faithful $G$-lattice $M$ with ${\rm rank}_\bm{Z} M=4$. In this way, there are exactly $710$ such lattices. Given a $G$-lattice $M$ with ${\rm rank}_\bm{Z} M=4$, the group $G$ acts on the rational function field $\bm{C}(M):=\bm{C}(x_1,x_2,x_3,x_4)$ by multiplicative actions, i.e. purely monomial automorphisms over $\bm{C}$. We are concerned with the rationality problem of the fixed field $\bm{C}(M)^G$. A tool of our investigation is the unramified Brauer group of the field $\bm{C}(M)^G$ over $\bm{C}$. A formula of the unramified Brauer group ${\rm Br}_u(\bm{C}(M)^G)$ for the multiplicative invariant field was found by Saltman in 1990. However, to calculate ${\rm Br}_u(\bm{C}(M)^G)$ for a specific multiplicatively invariant field requires additional efforts, even when the lattice $M$ is of rank equal to $4$. Theorem 1. Among the $710$ finite groups $G$, let $M$ be the associated faithful $G$-lattice with ${\rm rank}_\bm{Z} M=4$, there exist precisely $5$ lattices $M$ with ${\rm Br}_u(\bm{C}(M)^G)\neq 0$. In these situations, $B_0(G)=0$ and thus ${\rm Br}_u(\bm{C}(M)^G)\subset H^2(G,M)$. The {\rm GAP IDs} of the five groups $G$ are {\rm (4,12,4,12), (4,32,1,2), (4,32,3,2), (4,33,3,1), (4,33,6,1)} in {\rm \cite{BBNWZ}} and in {\rm \cite{GAP}}. Theorem 2. There exist $6079$ finite subgroups $G$ in $GL_5(\bm{Z})$. Let $M$ be the lattice with rank $5$ associated to each group $G$. Among these lattices precisely $46$ of them satisfy the condition ${\rm Br}_u(\bm{C}(M)^G)\neq 0$. A similar result for lattices of rank $6$ is found also.

math.AG↗

arXiv2017

Rationality problems for relation modules of dihedral groups

Let D_n be the dihedral group of order 2n where n \ge 2, 1 \to R \to F \to D_n \to 1 be a free presentation of D_n. R^{ab}:=R/[R,R] becomes a \bm{Z}[D_n]-lattice. We will study the module structure and the rationality problem of R^{ab}.

math.NT↗

arXiv2017

Noether's Problem for Some Semidirect Products

Let $k$ be a field, $G$ be a finite group, $k(x(g):g\in G)$ be the rational function field with the variables $x(g)$ where $g\in G$. The group $G$ acts on $k(x(g):g\in G)$ by $k$-automorphisms where $h\cdot x(g)=x(hg)$ for all $h,g\in G$. Let $k(G)$ be the fixed field defined by $k(G):=k(x(g):g\in G)^G=\{f\in k(x(g):g\in G): h\cdot f=f$ for all $h\in G\}$. Noether's problem asks whether the fixed field $k(G)$ is rational (= purely transcendental) over $k$. Let $m$ and $n$ be positive integers and assume that there is an integer $t$ such that $t\in (\bm{Z}/m\bm{Z})^\times$ is of order $n$. Define a group $G_{m,n}:=\langleσ,τ:σ^m=τ^n=1,τ^{-1}στ=σ^t\rangle$ $\simeq C_m \rtimes C_n$. We will find a sufficient condition to guarantee that $k(G)$ is rational over $k$. As a result, it is shown that, for any positive integer $n$, the set $S:=\{p: p$ is a prime number such that $\bm{C}(G_{p,n})$ is rational over $\bm{C} \}$ is of positive Dirichlet density; in particular, $S$ is an infinite set.

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arXiv2016

A note on Plans's paper of Noether's problem

Let $p$ be a prime number and $ζ_p$ be a primitive $p$-th root of unity in $\bm{C}$. Let $k$ be a field and $k(x_0,\ldots,x_{p-1})$ be the rational function field of $p$ variables over $k$. Suppose that $G=\langleσ\rangle \simeq C_p$ acts on $k(x_0,\ldots,x_{p-1})$ by $k$-automorphisms defined as $σ:x_0\mapsto x_1\mapsto\cdots\mapsto x_{p-1}\mapsto x_0$. Denote by $P$ the set of all prime numbers and define $P_0=\{p\in P:\bm{Q}(ζ_p)$ is of class number one$\}$. Theorem. If $k$ is an algebraic number field and $p\in P\backslash (P_0\cup P_k)$, then $k(x_0,\ldots,x_{p-1})^G$ is not stably rational over $k$ where $P_k=\{p\in P: p$ is ramified in $k\}$.

math.NT↗

arXiv2015

Cartan maps and projective modules

Let $R$ be a commutative ring, $π$ be a finite group, $Rπ$ be the group ring of $π$ over $R$. Theorem 1. If $R$ is a commutative artinian ring and $π$ is a finite group. Then the Cartan map $c:K_0(Rπ)\to G_0(Rπ)$ is injective. Theorem 2. Suppose that $R$ is a Dedekind domain with $\fn{char}R=p>0$ and $π$ is a $p$-group. Then every finitely generated projective $Rπ$-module is isomorphic to $F \oplus cA$ where $F$ is a free module and $cA$ is a projective ideal of $Rπ$. Moreover, $R$ is a principal ideal domain if and only if every finitely generated projective $Rπ$-module is isomorphic to a free module. Theorem 3. Let $R$ be a commutative noetherian ring with total quotient ring $K$, $A$ be an $R$-algebra which is a finitely generated $R$-projective module. Suppose that $I$ is an ideal of $R$ such that $R/I$ is artinian. Let $\{cM_1,\ldots,cM_n\}$ be the set of all maximal ideals of $R$ containing $I$. Assume that the Cartan map $c_i: K_0(A/cM_iA)\to G_0(A/cM_iA)$ is injective for all $1\le i\le n$. If $P$ and $Q$ are finitely generated $A$-projective modules with $KP\simeq KQ$, then $P/IP\simeq Q/IQ$.

math.GR↗

arXiv2015

Degree Three Unramified Cohomology Groups

Let $p$ be an odd prime number. Peyre shows that there is a group $G$ of order $p^{12}$ such that $H_{nr}^3(\bm{C}(G), \bm{Q}/\bm{Z})$ is non-trivial. Using Peyre's method, we are able to prove that the same conclusion is true for some groups of order $p^9$.

math.AG↗

arXiv2014

Invertible Lattices

Theorem. Let $π$ be a finite group of order $n$, $R$ be a Dedekind domain satisfying that (i) $\fn{char}R=0$, (ii) every prime divisor of $n$ is not invertible in $R$, and (iii) $p$ is unramified in $R$ for any prime divisor $p$ of $n$. Then all the flabby (resp.\ coflabby) $Rπ$-lattices are invertible if and only if all the Sylow subgroups of $π$ are cyclic. The above theorem was proved by Endo and Miyata when $R=\bm{Z}$ \cite[Theorem 1.5]{EM}. As applications of this theorem, we give a short proof and a partial generalization of a result of Torrecillas and Weigel \cite[Theorem A]{TW}, which was proved using cohomological Mackey functors.

math.NT↗

arXiv2014

Noether's problems for groups of order 243

Let $k$ be any field, $G$ be a finite group. Let $G$ act on the rational function field $k(x_g:g\in G)$ by $k$-automorphisms defined by $h\cdot x_g=x_{hg}$ for any $g,h\in G$. Denote by $k(G)=k(x_g:g\in G)^G$ the fixed field. Noether's problem asks, under what situations, the fixed field $k(G)$ will be rational (= purely transcendental) over $k$. According to the data base of GAP there are $10$ isoclinism families for groups of order $243$. It is known that there are precisely $3$ groups $G$ of order $243$ (they consist of the isoclinism family $Φ_{10}$) such that the unramified Brauer group of $\bm{C}(G)$ over $\bm{C}$ is non-trivial. Thus $\bm{C}(G)$ is not rational over $\bm{C}$. We will prove that, if $ζ_9 \in k$, then $k(G)$ is rational over $k$ for groups of order $243$ other than these $3$ groups, except possibly for groups belonging to the isoclinism family $Φ_7$.

math.AG↗

arXiv2014

Class Numbers and Algebraic Tori

Let $p$ be an odd prime number, $D_p$ be the dihedral group of order $2p$, $h_p$ and $h^+_p$ be the class numbers of $\bm{Q}(ζ_p)$ and $\bm{Q}(ζ_p+ ζ_p^{-1})$ respectively. Theorem. $h_p^+=1$ if and only if, for any field $k$ admitting a $D_p$-extension, all the algebraic $D_p$-tori over $k$ are stably rational. A similar result for $h_p=1$ and $C_p$-tori is valid also.

math.NT↗

arXiv2013

Invariants of wreath products and subgroups of S_6

Let $G$ be a subgroup of $S_6$, the symmetric group of degree 6. For any field $k$, $G$ acts naturally on the rational function field $k(x_1,...,x_6)$ via $k$-automorphisms defined by $σ\cdot x_i=x_{σ(i)}$ for any $σ\in G$, any $1\le i\le 6$. Theorem. The fixed field $k(x_1,...,x_6)^G$ is rational (=purely transcendental) over $k$, except possibly when $G$ is isomorphic to $PSL_2(\bm{F}_5)$, $PGL_2(\bm{F}_5)$ or $A_6$. When $G$ is isomorphic to $PSL_2(\bm{F}_5)$ or $PGL_2(\bm{F}_5)$, then $\bm{C}(x_1,...,x_6)^G$ is $\bm{C}$-rational and $k(x_1,...,x_6)^G$ is stably $k$-rational for any field $k$. The invariant theory of wreath products will be investigated also.

math.AG↗

arXiv2013

Rational invariants for subgroups of S_5 and S_7

Let $G$ be a subgroup of $S_n$, the symmetric group of degree $n$. For any field $k$, $G$ acts naturally on the rational function field $k(x_1,x_2,\ldots,x_n)$ via $k$-automorphisms defined by $σ\cdot x_i=x_{σ(i)}$ for any $σ\in G$, any $1\le i\le n$. Theorem. If $n\le 5$, then the fixed field $k(x_1,\ldots,x_n)^G$ is purely transcendental over $k$. We will show that $\bm{C}(x_1,\ldots,x_7)^G$ is also purely transcendental over $\bm{C}$ if $G$ is any transitive subgroups of $S_7$ other than $A_7$; a similar result is valid for solvable transitive subgroups of $S_{11}$.

math.AG↗

arXiv2013

The Bogomolov multiplier of rigid finite groups

The Bogomolov multiplier of a finite group $G$ is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of $G$. This invariant of $G$ plays an important role in birational geometry of quotient spaces $V/G$. We show that in many cases the vanishing of the Bogomolov multiplier is guaranteed by the rigidity of $G$ in the sense that it has no outer class-preserving automorphisms.

math.GR↗

arXiv2013

Action of dihedral groups

Let $K$ be any field and $G$ be a finite group. Let $G$ act on the rational function field $K(x_g: \ g \in G)$ by $K$-automorphisms defined by $g \cdot x_h=x_{gh}$ for any $g, \ h \in G$. Denote by $K(G)$ the fixed field $K(x_g: \ g \in G)^G$. Noether's problem asks whether $K(G)$ is rational (=purely transcendental) over $K$. We will give a brief survey of Noether's problem for abelian groups and dihedral groups, and will show that $\Bbb Q(D_n)$ is rational over $\Bbb Q$ for $n \le 10$.

math.AG↗

arXiv2012

Bogomolov multipliers and retract rationality for semi-direct products

Let $G$ be a finite group. The Bogomolov multiplier $B_0(G)$ is constructed as an obstruction to the rationality of $\bm{C}(V)^G$ where $G\to GL(V)$ is a faithful representation over $\bm{C}$. We prove that, for any finite groups $G_1$ and $G_2$, $B_0(G_1\times G_2)\xrightarrow{\sim} B_0(G_1)\times B_0(G_2)$ under the restriction map. If $G=N\rtimes G_0$ with $\gcd\{|N|,|G_0|\}=1$, then $B_0(G)\xrightarrow{\sim} B_0(N)^{G_0}\times B_0(G_0)$ under the restriction map. For any integer $n$, we show that there are non-direct-product $p$-groups $G_1$ and $G_2$ such that $B_0(G_1)$ and $B_0(G_2)$ contain subgroups isomorphic to $(\bm{Z}/p \bm{Z})^n$ and $\bm{Z}/p^n \bm{Z}$ respectively. On the other hand, if $k$ is an infinite field and $G=N\rtimes G_0$ where $N$ is an abelian normal subgroup of exponent $e$ satisfying that $ζ_e\in k$, we will prove that, if $k(G_0)$ is retract $k$-rational, then $k(G)$ is also retract $k$-rational provided that certain "local" conditions are satisfied; this result generalizes two previous results of Saltman and Jambor \cite{Ja}.

math.AG↗