代数学コロキウム

過去の記録 ~03/28次回の予定今後の予定 03/29~

開催情報 水曜日 17:00~18:00 数理科学研究科棟(駒場) 117号室
担当者 今井 直毅,ケリー シェーン

2006年12月20日(水)

16:30-18:45   数理科学研究科棟(駒場) 117号室
2講演です
Anna Cadoret 氏 (RIMS/JSPS) 16:30-17:30
On the profinite regular inverse Galois problem
[ 講演概要 ]
Given a field $k$ and a (pro)finite group $G$, consider the
following weak version of the regular inverse Galois problem:
(WRIGP/$G$/$k$) \\textit{there exists a smooth geometrically
irreducible curve $X_{G}/k$ and a Galois extension $E/k(X_{G})$
regular over $k$ with group $G$.} (the regular inverse Galois
problem (RIGP/$G$/$k$) corresponding to the case
$X_{G}=\\mathbb{P}^{1}_{k}$). A standard descent argument shows that
for a finite group $G$ the (WRIGP/$G$/$k$) can be deduced from the
(RIGP/$G$/$k((T))$). For
profinite groups $G$, the (WRIGP/$G$/$k((T))$) has been proved for
lots of fields (including the cyclotomic closure of characteristic $0$
fields) but the descent argument no longer works.\\\\
\\indent Let $p\\geq 2$ be a prime, then a profinite group
$G$ is said to be \\textit{$p$-obstructed} if it fits in a profinite group extension
$$1\\rightarrow K\\rightarrow G\\rightarrow G_{0}\\rightarrow 1$$
with $G_{0}$ a finite group and $K\\twoheadrightarrow
\\mathbb{Z}_{p}$. Typical examples of such profinite groups $G$ are
universal $p$-Frattini covers of finite $p$-perfect groups or
pronilpotent projective groups.\\\\
\\indent I will show that the (WRIGP/$G$/$k$) - even under
its weaker formulation: (WWRIGP/$G$/$k$) \\textit{there exists a
smooth geometrically irreducible curve $X_{G}/k$ and a Galois
extension $E/k(X_{G}).\\overline{k}$ with group $G$ and field of
moduli $k$.} - fails for the whole class of $p$-obstructed profinite
groups $G$ and any field $k$ which is either a finitely generated
field of characteristic $0$ or a finite field of characteristic
$\\not= p$.\\\\
\\indent The proof uses a profinite generalization of the cohomological obstruction
for a G-cover to be defined over its field of moduli and an analysis of the constrainsts
imposed on a smooth geometrically irreducible curve $X$ by a degree $p^{n}$
cyclic G-cover $X_{n}\\rightarrow X$, constrainsts which are too rigid to allow the
existence of projective systems $(X_{n}\\rightarrow
X_{G})_{n\\geq 0}$ of degree $p^{n}$ cyclic G-covers
defined over $k$. I will also discuss other implicsations of these constrainsts
for the (RIGP).
Eric Friedlander 氏 (Northwestern) 17:45-18:45
An elementary perspective on modular representation theory