## Number Theory Seminar

Seminar information archive ～08/08｜Next seminar｜Future seminars 08/09～

Date, time & place | Wednesday 17:00 - 18:00 117Room #117 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | Naoki Imai, Shane Kelly |

### 2006/12/20

16:30-18:45 Room #117 (Graduate School of Math. Sci. Bldg.)

On the profinite regular inverse Galois problem

An elementary perspective on modular representation theory

**Anna Cadoret**(RIMS/JSPS) 16:30-17:30On the profinite regular inverse Galois problem

[ Abstract ]

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).

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**(Northwestern) 17:45-18:45An elementary perspective on modular representation theory