TITLE: Obstruction in modular towers.
(Darren Semmen)
SLIDES:
semmen-RedLodgeSlides.pdf
ABSTRACT: The
difficulty in proving the regular version of the Inverse Galois Problem
would be explained by the (conjectural) disappearance of rational
points on high levels of any modular tower. It is already known
that there are no projective sequences of rational points, so the
current strategy to prove this conjecture is to characterize infinite
sequences of components and (for r =
4) show that their genera are unbounded. This talk surveys tools
required for the first task (i.e. the analysis of obstruction of
connected components):
- The modular representations
that define the sequence of finite groups used to create a modular
tower.
- Cohomological dimension and
Poincarè duality groups.
REFERENCES: The two
survey
talks of Anna
Cadoret and Pierre
Dèbes.
D. Semmen, The group theory behind
modular towers, in Groupes
de
Galois arithmétiques et différentiels (Luminy
2004; eds. D. Bertrand and P. Dèbes), Séminaires et
Congrès, 13, (to
appear).
At http://members.cox.net/dsemmen/
K.S.
Brown, Cohomology of groups,
Graduate Texts in Mathematics, 87.
Springer-Verlag, New
York-Berlin, 1982.
M.D. Fried, The Main Conjecture of
Modular Towers and its higher
rank generalization, in Groupes
de
Galois arithmétiques et différentiels (Luminy
2004; eds. D. Bertrand and P. Dèbes), Séminaires et
Congrès, 13, (to
appear). At http://www.math.uci.edu/~mfried/talkfiles/lum03-12-04.html