Sharpening `Manin-Mumford' for certain algebraic groups of dimension 2
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Thu, 01/12/2011 16:00 |
Umberto Zannier (Pisa) |
Logic Seminar  Number Theory Seminar |
L3 |
(Joint work with P. Corvaja and D.
Masser.)
The topic of the talk arises from the
Manin-Mumford conjecture and its extensions, where we shall
focus on the case of (complex connected) commutative
algebraic groups  of dimension  . The `Manin-Mumford'
context in these cases predicts finiteness for the set of
torsion points in an algebraic curve inside , unless the
curve is of `special' type, i.e. a translate of an algebraic
subgroup of .
In the talk we shall consider not merely the set of torsion
points, but its topological closure in (which turns out
to be also the maximal compact subgroup). In the case of
abelian varieties this closure is the whole space, but this is
not so for other ; actually, we shall prove that in certain
cases (where a natural dimensional condition is fulfilled) the
intersection of this larger set with a non-special curve
must still be a finite set.
We shall conclude by stating in brief some extensions of
this problem to higher dimensions. |
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