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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Torsion of abelian schemes and rational points on
moduli spaces (joint work with Anna Cadoret) - Tam
agawa\, A (Kyoto)
DTSTART;TZID=Europe/London:20090827T093000
DTEND;TZID=Europe/London:20090827T103000
UID:TALK19593AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/19593
DESCRIPTION:We show the following result supporting the unifor
m boundedness conjecture for torsion of abelian va
rieties: Let k be a field finitely generated over
the rationals\, X a smooth curve over k\, and A an
abelian scheme over X. Let l be a prime number an
d d a positive integer. Then there exists a non-ne
gative integer N\, such that\, for any closed poin
t x of X with [k(x):k] leq d and any k(x)-rational
\, l-primary torsion point v of A_x\, the order of
v is leq l^N. (Here\, A_x stands for the fiber of
the abelian scheme A at x.) As a corollary of thi
s result\, we settle the one-dimensional case of t
he so-called modular tower conjecture\, posed by F
ried in the context of the (regular) inverse Galoi
s problem. \n\nThe above result is obtained by com
bining geometric results (estimation of genus/gona
lity) and Diophantine results (Mordell/Mordell-Lan
g conjecture\, proved by Faltings) for certain ``m
oduli spaces''. If we have time\, we will also exp
lain our recent progress on a variant of these geo
metric results\, where the set of powers of the fi
xed prime l is replaced by the set of all primes.
\n\nFor an extension of the above results to more
general l-adic representations\, see Cadoret's tal
k on Friday.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:Mustapha Amrani
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