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SUMMARY:Torsion of abelian schemes and rational points on moduli spaces (j
 oint work with Anna Cadoret) - Tamagawa\, A (Kyoto)
DTSTART:20090827T083000Z
DTEND:20090827T093000Z
UID:TALK19593@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:We show the following result supporting the uniform boundednes
 s conjecture for torsion of abelian varieties: Let k be a field finitely g
 enerated over the rationals\, X a smooth curve over k\, and A an abelian s
 cheme over X. Let l be a prime number and d a positive integer. Then there
  exists a non-negative integer N\, such that\, for any closed point 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 a
 belian scheme A at x.) As a corollary of this result\, we settle the one-d
 imensional case of the so-called modular tower conjecture\, posed by Fried
  in the context of the (regular) inverse Galois problem. \n\nThe above res
 ult is obtained by combining geometric results (estimation of genus/gonali
 ty) and Diophantine results (Mordell/Mordell-Lang conjecture\, proved by F
 altings) for certain ``moduli spaces''. If we have time\, we will also exp
 lain our recent progress on a variant of these geometric results\, where t
 he set of powers of the fixed prime l is replaced by the set of all primes
 . \n\nFor an extension of the above results to more general l-adic represe
 ntations\, see Cadoret's talk on Friday. 
LOCATION:Seminar Room 1\, Newton Institute
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