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Upper bounds on $\char93 \mbox{\cyrbig X}$

V. Kolyvagin and K. Kato [12,23] obtained upper bounds on  $\char93 \mbox{\cyr X}(A)$. To verify the full BSD conjecture for certain abelian varieties, it is necessary is to make these bounds explicit. Kolyvagin's bounds involve computations with Heegner points, and Kato's involve a study of the Galois representations associated to A. I plan to carry out such computations in many specific cases.