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By Kusahara T.

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2) A ~ G = (~+~', V) is made where on the "Baer sum" this group by ~ of into a group defines (¢~) and the (~,). (~)-$Hom(A,G). 2) (A,G) ~ Extrig(A,G). one then obtains (¢)-~ Hom(A,G) Later we shall prove isomorphism G = Gm if and A a homomorphism a homomorphism ~ (~)-Homrig(A,G). 2) is an isomorphism is an if scheme. Let the ~-extension, (~) 0 ~ G ~ E ~ A ~ 0 be given. Denote by i , the inclusion the structural morphism determined plo~ = eAov , p2°~ = i . by Since the V: p~(E) ~ and by Infl(A) ~ ~-structure p~(E) on ~: Infl(A) E is given , we can "pull back" T*(V): v*oe~(E) ~ i*(E) V A , ~: ~nfl(A) ~ ~l(A) the morphism by an isomorphism via ~ S T to obtain: 35 Since E is a group e~(E) and hence with an obvious choice of section, ~*e~(E) is equipped the unit section.

1) of Let interesting over the base to consider the canonical S = Spec{~). M = N(Z) ~ N ( Q ) abelian variety Let of the canonical extension a N~ron model. It is especially extension characterization denote the Mordell-Well group of the NQ • This is a finitely generated group. M* = E(N)(Z). ~N,(~) ~ M* ~ M ~ O _~,(~) we see that is a free abelian group whose rank is M* d + rank(M). to be free. 3) Theorem: of If then either p divides the order of the torsion p = 2 or p subgroup is a prime of bad reduction N .

One then obtains (¢)-~ Hom(A,G) Later we shall prove isomorphism G = Gm if and A a homomorphism a homomorphism ~ (~)-Homrig(A,G). 2) is an isomorphism is an if scheme. Let the ~-extension, (~) 0 ~ G ~ E ~ A ~ 0 be given. Denote by i , the inclusion the structural morphism determined plo~ = eAov , p2°~ = i . by Since the V: p~(E) ~ and by Infl(A) ~ ~-structure p~(E) on ~: Infl(A) E is given , we can "pull back" T*(V): v*oe~(E) ~ i*(E) V A , ~: ~nfl(A) ~ ~l(A) the morphism by an isomorphism via ~ S T to obtain: 35 Since E is a group e~(E) and hence with an obvious choice of section, ~*e~(E) is equipped the unit section.