Universal Extensions and One Dimensional Crystalline by Barry Mazur

By Barry Mazur

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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 .

X ) Infs1 >a ) >s g and the isomorphisms the composition (*), (**) above one can see that of the two top horizontal arrows ~ is in the f o l l o w i n g diagram: ~G 1 " dg ~ 1 > ~*~/s ~ *nxls ~ * ~2/ s ' > ~ ~ ' x2 / s Since the image of ~G in i *~/S is killed by d , the 31 lemma follows, of the and our construction G-torseur P is concluded. ~ ~X/s@LIz~G, Xet. we obtain the ordinary de Rham complex ~X -+ ~ / S If is zero we say that is integrable. Consider the map of sheaves of of the global curvature ~ ....

Namely S -scheme and is an extension, we can take any structure of rigidified extension on it, then by put a connection on E • Notice this E' the above procedure and hence finally obtain the curvature ~'~s,/~) tensor which lies in r(S',~As,. (~A/S)S,). (OA/S). (QA/S) is a vector group, this morphism is constant. Clearly the image of the trivial extension is zero and thus the map is identically zero implying that the connection V' is integrab le. 5) To show the connection structure let us replace gE" V' E is compatible with the group by the corresponding line bundle Then we are to show the isomorphism s*(gE) ~ , v~(~E)® v~(gE) is horizontal.

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