Download PDF by B.H. Gross, B. Mazur: Arithmetic of elliptic curves with complex multiplication

By B.H. Gross, B. Mazur

ISBN-10: 3540097430

ISBN-13: 9783540097433

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Compare this with the blow-up H, In dimension 1 the simplest closed string is given by, R = k[x, y]/(f ), with f = x2 + y 2 − r2 , such that P hR = k < x, y, dx, dy > /(f, [x, y], d[x, y], df ), and with the two points, i : P hR → k(pi ), defined by the actions on k(pi ) := k, given by, xi , yi , (dx)i , (dy)i , i = 1, 2. It is easy to see that the vectors, ξi := ((dx)i , (dy)i ) are tangent vectors to the circle at the ws-book9x6 January 25, 2011 11:26 World Scientific Book - 9in x 6in Deformations and Moduli Spaces ws-book9x6 49 points pi , and if p1 = p2 we find that Ext1P hR (k(p1 ), k(p2 ) = k.

Since Ext2A (V, V ) = 0 we find H(V ) = k << t1 , t2 , t3 , t4 , t5 >> and so H(V )com k[[t1 , t2 , t3 , t4 , t5 ]]. The formal versal family V˜ , is defined by the actions of x1 , x2 , given by, 0 1 + t3 t1 t2 . , t5 )2 ⊂ H(V ), for all i, p, q = 1, 2. This proves that ˆ v must be isomorphic to H(V ), and that the composition, C(2) A −→ A(2) −→ M2 (C(2)) ⊂ M2 (H(V ))) is topologically surjective. By the construction of C(n) this also proves that C(2) k[t1 , t2 , t3 , t4 , t5 ]. locally in a Zariski neighborhood of the origin.

Moreover, the tangent space of H is isomorphic to Ext1A (M, M ), and H can be computed in terms of ExtiA (M, M ), i = 1, 2 and their matric Massey products, see [15], [16], [21]. In the general case, consider a finite family V = {Vi }ri=1 of right Amodules. Assume that, dimk Ext1A (Vi , Vj ) < ∞. Any such family of A-modules will be called a swarm. We shall define a deformation functor, DefV : ar → Sets generalizing the functor Def M above. Given an object π : R = (Ri,j ) → k r of ar , consider the k-vector space and left R-module (Ri,j ⊗k Vj ).

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Arithmetic of elliptic curves with complex multiplication by B.H. Gross, B. Mazur

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