By B.H. Gross, B. Mazur

Publication by way of Gross, B.H.

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**Additional info for Arithmetic on Elliptic Curves with Complex Multiplication**

**Example text**

First suppose that f actually coincides with g throughout an entire neighborhood V of N. Let h : sv - y -+ Rp be stereographic projection. Then the homotopy F(x, t) = f(x) F(x, t) = h - l [t · h(f(x)) + (I - t) · h(g(x))] for x t V for x £ lJ1 - N proves that f is smoothly homotopic to g. Thus is suffices to deform f so that it coincides with g in some small neighborhood of N, being careful not to map any new points into y during the deformation. Choose a product representation N X Rv-. V C M for a neighborhood V of N, where Vis small enough so that f(V) and *For example,

0 for t < 1 and ;\(t) = 0 fort 2: 1.

Thus is suffices to deform f so that it coincides with g in some small neighborhood of N, being careful not to map any new points into y during the deformation. Choose a product representation N X Rv-. V C M for a neighborhood V of N, where Vis small enough so that f(V) and *For example,

0 for t < 1 and ;\(t) = 0 fort 2: 1. 49 The Pontryagin construction g(V) do not contain the antipode y of y.

Two framed submanifolds (N, b) and (N', ltl) are framed cobordant if there exists a cobordism X C M X [0, 1] between N and N' and a framing u of X, so that u;(x, t) = (v;(x), 0) u;(x, t) = (w'(x), 0) for (x, t) t: N X [0, e) for (x, t) t: N' X (1 - e, 1]. Again this is an equivalence relation. Now consider a smooth map f : M-+ sv and a regular value y t: sv. The map f induces a framing of the manifold 1 (y) as follows: Choose a positively oriented basis b = (v\ · · · , vv) for the tangent space T(Sv) •.