Two Longer Corrections to Elliptic Curves from Langlands First Correction

Difficulty. The use of Proposition 5.5 to obtain Proposition 5.6 is inadequate if the reduction map rp on the set of distinct points among {P,Q,PQ} is not oneone. For example, if P , Q, and PQ are distinct and rp(P ) = rp(Q) = rp(PQ), then Proposition 5.5 shows that the intersection multiplicity for rp(P ) is ≥ 1, but it does not produce either a second or a third point on rp of the line. Thus we cannot obtain the desired conclusion that rp(P )rp(P ) = rp(P ), i.e., that rp(P ) has intersection multiplicity 3. What is needed is an improved version of Proposition 5.5 and then a little extra argument in Proposition 5.6 to show that all cases have been handled. The improved version below is actually more than is needed; only the cases k ≤ 2 are needed with elliptic curves, and a page of matrix calculations are unncessary for such cases. However, the principle is a little clearer with the version of Proposition 5.5 given below.