----------------------------------------------------------------- The following I have already incorporated in June 2006: ----------------------------------------------------------------- It might be worth mentioning explicitly that if f is a birational transformation over Q from E to E' (both in cubic form) then the group law is preserved and E(Q) is isomorphic under f to E'(Q). ----------------------------------------------------------------- (1) Already corrected: From george.walker@jesus.ox.ac.uk Wed Nov 23 09:36:54 2005 Date: Wed, 23 Nov 2005 09:36:53 -0000 (GMT) Was just flicking through the intro to elliptic curves and on page 14 there's a couple of small typos where you've reverted to 3 being a quadratic non-residue mod 1019, rather than 6 as the proof starts. Not really important, but it may confuse someone who has no idea what's going on! George ---------------------------------------------------------------- From Christopher Grattoni: 1) In Example 0.109, you wrote Res(x^2 + ax + b, 2x + a)/a instead of Res(ax^2 + bx + c, 2ax + b)/a. 2) In Example 0.111, you wrote ``sigularities'' instead of ``singularities.'' 3) In Example 0.121, you wrote ``the/ x/x-coordinate of any point'' instead of ``the /x/-coordinate of any point.'' 4) In Comment 0.122, you forgot to italicize the letter ``y'' in the fifth line of the comment. 5) In Example 0.127, you wrote ``projective cure'' instead of ``projective curve.'' ---------------------------------------------------------------- From: Bjoern Assmann So here is one ( I think :-) ). Theorem 0.119: Line 4 of the Proof. Instead of g(t,tx) it should be g(x,tx). Def 0.126 Line 1: "an (affine)" instead of "ab (affine) Line 3: Bracket missing before X_0,Y_0,Z_0) Example 0.127 Line 3: "at infinity are" instead of "at infinity at" ---------------------------------------------------------------- From bjoern@mcs.st-andrews.ac.uk Fri Feb 10 09:56:11 2006 Subject: Some more typos 2 Dear Victor, I found another typo. Example 0.30. (b): Line 4: 6=24/25 times 25/4 instead of 6=24/25 times 25/24 And I have a question concerning Def. 0.136, where you define the group law. Do you need Bezout's Theorem for the existence of the third point of intersection, or is there a straightforward direct argument ? In the first case, it might be worth mentioning Bezout. Cheers Bjoern ---------------------------------------------------------------- From bjoern@mcs.st-andrews.ac.uk Mon Feb 13 11:33:00 2006 Date: Mon, 13 Feb 2006 11:32:52 +0000 From: Bjoern Assmann To: flynn@maths.ox.ac.uk Subject: Further remaks Dear Victor, in Comment 0.100 I found a typo: Line 6: translated by (-x_0, -y_0) instead of translated by (-x_0, y_0) Further I think that the "Step 2" in the algorithm is not correct. R_k(x,y) does not have to factorize completely into linear factors. Take for example X^3 - Y^3 over Q (hast just one tangent in (0,0)) X^2 + Y^2 + 2*Y^3 over Q (has no tangent in (0,0)). Cheers Bjoern ------------------------------------------------------------------- Also: Page 23, "crossing points typified" should be "crossing points are typified" ------------------------------------------------------------------- Morrow points out: defn of diff form on a power series should include: d(x+y) = dx + dy, d(xy) = xdy + ydx, da = 0 (a constant). Also, when I start talking later about omega composed with F(S,T), I am regarding F(S,T) as a member of R[[S]][[T]], so in talking about d F(S,T) I should mention that I am also extending the defn to diff forms over (R_S)_T. ------------------------------------------------------------------- Follow-up: Dear Matthew, Thanks - yes, that looks like it'll do the trick! (plus the amendment later, that when I discuss omega composed with F(S,T), I am regarding F(S,T) as a member of R[[S]][[T]], and so in talking about d F(S,T) I should mention that I am also extending the defn to diff forms over (R_S)_T). For your question, it's not necessary for p to be prime in R. We have that p divides n*a_n (in R). My conclusion "p divides n or a_n" should be interpreted as "(p divides n in Z) or (p divides a_n in R)". This is how it is used in the follow-up result that the power series can be written as p*f(T) + g(T^p). We have that p divides n*a_n (in R), so that there exists b in R such that n*a_n = p*b. Suppose that p doesn't divide n in Z. Then p and n are coprime in Z, and so there exist lambda, mu in Z such that: lambda*p + mu*n = 1. Then: a_n = lambda*p*a_n + mu*n*a_n = lambda*p*a_n + mu*p*b = p*(lambda*a_n + mu*b), and so p divides a_n in R, as required. - Victor On Mon, 22 May 2006, Matthew Morrow wrote: > Dr. Flynn, > > Following on from our discussion in the 3rd week Elliptic Curves consultation > session regarding invariant differentials, I am certain that the following > definition covers all the desired relations:- > > Given a commutative ring R with unity, let the space of differential forms be > the free R[[T]]-module spanned by the symbols > {df : f \in R[[T]]} > quotiented out by the submodule spanned by > {f'dT - df : f /in R[[T]]}. > > All the required relations easily follow, the space is non-trivial, and > f dg = h dg if, and only if, f = h (something which was implicitly used in > lectures). > > May I take this opportunity to ask another question please. In Corollary 4.13, > you deduce that the rational prime p divides na_n, where [p]=/sum_{n /ge 0} > a_nT^n is the 'multiplication-by-p' map on a formal group. I don't understand > how it follows that p divides one of n and a_n, for we do not know that p is > prime in the ring R. I have tried, without success, to construct a > counterexample to the result. > > Yours, > Matthew Morrow -------------------------------------------------------------------