On Jan 7, 7:34 am, David Bernier <david53...@gmail.com> wrote: > On Jan 6, 10:53 pm, Rupert <rupertmccal...@yahoo.com> wrote: > > > > > > > On Jan 7, 12:01 am, RJB <Robert_Be...@alum.umb.edu> wrote: > > > > (1) Let E1, E2 be two elliptic curves isomorphic over the same > > > extension field L. Then their torsion subgroups are equal, up to > > > isomorphism. > > > > (2) If E1,E2 satisfy the conditions specified in (1) above, then they > > > both have the same rank. > > > > Question 1: Are the statements in (1) and (2) above true or false? > > > > Question 2: Please cite a textbook, paper or ArXiv preprint that > > > either justifies the truthfulness or proves the falsity of (1) and > > > (2). > > > > (Serious respondents only and no flaming. An undergraduate complex > > > analysis professor once told me the only ignorant question was one > > > that went unasked. Whoever does not ask questions does not learn.) > > > It seems to me that if you have an elliptic curve defined over a field > > L, and another elliptic curve defined over L, and they are isomorphic > > over L, then the groups of L-rational points in each elliptic curve > > must obviously be isomorphic. If L is a finite extension of Q, then > > these groups are finitely generated by the Mordell-Weil theorem, and > > since they are isomorphic, they have the same torsion and the same > > rank. This would not be justified in detail in any textbook, paper, or > > ArXiv preprint because it is so trivial. > > To get a group, I think we need to include the > point at infinite, in projective geometry terminology. > > I was wondering if there is just one point at infinity. > > Let's say we take the projective real 2D plane. > One model is lines in R^3 through the origin, > with "closeness" (topology) measured by the angular > separation. Then I think we get S^2 but with > antipodal points being glued together as one. > > Anyway, removing any point of S^2, what's left is > homeomorphic to R^2. For S^2 with antipodal points > identified, removing the North pole removes the > South pole; what's left is a punctured Northern hemisphere > with half of the equator gone. I can't quite visualize this. > > So, I still think projectively we get something simple for > elliptic curves; actually, perhaps a torus S^1 x S^1 > if the field were the reals. > > x + x = (a) point at infinity, by Fermat's geometric > addition method for y^2 = x^3 + ax + b. > > where x is an ordinary point. > x + x = y + y for ordinary point x, y. > > And, P.S.: does "isomorphic elliptic curves" go back > or refer to algebraic geometry? > > David Bernier-
I don't think it really matters whether there's more than one option for the point in infinity to define the neutral element in the Poincare group of an elliptic curve (over C, say): an elliptic (complex) curve is a torus C/L, with L a lattice, and it is isomorphic to another torus C/L' iff L' = cL, with c a non-zero complex number.
Isomorphic curves thus defined have, of course, the same j-invariant and the same group structure, and I suppose this is true whether we're over Q or C (or even a field of positive characteristic).
And I think "isomorphic elliptic curves" take within the definition the whole pack (but I'm not 100% sure), i.e. isomorphism as curves (alg. geometry), the group laws defined on them, their invariants, etc., of course when they're defined over the same field.