By William R. Schmitt

(15 pages, pdf). Notes written to supply scholars with priceless heritage for a path on Hopf algebras in combinatorics.

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**Extra resources for Notes on modules and algebras**

**Example text**

Let d ∈ Z, d = 0 and let E be the elliptic curve given by the cubic equation X 3 + Y 3 = dZ 3 with O = [1, −1, 0]. The reader should verify that E is a smooth curve. We wish to ﬁnd a Weierstrass equation for E and, indeed, one can ﬁnd a change of variables ψ : E → E given by ψ([X, Y, Z]) = [12dZ, 36d(X − Y ), X + Y ] = [x, y, z] such that zy 2 = x3 − 432d2 z 3 . The map ψ is invertible; the inverse map ψ −1 : E → E is ψ −1 ([x, y, z]) = 36dz + y 36dz − y x , , . 72d 72d 12d In aﬃne coordinates, the change of variables is going from X 3 + Y 3 = d to the curve y 2 = x3 − 432d2 : ψ(X, Y ) = ψ −1 (x, y) = 12d 36d(X − Y ) , X +Y X +Y 36d + y 36d − y , .

3) Let P ∈ E(Q). Then h(P ) ≥ 0, and h(P ) = 0 if and only if P is a torsion point. For the proofs of these properties, see [Sil86], Ch. VIII, Thm. 3, or [Mil06], Ch. IV, Prop. 5 and Thm. 7. As we mentioned at the beginning of this section, we can calculate upper bounds on the rank of a given elliptic curve (see [Sil86], p. 2). 4 ([Loz08], Prop. 1). Let E/Q be an elliptic curve given by a Weierstrass equation of the form E : y 2 = x3 + Ax2 + Bx, with A, B ∈ Z. Let RE be the rank of E(Q). For an integer N ≥ 1, let ν(N ) be the number of distinct positive prime divisors of N .

3 (Néron-Tate). Let E/Q be an elliptic curve and let h be the canonical height on E. (1) For all P, Q ∈ E(Q), h(P +Q)+ h(P −Q) = 2h(P )+2h(Q). 7. The rank and the free part of E(Q) 45 (2) For all P ∈ E(Q) and m ∈ Z, h(mP ) = m2 · h(P ). ) (3) Let P ∈ E(Q). Then h(P ) ≥ 0, and h(P ) = 0 if and only if P is a torsion point. For the proofs of these properties, see [Sil86], Ch. VIII, Thm. 3, or [Mil06], Ch. IV, Prop. 5 and Thm. 7. As we mentioned at the beginning of this section, we can calculate upper bounds on the rank of a given elliptic curve (see [Sil86], p.