Math 5020 - Elliptic Curves
Homework - Chapter 1 (1.3, 1.6)
1.3 Let V An be a variety given by a single equation. A point P V is nonsingular if and only
if
dimK mP /m2 = dim V,
P
where mP is the maximal ideal of P in the ane ring of V .
Proof (by Alvaro a
Math 5020 - Elliptic Curves
Homework Chapter 2 (2.3, and 2.6)
2.3 (a) Prove proposition 2.6 for the special case of a non-constant map : P1 P1 .
Proof (by Alvaro). Let : P1 P1 be a non-constant separable rational map. Thus,
it is determined by ([x, 1]) =
Math 5020 - Elliptic Curves
Homework Chapter 3 (3.4 (use SAGE), 3.5, 3.8)
3.4 Referring to example (2.4), express each of the points P2 , P4 , P5 , P6 , P7 , P8 in the form
[m]P1 + [n]P3 with m, n Z.
Proof by Ryan Schwarz. Using SAGE and some trial and er
Math 5020 - Elliptic Curves
Homework 2 (3.4 (use SAGE), 3.5, 3.8, and the exercise below)
3.4 Referring to example (2.4), express each of the points P2 , P4 , P5 , P6 , P7 , P8 in the form
[m]P1 + [n]P3 with m, n Z.
3.5 Let E/K be given by a singular Weie
Math 5020 - Elliptic Curves
Homework 2 and 3 - Word problems
Note: At the beginning of each proof, the author is indicated. In most cases, I have only added
some minor comments to the original submitted proof.
P1-Hw2. (Proofs by Gagan Sekhon) Let E/Q be a
Math 5020 - Elliptic Curves
Homework 3 (the exercise below)
Problem 1 The elliptic curve y 2 = x3 + 2x2 3x satises E (Q)[4] = Z/4Z Z/2Z, i.e. the full
2-torsion is dened over Q and there is also a point of order 4 dened over Q. The goal of this
exercise i
MATH 5020 - Elliptic Curves
Homework 4
Problem 1 As you know, the elliptic curve y 2 = x3 +2x2 3x satises E (Q)[4] = Z/4ZZ/2Z. In previous
exercises, it has been shown that Q(E [4]) = Q(i, 3) and Gal(Q(E [4])/Q) Z/2Z Z/2Z.
=
Z/4Z Z/2Z,
In the rest of the
Math 5020 - Elliptic Curves
Homework 5
Problem 1. (Silvermans VIII: 8.1, 8.2)
(a) Let K be a number eld, E/K and elliptic curve, m 2 and integer, Cl(K ) the ideal
class group of K and
0
0
S = cfw_ MK : E has bad reduction at cfw_ MK : (m) = 0 MK .
Assumi