Problems

Try all the problems but definitely do the ones with your name in front of them.

1. (Mauro) Look at Figure 2.6 in Silverman-Tate. It is the graph of an elliptic curve with one real component along with the corresponding graph in the $s$-$t$ plane. Choose an elliptic curve with two real components and draw its graph in the $s$-$t$ plane.

2. (Alex) The third paragraph on page 52 of Silverman-Tate begins: ``Let $P_1=(t_1,s_1)$ and $P_2=(t_2,s_2)$ be distinct points. If $t_1=t_2$, then $P_1=-P_2$, so $P_1+P_2$ is certainly in $C(p^{\nu})$'' (i.e., the $t$-coordinate of the sum is divisible by $p^\nu$). I think this is a mistake in the proof, because $P_1=-P_2$ if and only if $t_1=-t_2$ and $s_1=-s_2$, as discussed at the bottom of page 53. Repair the mistake; that is, give a proof that if $t_1=t_2$ then $P_1+P_2$ is in $C(p^{\nu})$.

3. (Jeff) Let $p\geq 2$ be a prime and let $E$ be the elliptic curve
   $$y^2 = x^3 + px.$$
   Find all points of finite order in $E({\mathbb{Q}})$.

4. (Jenna) Let $p$ be a prime and let $S=S_p={\mathbb{Z}}[\frac{1}{p}]$ be the set of rational numbers of the form $a/p^r$ for $a\in{\mathbb{Z}}$ and $r\geq 0$.
   1. Prove that $S$ is a subring of ${\mathbb{Q}}$.
   2. Prove that the group of units in $S$ is $\{\pm p^{\nu} : \nu \in {\mathbb{Z}}\}$.
   3. Let $q\neq p$ be a prime. Prove that $q$ generates a maximal ideal of $S$ and describe the quotient field $S/(q)$. Prove that every maximal ideal of $S$ is of this form.

5. (Jennifer) For each of the following elliptic curves $E$, determine the torsion subgroup of $E({\mathbb{Q}})$. You may use the stronger form of Nagell-Lutz (i.e., $2P=0$ or $y^2\mid D$) and you may use a computer to automate use of Nagell-Lutz (but don't just write TorsionSubgroup(EllipticCurve(...)) in MAGMA). By Mazur's theorem, the groups you get will represent all possibilities for $E({\mathbb{Q}})_{{\rm tor}}$ for any elliptic curve $E$ over  ${\mathbb{Q}}$.
   1. $y^2=x^3-2$
   2. $y^2=x^3+8$
   3. $y^2=x^3+4$
   4. $y^2=x^3+4x$
   5. $y^2-y=x^3-x^2$
   6. $y^2=x^3+1$
   7. $y^2=x^3-43x+166$
   8. $y^2+7xy=x^3+16x$
   9. $y^2+xy+y=x^3-x^2-14x+29$
   10. $y^2+xy=x^3-45x+81$
   11. $y^2+43xy-210y=x^3-210x^2$
   12. $y^2=x^3-4x$
   13. $y^2+xy-5y=x^3-5x^2$
   14. $y^2+5xy-6y = x^3-3x^2$
   15. $y^2+17xy-120y=x^3-60x^2$
   > function t(e) return Invariants(TorsionSubgroup(EllipticCurve(e))); end function; > t([0,-2]); > t([0,8]); [ 2 ] > t([0,4]); [ 3 ] > t([4,0]); [ 4 ] > t([0,-1,-1,0,0]); [ 5 ] > t([0,1]); [ 6 ] > t([-43,166]); [ 7 ] > t([7,0,0,16,0]); [ 8 ] > t([1,-1,1,-14,29]); [ 9 ] > t([1,0,0,-45,81]); [ 10 ] > t([43,-210,-210,0,0]); [ 12 ] > t([-4,0]); [ 2, 2 ] > t([1,-5,-5,0,0]); [ 2, 4 ] > t([5,-3,-6,0,0]); [ 2, 6 ] > t([17,-60,-120,0,0]); [ 2, 8 ]

6. (Mauro)Use mwrank to find generators for a subgroup of finite index of the group of rational points on the following elliptic curves:
   1. $y^2 + y = x^3 - x^2 - 10x - 20$
   2. $y^2 + y = x^3 - x$
   3. $y^2 + y = x^3 + x^2 - 2x$
   4. $y^2 + y = x^3 - 7x + 6$
   5. $y^2 + xy = x^3 - x^2 - 79x + 289$
   6. $y^2 + y = x^3 - 79x + 342$

7. (Jenna)Use gp (PARI) to do the following elliptic curve arithmetic. Let $P=(1,0)$ and $Q=(-1,1)$ on $y^2 + y = x^3 + x^2 - 2x$.
   1. Compute $4P + 5Q$.
   2. Find the smallest multiple $nP$ of $P$ such that the $x$ and $y$-coordinates of $nP$ are not both integers, and hence prove that $P$ has infinite order. Do the same for $Q$.
   3. Find five distinct right triangles with rational side lengths and area $5$ using arithmetic on an elliptic curve and Proposition 4.2 and Example 4.4 from the notes for 02/11/03. Use Nagell-Lutz to prove that there are infinitely many right triangles with rational side lengths and area $5$ (assuming the truth of Proposition 4.2).

8. (Alex) Use magma to do the same arithmetic as in Exercise 7.

9. (Jennifer) Part (c) of the proposition on page 55 asserts that the map
   $$\frac{C(p^{\nu})}{C(p^{3\nu})} \to \frac{p^\nu R}{p^{3\nu}R}, \qquad P=(x,y) \mapsto t(P)=\frac{x}{y}$$
   is a one-to-one homomorphism. Let $p=3$ and $\nu=1$. Determine the size of the image of this map for the first 3 curves in Problem 6 (assume that the subgroup of finite index output by mwrank is actually of index $1$).

10. (Jeff) Prove that for every rational number $t\neq 0,\frac{1}{4}$, the point $(t,t)$ on the elliptic curve defined by
    $$y^2=x^3-(2t-1)x^2+t^2x$$
    is a point of order four. (See the discussion on page 57 of [Silverman-Tate], and feel free to use a computer to simplify the algebra.)