Math 522 Sample Final Exam Problems

1.  Describe a correspondence between lattices in the complex plane C
and elliptic curves over C.  Introduce the Weierstrass p-function and
explain the role it plays in this correspondence.

2.  Suppose b,c are non-zero integers.  Consider the family of
elliptic curves y^2 = x^3 + c.  Prove that for any prime p = 2 mod 3,
this curve has p+1 points mod p.  For the family y^2 = x^3 + b*x, and
prime p = 3 mod, again prove that the reduction of the curve mod p has
p+1 points.

3. (a) Suppose an elliptic curve E: y^2 = f(x) , where
f(x)=x^3+a*x^2+b*x+c, a,b,c rational numbers, has all its 2-torsion
points defined over Q.  Explain why this is the same as requiring that
f(x) has three distinct roots r,s,t let's call them, in Q.

(b) Now suppose the curve is of the form y^2 = x^3+a*x^2+b*x.  The
roots of f(x) now are 0,r,s, say.  We know that the curve

y^2 = x^3 + a' x^2 + b'*x where a = -2*a, b =a^2-4*b

is 2-isogenous to E.  By moving the points (r,0) and (s,0) to (0,0),
find two other 2-isogenous curves to E.

4.  The elementary sounding question "When is the product of three
consecutive integers equal to the product of two consecutive integers"
can be converted to "What are the integral points on the elliptic
curve y^2+y = x^3 -x." Explain why this is so.  Convert this curve
into the form

Y^2 = X^3 + a*X^2 + b*X + c with a,b,c integers.

5.  Give an example of an elliptic curve E such that its real points
{(x,y) on E such that x,y are real numbers} form two disjoint
components.  Give another example where the real points form a single
connected component.  (Draw some examples on a piece of paper).

6.  Parametrize all primitive integer solutions of x^2 + 2y^2 = 3 z^2.

7.  Consider y^2 = x^3 + a*x^2 + b*x, for some small values of a, b
such that a*(a^2-4*b) is non-zero.  (Say a=3, b=1).  Let E' be the
2-isogenous curve as in Problem 3 and in our book. (I'm using '
instead of bar because it is easier in this format).

Compute a_p(E) for your specific curve (a=3, b= 1 say) as well as
a_p(E') for 1 < p < 20.  Do you note any patterns?  Formulate a
conjecture based on that pattern.  Any ideas for a proof for this
particular case (a=3,b=1)?  Any ideas for general a,b?

8.  One of the key points in the proof of the Mordell-Weil theorem is
the construction of a certain homomorphism $\alpha$ into the non-zero
rationals modulo squares.  Give the precise definition of this map if
you remember it.

9.  Prove that the curve y^2=x^3+1 has no rational points other than a
few small obvious ones.

10.  A positive integer k is called a congruent number if it is the
area of a right triangle whose sides have rational length.

(a) Show that 6 and 5 are congruent numbers.  (Hint: use your
knowledge of Phythagorean triples!)

(b) Show that k is a congruent number if and only if there exist 3
rational numbers a,b,c such that b-a=k and c-b=k, i.e. there are three
rational numbers in arithmetic progression with common difference k.

(c) Show that if k is a congruent number, then the curve 

y^2 = x^3 - k^2*x 

has a rational point other than the obvious ones O, (n,0), (0,0), (-n,0).

(d)* Extra Credit Prove the converse: if y^2 = x^3 -k^2*x has a
non-obvious rational point, then k is a congruent number.

11.  Find the torsion subgroup over Q of y^2 = x^3 +5*x^2+4*x.