MATH 845: HOMEWORK 6, DUE MAY 4.
11. Every year, the London Sunday Telegraph has a New Year’s Quiz. In 1995,
two of the questions were the following:
(a) Solve the equation A3/B3+C3/D3= 6, where A, B, C, D are all positive
whole numbers below 100.
(b) (A special question with a 450 pound prize.) Either give a second solution to
the above equation where the four variables are all whole numbers above 100 (A, B
and C, D relatively prime), or demonstrate that no such second solution can exist.
[It’s too late to claim the prize.]
12. The integer equation a4+ma2b2+b4=c2(∗) (a, b) = 1, a, b > 0 was studied
by Fermat and Euler. A solution is called trivial if either ab = 0 or a=b= 1.
(a) Let Ebe the elliptic curve over Qgiven by y2=x3+mx2+x. Show that
(∗) has a nontrivial solution if and only if the Mordell-Weil rank of Eis nonzero.
(b) Euler showed that for m= 14, there are only trivial solutions of (∗). Prove
this.
(c) Suppose L(E, s) = Pcn/ns. Since Eis modular (why?), work of Buhler,
Gross, et al. gives the formula:
L(E, 1) = Xcn(exp(−2πnx/√N) + exp(−2πn/(x√N)))/n
where xis any positive real number, Nis the conductor of E, and =±1 its
root number.
Explain why this formula gives a means of computing . In the case = 1, obtain
a simpler formula for L(E, 1).
(d) For m= 145, Euler claimed that (∗) had a nontrivial solution, namely
(159,40). Show that he was mistaken.
(e) Kolyvagin proved the weak Birch Swinnerton-Dyer conjecture for modular
elliptic curves over Qwhose L-functions vanish to order at most 1 at s= 1. Show
how this gives a way to prove that for a given mthere are no nontrivial solutions.
For m= 145 compute the coefficients of L(E, s) up to n= 10 (the conductor of E
is 48048 and root number 1) - using (c), is this enough to determine whether (∗)
has nontrivial solutions for m= 145?
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