578 XIII Connections with Number Theory
Moreoverx′,x′+nandx′−nare allnonzerorational squares. Since 2Pis of finite
order, the theorem of Nagell and Lutz mentioned in§5 implies thatx′is an integer.
Consequently
x′=r^2 , x′+n=s^2 , x′−n=t^2for some positiveintegers r,s,t. Hencenis even, since
2 n=s^2 −t^2 =(s−t)(s+t)and if one ofs−tands+tis even, so also is the other. Sincen=s^2 −r^2 and
any integral square is congruent to 0 or 1 mod 4, we cannot haven≡2 mod 4. Hence
n≡0 mod 4. But then(x′/ 4 ,y′/ 8 )is a rational point of finite order ofCn/ 4 ,which
contradicts the minimality ofn.
Ifnis congruent, then so also ism^2 nfor any positive integerm. Thus it is enough
to determine which square-free positive integers are congruent. By what we have just
proved and Proposition 13, a square-free positive integernis congruent if and only if
the elliptic curveCnhas positive rank. SinceCnadmits complex multiplication, a re-
sult of Coates and Wiles (1977) shows that ifCnhas positive rank, then itsL-function
vanishes ats=1. (According to theBSD-conjecture,Cnhas positive rank if and only
if itsL-function vanishes ats=1.)
By means of the theory of modular forms, Tunnell (1983) has obtained a practical
necessary and sufficient condition for theL-functionL(s,Cn)ofCnto vanish ats=1:
ifnis a square-free positive integer, thenL( 1 ,Cn)=0 if and only ifA+(n)=A−(n),
whereA+(n),resp.A−(n), is the number of triples(x,y,z)∈Z^3 withzeven, resp.z
odd, such that
x^2 + 2 y^2 + 8 z^2 =n ifnis odd, or 2x^2 + 2 y^2 + 16 z^2 =n ifnis even.It is not difficult to show thatA+(n)=A−(n)whenn≡ 5 ,6or7mod8,butthere
seems to be no such simple criterion in other cases. With the aid of a computer it has
been verified that, for everyn<10000,nis congruent if and only ifA+(n)=A−(n).
The arithmetic of elliptic curves also has a useful application to the class number
problem of Gauss. For any square-free integerd<0, leth(d)be theclass numberof
the quadratic fieldQ(
√
d). As mentioned in§8 of Chapter IV, it was conjectured by
Gauss (1801), and proved by Heilbronn (1934), thath(d)→∞asd→−∞.How-
ever, the proof does not provide a method of determining an upper bound for the values
ofdfor which the class numberh(d)has a given value. As mentioned in Chapter II,
Stark (1967) showed that there are no other negative values ofdfor whichh(d)= 1
besides the nine values already known to Gauss. Using methods developed by Baker
(1966) for the theory of transcendental numbers, it was shown by Baker (1971) and
Stark (1971) that there are exactly 18 negative values ofdfor whichh(d)=2. A
simpler and more powerful method for attacking the problem was found by Goldfeld
(1976). He obtained an effective lower bound forh(d), provided that there exists a
modular elliptic curve overQwhoseL-function has a triple zero ats=1. Gross and
Zagier (1986) showed that such an elliptic curve does indeed exist. However, to show
that this elliptic curve was modular required a considerable amount of computation.
The proof of theTW-conjecture makes any computation unnecessary.