1.4. Exercises 15

by

L(s, χn) =

∞

a=1

χn(a)

as

.

We are ready to write down the formula for S3(n), due to Gauss,

Dirichlet and Shimura (there are also formulas for S5(n), due to Eisen-

stein, Smith, Minkowski and Shimura, and a formula for S7(n), also

due to Shimura). For simplicity, let us assume that n is odd and

square free (for the utmost generality, please check [Shi02]):

S3(n) =

0 if n ≡ 7 mod 8,

24

√

n

π

L(1,χn) otherwise.

The reader is encouraged to investigate this problem further by at-

tempting Exercises 1.4.6 and 1.4.7.

1.4. Exercises

Exercise 1.4.1. Use the divisibility properties of integers to show

that the only solutions to

y2

= x(x + 1)(x + 2) with x, y ∈ Z are

(0,0), (−1,0) and (−2,0). (Hint: If a and b are relatively prime and

ab is a square, then a is a square and b is a square.)

Exercise 1.4.2. Find all the Pythagorean triples (a, b, c), i.e., a, b, c ∈

Z and

a2

+

b2

=

c2,

such that

b2

+

c2

=

d2

for some d ∈ Z. In other

words, find all the integers a, b, c, d such that (a, b, c) and (b, c, d)

are both Pythagorean triples. (Hint: You may assume that y2 =

x(x + 1)(x + 2) has no rational points other than (0,0), (−1,0) and

(−2,0).)

Exercise 1.4.3. Prove Proposition 1.1.3; i.e., show that f((a, b, c)) is

a point in En, that g((x, y)) is a triangle in Cn and that f(g((x, y))) =

(x, y) and g(f((a, b, c))) = (a, b, c).

Exercise 1.4.4. Calculate S4(n), for n = 1,3,5,6, by hand, using

Jacobi’s formula and also by finding all possible ways of writing n as

a sum of 4 squares.

Exercise 1.4.5. The goal of this problem is to find the q-expansion

of

Θ6(q):