2.4 Vanishing Sums of Roots of Unity
2.4.1 The Results of Conway and Jones
We shall now present the theory of formal sums of roots of unity introduced by Conway and Jones [2].
Definition. Define the collection of roots of unity as R = {exp(2πq) : q ∈ Q}
and denote the space of maps f :R → Q by V. For each root of unity α∈ R we define a map Lα ∈ V by Lα :α 7→ 1 and Lα :β 7→0 for β 6= α. Therefore Lα is the delta function atα.
The space V inherits a vector space structure from Q which gives the natural addition onL, K ∈ V by:
(L+K) :α7→L(α) +K(α).
2.4. VANISHING SUMS OF ROOTS OF UNITY 93
roots α and β we would have LαLβ ≡ 0. Therefore we define multiplication by LαLβ =Lγ, where γ =αβ. Thus we have an addition and a multiplication onV. Definition. Define thering of formal sums of roots of unity F to be all functions S ∈ V of the form S = P
cαLα, where Lα ranges over all α ∈ R and cα ∈ Q with cα = 0 for all but finitely many α ∈ R, with the addition and multiplication defined above.
Definition. The formal sum S is said to involve α when the expression for S in terms of the above basis has non-zero coefficient of Lα.
Definition. Thelength l(S), of S is the number of roots involved in S.
Definition. The sum S is called similar to kLαS for any root α and any k ∈ Q\ {0}.
Definition. Theexponente(S), of S is the lowest common multiple of the orders of the roots involved in S.
Definition. Thereduced exponentr(S), of Sis the smallest exponent of any sum similar to S.
There is a vector space homomorphism v from F to C given by evaluation, that is v :Lα 7→α for all rootsα ∈ R.
Definition. The value v(S), of the sum S = P
cαLα is the complex number P
cαα.
Definition. We say that the sum S is a vanishing sumif it is in the kernel of v, that is v(S) = 0.
Definition. A vanishing sum S is said to be minimal if no proper subsum van- ishes.
94CHAPTER 2. TRIANGLE GROUPS WITH TWO ELLIPTIC ELEMENTS
Suppose S is monic, that is we can write
S=c1L1+cαLα+cβLβ+...
whereα and β are roots of unity. Now the exponent of S is e(S) = lcm(ord(α), ord(β), ...). If we take any sum similar toS, say kLωS then
e(kLωS) = lcm(ord(ω), ord(ωα), ord(ωβ), ...)
= lcm(ord(ω), lcm(ord(ω), ord(α)), lcm(ord(ω), ord(β)), ...) = lcm(ord(ω), ord(α), ord(β), ...)
= lcm(ord(ω), e(S))
≥ e(S).
Therefore if S is monic then the exponent coincides with its reduced exponent, i.e. e(S) =r(S). We can now give some properties of formal sums.
Proposition 2.4.1.1. Let S be a formal sum of roots of unity. If S is a minimal vanishing sum then r(S) is square-free.
Proof. See Theorem 1 of [2].
Suppose that S =P
cαLα is a minimal vanishing sum, then any sum similar toSis also a minimal vanishing sum. Therefore we may assume thate(S) = r(S). Hence from Proposition 2.4.1.1 we may assume that every root α involved in S has square-free order. Now suppose that we have a root β involved in S which has even order. This implies that the order of β is 2n, where n is odd, since if n was even then β would not be of square-free order. Then the root −β has odd
2.4. VANISHING SUMS OF ROOTS OF UNITY 95
order, n. Therefore we define a projection map Π :F → F given by: Π :cαLα+c−αL−α 7→(cα−c−α)Lα, if α is of odd order and Π :cαLα+c−αL−α 7→(−cα+c−α)L−α, if α is of even order. for any root α. Thus ΠS contains only roots of odd order and so e(ΠS) is odd. Remark 2.4.1.2. The projection Π does not alter the value of the sum, that is v(ΠS) = v(S). Therefore without loss of generality we only consider minimal vanishing sums that are inΠF. Such sums have odd, square-free reduced exponent.
Example. Consider the vanishing sum S=Lω5+Lω−L1, whereω = 1 2 +i
√
3 2 is a primitive sixth root of unity. As ω has even order, when we apply the projection Π we replace Lω by −(L−ω). We can now use that −ω =ω4, which has order 3, to replace L−ω by Lω4. Similarly we replace Lω5 by −(L−ω5) = −Lω2 and obtain ΠS=−Lω2 −Lω4 −L1, which involves only roots of odd order.
Proposition 2.4.1.3. Let ω be a primitive pth root of unity, where p is prime and suppose
S =L1S0+LωS1+Lω2S2+...+Lωp−1Sp−1
is a vanishing sum, with each Si having exponents prime to p. Then Si all have
the same value.
Proof. See Lemma 1 of [2].
Proposition 2.4.1.4. LetSbe a minimal vanishing sum with lengthland reduced exponent r. Then
l≥2 +X p|r
96CHAPTER 2. TRIANGLE GROUPS WITH TWO ELLIPTIC ELEMENTS
the sum is taken over all prime p such that p|r. Proof. See Theorem 5 of [2].