MA342J: Introduction to Modular Forms
Tutorial 1, January 26
1. Triangulation of H by the images of the fundamental domain.
In class we constructed a fundamental domain D for the modular group
Γ = SL(2, Z)/{∓1} in the upper half-plane H. Draw its images g(D) for
the following elements g ∈ Γ:
g = 1 , S , T , T −1 , T −1 S , T S , ST S , ST , ST −1 , ST −1 S
Remark: The following property of fractional linear transformations is
useful to draw these images. The image of any straight line or a circle is
again either a straight line or a circle.
Find some trivial combinations of the generators S and T , e.g. (ST )3 = 1.
2. Action of Γ on the “boundary”. The projective line P 1 (Q) can be
constructed as the quotient of the set Q × Q \ {(0, 0)} by the multiplicative
group Q× = Q \ {0} acting by
α(x1 , x2 ) = (αx1 , αx2 ) .
Its elements are denoted (x1 : x2 ) ∈ P 1 (Q), e.g. (3 : 2) = (3/2 : 1) = (1 :
2/3). This is called homogenous coordinates on P 1 (Q).
Also one can view P 1 (Q) as Q with one point added “at infinity”, i.e.
= Q ∪ {∞}. Check that these two sets are identified by the map
(
x1
if x2 6= 0 ,
(x1 : x2 ) 7→ x2
∞ otherwise.
Define the action of Γ on P 1 (Q) by fractional linear transformations. Show
that this action is transitive by finding for every α ∈ Q an element g ∈ Γ
such that α = g∞. Is the element g with this property unique? Describe
the set I(∞) = {g ∈ Γ|g∞ = ∞}.
Remark 1: Analogously, one can construct the set P 1 (k) for any field k.
Observe that P 1 (C) is the Riemann sphere we constructed in class.
Remark 2: If H∗ = H ∪ P 1 (Q) then
Γ\H∗ ∼
= Γ\H ∪ {∞} ∼
= P 1 (C)
is the compactification of the quotient Γ\H we constructed in class.
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