EXERCISES DAY 2
DAVID ROE
(1) Let GL+
2 (Q) denote the rational 2 × 2 matrices with positive determinant. For γ =
+
a c
a
b
a
b
( c d ) ∈ GL+
2 (Q), let [γ] = { b , d } be the path from c to d . If α, β ∈ GL2 (Q), show
that [βα] = β · [α]. Conclude that all values of a modular symbol Φ ∈ SymbΓ (V ) are
determined by its values on [αi ], where αi ranges over a set of coset representatives
for Γ in SL2 (Z).
(2) Find a presentation for ∆0 as a Z[Γ0 (23)]-module by finding a fundamental domain
for Γ0 (23).
(3) Recall that GL+
2 (Q) acts on SymbΓ (Q) on the right by (Φ|γ)(D) = Φ(γD)|γ. The
Hecke operators act on modular symbols as follows:
Φ|T` =
Φ| ( 0` 01 )
+
`−1
X
Φ| ( 10 a` )
a=0
q−1
Φ|Uq =
X
1 a
0 q
Φ|
a=0
for q dividing the level and ` not doing so. Using the presentation we found in the lecture of ∆0 as a Γ0 (11)-module, compute the action of T2 on a basis for SymbΓ0 (11) (Q).
(4) Let Vk (C) = Symk−2 (C) = C[X, Y ]k−2 be the space of homogeneous polynomials of
degree k − 2, with action given as in the lecture by P |γ(X, Y ) = P (dX − cY, −bX +
aY ). Suppose f is a modular form of weight k and level Γ0 (N ). Define a function
Φf ∈ Hom(∆0 , Vk (C)) by
Z s
Φf ({r, s}) = 2πi
f (z)(zX + Y )k−2 dz ∈ Vk (C).
r
Show that Φf is a modular symbol.
1