1. The Modular Group and its Action on the Upper Half Plane

Question 1

Define the group G

Answer 1

SL2(R) = {g; detg =1}

Question 2

Define the ‘modular group’ Γ

Answer 2

SL2(Z) = {γ ∈ M2(Z); detγ = 1}

Question 3

Define the group GL2(Z)

Answer 3

GL2(Z)= {γ ∈ M2(Z); detγ = ±1}

Question 4

Define the ‘upper half plane’ H

Answer 4

H = {τ = u+iv; v>0}

Question 5

Define the mobius transformation

Answer 5

gτ = aτ + b /cτ + d

Question 6

Define Im(gτ)

Answer 6

= Im(τ)/|cτ+d|^2

Question 7

Define the automophy factor

Answer 7

j(g,τ) = cτ + d

Question 8

Define the co-cyle relation j(gg’,τ)

Answer 8

= j(g,g’τ) ̇ j(g’,τ)

Question 9

Define equivalence for elements of H

Answer 9

If there exists some γ ∈ SL2(Z) such that
τ’ = γτ

Question 10

Define S ∈ SL2(Z)

Answer 10

Clockwise from top left

S = (0, -1, 1, 0)

S: τ → -1/τ

Question 11

Define T ∈ SL2(Z)

Answer 11

T = (1, 1, 1, 0)
T: τ → τ + 1

Question 12

Define fundamental domain F for SL2(Z)
(words)

Answer 12

The closed region of H such that no two interior points are equivalent.

Question 13

Define fundamental domain F for SL2(Z)
(region)

Answer 13

F = {τ=u+iv; |τ|>1, -1/2 ≤ Re(τ) ≤ 1/2}