MODULAR FORMS-page83

# MODULAR FORMS-page83 - 79 where c is a positive real number...

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79 where c is a positive real number. Since SL(2 , Z ) acts transitively on H * \ H we can take for a basis of open neighborhoods of each x Q the set of g -translates of the sets U c for all c > 0 and all g SL(2 , Z ) such that g · ∞ = x . Each g ( U c ) is equal to the union of the point x and the interior the circle of radius r = 1 2 γ 2 c touching the real line at the point x . In fact, if g = α β γ δ « , we have x = α/γ and g ( U c ) = { τ ∈ H : Im g - 1 · τ > c } = { τ ∈ H : Im τ | - γτ + α | 2 > c } = { τ = x + iy : ( x - α γ ) 2 + ( y - 1 2 γ 2 c ) 2 < 1 4 γ 4 c 2 } . (8.7) Now the topology on H * / Γ is deﬁned as the usual quotient toplogy: an open set in H * / Γ is open if and only if its pre-image in H is open. Since | γ | ≥ 1 in (8.7) unless g Γ , we can ﬁnd a suﬃciently large c such that Γ = { g Γ : g ( U c ) U c 6 = ∅} . Now, if
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