MODULAR FORMS-page83

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.
Ask a homework question - tutors are online