MODULAR FORMS-page64

# MODULAR FORMS-page64 - 60 LECTURE 6 MODULAR FORMS In...

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Unformatted text preview: 60 LECTURE 6. MODULAR FORMS In particular, 120 Î¶ (4) = (2 Ï€ ) 4 / 12 , 280 Î¶ (6) = (2 Ï€ ) 6 / 216 and we can write g 2 = (2 Ï€ ) 4 [ 1 12 + 20 X ] , g 3 = (2 Ï€ ) 6 Ë† 1 216- 7 Y 3 Ëœ . This gives g 3 2- 27 g 2 3 = (2 Ï€ ) 12 [(5 X + 7 Y ) / 12 + 100 X 2 + 20 X 3- 42 Y 2 ] = (2 Ï€ ) 12 q + q 2 ( ... ) . Now from Example 6.1 we have Î”( Ï„ ) = q + q 2 ( ... ). This shows that the ratio R = g 3 2- 27 g 2 3 / Î” is holomorphic at âˆž too. This implies that R is bounded on the fundamental domain D of Î“(1). Since R is invariant with respect to Î“(1) we see that R is bounded on the whole upper half-plane. By Liouvilleâ€™s theorem it is constant. Comparing the coefficients at q , we get the assertion. 6.6 Recall that we constructed the modular forms g 2 and g 3 as the coefficients of the elliptic function â„˜ ( z ; Ï„ ) in its Taylor expansion at z = 0. The next theorem gives a generalization of this construction providing a convenient way to construct modular forms with respect to a subgroup of finite index Î“ of SL(2...
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