MODULAR FORMS-page116

MODULAR FORMS-page116 - 112 LECTURE 11. HECKE OPERATORS pT...

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Unformatted text preview: 112 LECTURE 11. HECKE OPERATORS pT ( p n- 1 ) R p () = pT ( p n- 1 )( p ) = p X [: ]= p n +1 b , where b = ( 1 if p . 0 if 6 p . (11.7) Comparing the coefficients at we have to show that (a) a = 1 if 6 p ; (b) a = p + 1 if p . Recall that a counts the number of 00 of index p in which contain as a sublattice of index p n . We have p 00 . Thus the image of 00 in /p is a subgroup of index p . In case (a) the image of in the same group is a non- trivial group contained in . Since the order of is equal to p , they must coincide. This shows that 00 in /p is defined uniquely, hence there is only one such 00 , i.e. a = 1. In case (b), 00 could be any subgroup of order p in /p . The number of subgroups of order p in ( Z /p Z ) 2 is obviously equal to p + 1....
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