Unformatted text preview: 112 LECTURE 11. HECKE OPERATORS pT ( p n 1 ) ◦ R p (Λ) = pT ( p n 1 )( p Λ) = p X [Λ:Λ ]= p n +1 b Λ Λ , where b Λ = ( 1 if Λ ⊂ p Λ. 0 if Λ 6⊂ p Λ. (11.7) Comparing the coefficients at Λ we have to show that (a) a Λ = 1 if Λ 6⊂ p Λ; (b) a Λ = p + 1 if Λ ⊂ p Λ. Recall that a Λ counts the number of Λ 00 of index p in Λ which contain Λ as a sublattice of index p n . We have p Λ ⊂ Λ 00 ⊂ Λ. Thus the image ¯ Λ of Λ 00 in Λ /p Λ is a subgroup of index p . In case (a) the image of Λ in the same group is a non trivial group contained in ¯ Λ . Since the order of ¯ Λ is equal to p , they must coincide. This shows that Λ 00 in Λ /p Λ is defined uniquely, hence there is only one such Λ 00 , i.e. a λ = 1. In case (b), ¯ Λ 00 could be any subgroup of order p in Λ /p Λ. The number of subgroups of order p in ( Z /p Z ) 2 is obviously equal to p + 1....
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This note was uploaded on 01/08/2012 for the course MATH 300 taught by Professor Ontonkong during the Fall '09 term at SUNY Stony Brook.
 Fall '09
 ONTONKONG

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