Unformatted text preview: 106 LECTURE 10. THE MODULAR EQUATION Observe also that F 2 =- 1 so that Fr 2 = identity. It is called the Fricke involution . By taking the inverse transform of functions, the Fricke involution acts on modular functions of weight k by Fr * ( f )( τ ) = f (- 1 /Nτ ) = ( Nτ ) 2 k f ( Nτ ) . In particular, Fr * ( j ( τ )) = j ( Nτ ) , Fr * ( j ( Nτ )) = j (- 1 /τ ) = j ( τ ) . This implies that the Fricke involution acts on the modular equation by switching X and Y . Remark 10.4 . Let X ( N ) + = X ( N ) / ( Fr ) be the quotient of the curve X ( N ) by the cyclic group generated by the Fricke involution. One can find all numbers N such that the genus of this curve is equal to 0. It was observed by A. Ogg that the list of corresponding primes is the same as the list of all prime divisors of the order of the Monster group, the largest simple sporadic finite group. This has been explained now....
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- Fall '09
- Cyclic group, Elliptic Curve, Modular form, Fricke, Fricke involution