Zeta Functions - Elliptic Curves: Number Theory and Cryptography - page 441

(a) Show that ( f | M 1 ) | M 2 = f | ( M 1 M 2 ).

(b) Let W = 0

. Show that W Γ 0 ( N ) W − 1 =Γ 0 ( N ).

1
N 0

−

Γ 0 ( N ). Let
g ( z )=( f |W )( z ). Show that g|M = g for all M ∈ Γ 0 ( N ). ( Hint:
Combine parts (a) and (b).)
(d) Suppose that f is a function with f |M = f for all M ∈ Γ 0 ( N ). Let
f + =

(c) Suppose that f is a function with f | M = f for all M ∈

1 1

|W = f +
and f − |W = −f − . This gives a decomposition f = f + + f − in
which f is written as a sum of two eigenfunctions for W .
14.3 It is well known that a product (1 + b n )convergesif |

2 ( f + f |W )and f − = 2 ( f − f |W ). Show that f +

converges.
Use this fact, plus Hasse's theorem, to show that the Euler product
defining L E ( s )convergesfor

b n |

( s ) > 3 / 2.