2-adic properties of modular functions associated to Fermat curves

Abstract

For an odd integer N N , we study the action of Atkin’s U ( 2 ) U(2) -operator on the modular function x ( τ ) x(\tau ) associated to the Fermat curve: X N + Y N = 1 X^N+Y^N=1 . The function x ( τ ) x(\tau ) is modular for the Fermat group Φ ( N ) \Phi (N) , generically a noncongruence subgroup. If x ( τ ) = q − 1 + ∑ i = 1 ∞ a ( i N − 1 ) q i N − 1 x(\tau )=q^{-1}+\sum _{i=1}^\infty a(iN-1)q^{iN-1} , we essentially prove that lim n → 0 a ( n ) = 0 \lim _{n \rightarrow 0}a(n)=0 in the 2 2 -adic topology.