2-adic properties of modular functions associated to Fermat curves

Kazalicki, Matija

doi:10.1090/S0002-9939-2011-10854-2

$2$-adic properties of modular functions associated to Fermat curves
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by Matija Kazalicki

Proc. Amer. Math. Soc. 139 (2011), 4265-4271

DOI: https://doi.org/10.1090/S0002-9939-2011-10854-2

Published electronically: April 29, 2011

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Abstract:

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

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