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The arithmetic of modular forms associated to Fermat curves

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Matija Kazalicki
U of Zagreb
Mit, 13/10/2010 - 14:15 - 15:15
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

For an odd integer $N$, we study the action of Atkin's $U(2)$-operator on the modular function $x(t)$ associated to the Fermat curve: $X^N+Y^N=1$. The function $x(t)$ is modular for the Fermat group $\Phi(N)$, generically a noncongruence subgroup. If $x(t)=q^{-1}+\sum_{i=1}^\infty a(iN-1)q^{iN-1}$, we essentially prove that $\lim a(n)=0$ in the 2-adic topology as n tends to zero. If time permits, we'll mention a conjecture related to Atkin and Swinnerton-Dyer congruences for certain cusp form of weight 3 for Fermat group $\Phi(3)$.

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