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'Holomorphic eta quotients' are certain classical
modular forms on suitable Hecke subgroups of the full modular
group. We call a holomorphic eta quotient $f$ 'reducible' if for
some holomorphic eta quotient $g$ (other than 1 and $f$), the
eta quotient $f/g$ is holomorphic. We show that for any positive
integer $N$, there are only finitely many irreducible holomorphic
eta quotients of level $N$. In particular, the weights of such eta
quotients are bounded above by a function of $N$. We shall
provide such an explicit upper bound. This is an analog of a
conjecture of Zagier which says that for any positive integer $k$,
there are only finitely many irreducible holomorphic eta quotients
of weight $k/2$ which are not integral rescalings of some other eta
quotients. This conjecture was established in 1991 by Mersmann.
We shall sketch a short proof of Mersmann's theorem and we shall
show that these results have their applications in factorizing
holomorphic eta quotient.
This talk will also be suitable for non-experts: We shall define
all the relevant terms and we shall state clearly the classical results
which we use in the proof. |