Seminar

Holomorphic eta quotients' are certain explicit classicalmodular forms on suitable Hecke subgroups of the full modular group.We call a holomorphic eta quotient $f$ 'reducible' if for someholomorphic eta quotient $g$ (other than 1 and $f$), the eta quotient$f/g$ is holomorphic. An eta quotient or a modular form in generalhas two parameters: Weight and level. We shall show that for anypositive integer $N$, there are only finitely many irreducible holomorphiceta quotients of level $N$. In particular, the weights of such eta quotientsare bounded above by a function of $N$. We shall provide such an explicitupper bound. This is an analog of a conjecture of Zagier which says thatfor any positive integer $k$, there are only finitely many irreducibleholomorphic eta quotients of weight $k/2$ which are not integral rescalingsof some other eta quotients. This conjecture was established in 1991 byMersmann. We shall sketch a short proof of Mersmann's theorem and weshall show that these results have their applications in factorizingholomorphic eta quotient. In particular, due to Zagier and Mersmann's work,holomorphic eta quotients of weight $1/2$ have been completely classified.We shall see some applications of this classification and we shall discussa few seemingly accessible yet longstanding open problems about etaquotients. This talk will be suitable also for non-experts: We shall define all therelevant terms and we shall clearly state the classical results which we use.