#### Alladi Ramakrishnan Hall

#### Factorization of holomorphic eta quotients

#### Soumya Bhattacharya

##### CIRM, Trento

*Unlike integer factorization, a reducible holomorphic eta*

quotient may not factorize uniquely

as a product of irreducible holomorphic eta quotients. But whenever such an

eta quotient is reducible,

the occurrence of a certain type of factor could be observed: We conjecture

that if a holomorphic eta

quotient f of level M is reducible, then f has a factor of level M. In

particular, it implies that rescalings

and Atkin-Lehner involutions of irreducible holomorphic eta quotients are

irreducible. We prove a number

of results towards this conjecture: For example, we show that a reducible

holomorphic eta quotient of

level M always factorizes nontrivially at some level N which is a multiple

of M such that rad(N) = rad(M)

and moreover, N is bounded above by an explicit function of M. This implies

a new and much faster

algorithm to check the irreducibility of holomorphic eta quotients. In

particular, we show that our conjecture

holds if M is a prime power. We also show that the level of any factor of a

holomorphic eta quotient f of

level M and weight k is bounded w.r.t. M and k. Further, we show that there

are only finitely many irreducible

holomorphic eta quotients of a given level and provide a bound on the

weights of such eta quotients. Finally,

we give an example of an infinite family of irreducible holomorphic eta

quotients of prime power levels.

Done