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.