Elementary divisors or the order matrix and irreducibility of holomorphic eta quotients

20 ottobre 2020
20 Ottobre 2020
Staff Dipartimento di Matematica

Università degli Studi Trento
38123 Povo (TN)
Tel +39 04 61/281508-1625-1701-3898-1980.
dept.math [at] unitn.it

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Ore: 14:00


  • Soumya Batthacharya  (Institute of Analysis and Number Theory, Graz, Austria)

The order matrix of level $N$ associated with the Dedekind eta function is the matrix whose $(i,j)$-th entry is the order of vanishing of $\eta(iz)$ at the cusp $1/j$ of the Hecke subgroup of level $N$ of the full modular group. Here $\eta$ denotes the Dedekind eta function. In this talk, we shall provide for all $N$ a canonical decomposition of this matrix of level $N$ into the elementary divisors form. As an application, we shall deduce an irreducibility criterion for eta quotients. A holomorphic eta quotient is called irreducible if it is not a product of two other holomorphic eta quotients. Like primality of natural numbers, irreducibility of holomorphic eta quotients also have some nice applications. Indeed, we shall begin with a short discussion about why we care about irreducibility of holomorphic eta quotients at all and why given any such eta quotient $f$, in general it is a rather difficult problem to determine whether $f$ is irreducible. 

Referente: Marco Andreatta