Factorization tests and algorithms arising from counting modular forms and automorphic representations

TL;DR

This paper characterizes squarefree integers and primes using counts of automorphic representations and Hecke newforms, proposing algorithms that could efficiently test and factor numbers based on these counts.

Contribution

It provides the first characterization of squarefree integers and primes via automorphic form counts and develops algorithms for number testing and factorization based on these properties.

Findings

Characterization of squarefree integers via automorphic representations

Algorithmic approach to test if a number is squarefree using automorphic form counts

Probabilistic methods to factor the squarefull part of a number

Abstract

A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight $k$ on $\Gamma_0(N)$ to a simpler function of $k$ and $N$, showing that the two are equal whenever $N$ is squarefree. We prove the converse of this theorem (with one small exception), thus providing a characterization of squarefree integers. We also establish a similar characterization of prime numbers in terms of the number of Hecke newforms of weight $k$ on $\Gamma_0(N)$. It follows that a hypothetical fast algorithm for computing the number of such automorphic representations for even a single weight $k$ would yield a fast test for whether $N$ is squarefree. We also show how to obtain bounds on the possible square divisors of a number $N$ that has been found to not be squarefree via this test, and we show how to probabilistically obtain the complete factorization of the squarefull part of $N$ from the number of such automorphic representations for two different weights. If in addition we have the number of such Hecke newforms for even a single weight $k$, then we show how to probabilistically factor $N$ entirely. All of these computations could be performed quickly in practice, given the number(s) of automorphic representations and modular forms as input.