Factorization tests and algorithms arising from counting modular forms and automorphic representations · arXivDesk
1709.02411Sep 7, 201715 pages. The original version of this manuscript was entitled "A characterization of squarefree numbers using automorphic representations"; the title has been changed to reflect additional theorems contained in the current version
Factorization tests and algorithms arising from counting modular forms and automorphic representations
A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight k on Γ0(N) to a simpler function of k and N, showing that the two are equal whenever N
Nearby in the stack
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
Γ0(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.