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Factorization Tests and Algorithms Arising from Counting Modular Forms and Automorphic Representations

Published online by Cambridge University Press:  09 January 2019

Miao Gu
Affiliation:
Department of Mathematics, Duke University, Durham, NC 27708, USA Email: miao.gu@duke.edu
Greg Martin
Affiliation:
Mathematics Department, University of British Columbia, Vancouver, BC V6T 1Z2 Email: gerg@math.ubc.ca
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Abstract

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A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight $k$ on $\unicode[STIX]{x0393}_{0}(N)$ 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 $\unicode[STIX]{x0393}_{0}(N)$.

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 not to 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.

Type
Article
Copyright
© Canadian Mathematical Society 2018 

Footnotes

The second author’s work is partially supported by a National Sciences and Engineering Research Council of Canada Discovery Grant.

References

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