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

  • Miao Gu (a1) and Greg Martin (a2)
Abstract

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.

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

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References
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[1] Agrawal, M., Kayal, N., and Saxena, N., PRIMES is in P . Ann. of Math. (2) 160(2004), no. 2, 781–793. https://doi.org/10.4007/annals.2004.160.781.
[2] Casandjian, C., Challamel, N., Lanos, C., and Hellesland, J., Reinforced concrete beams, columns and frames: mechanics and design. John Wiley & Sons, Hoboken, NJ, 2013, 267–276. https://doi.org/10.1002/9781118639511.
[3] Gekeler, E.-U., A remark on dimensions of spaces of modular forms . Arch. Math. (Basel) 65(1995), no. 6, 530–533. https://doi.org/10.1007/BF01194172.
[4]The LMFDB Collaboration, The L-functions and Modular Forms Database. http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic.
[5] Martin, G., Dimensions of the spaces of cusp forms and newforms on 𝛀0(N) and 𝛀1(N) . J. Number Theory 112(2005), no. 2, 298–331. https://doi.org/10.1016/j.jnt.2004.10.009.
[6] Rosser, J. B. and Schoenfeld, L., Approximate formulas for some functions of prime numbers . Illinois J. Math. 6(1962), 64–94.
[7] Shoup, V., A computational introduction to number theory and algebra . Cambridge University Press, Cambridge, 2005. https://doi.org/10.1017/CBO9781139165464.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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