Abstract
Let \(\ell \) a prime number and \(\Phi _{\ell }(X,Y)\) the modular polynomial of level \(\ell \). Since this polynomial has integer coefficients one may compute it modulo primes. Petr Lisonek and Yung-Jung Kim compute the polynomial \(\Phi _{\ell }(X,Y)\) modulo \(2\) explicitly for several \(\ell \) and they conjecture that the coefficients under the diagonal vanish. In this note we prove their conjecture and that the same property holds modulo the primes \(3\) and \(5\).
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C.-S. Radu—The research was supported by the strategic program “Innovatives OÖ 2010 plus” by the Upper Austrian Government in the frame of project W1214-N15-DK6 of the Austrian Science Fund (FWF).
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Kim, Y.-J.: Algorithms for Kloosterman Sums. Master’s thesis, Simon Fraser University (2011)
Lisonek, P.: Classical Modular Polynomials over GF(2). RISC Seminar Talk, 09 (2012)
Schoeneberg, B.: Elliptic Modular Functions. Springer, New York (1974)
Serre, J.P.: A Course in Arithmetic. Springer, New York (1996)
Acknowledgments
I would like to thank the anonymous referee for the suggestions and corrections which led to improvements of this paper.
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© 2015 Springer International Publishing Switzerland
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Radu, CS. (2015). A Note on a Problem Proposed by Kim and Lisonek. In: Gutierrez, J., Schicho, J., Weimann, M. (eds) Computer Algebra and Polynomials. Lecture Notes in Computer Science(), vol 8942. Springer, Cham. https://doi.org/10.1007/978-3-319-15081-9_9
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DOI: https://doi.org/10.1007/978-3-319-15081-9_9
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