Abstract
In this paper, we show several connections between the L-conjecture, proposed by Burgisser [3], and the boundedness theorem for the torsion of elliptic curves. Assuming the L-conjecture, a sharper bound is obtained for the number of torsions over extensions of k on an elliptic curve over a number field k, which improves Masser’s result [6]. It is also shown that the Torsion Theorem for elliptic curves [10] follows directly from the WL-conjecture, which is a much weaker version of the L-conjecture. Since the WL-conjecture differs from the trivial lower bound only at the coefficient, this result provides an interesting example where increasing the coefficient in a trivial lower bound of straight-line complexity is difficult and important.
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Cheng, Q. (2002). Some Remarks on the L-Conjecture. In: Bose, P., Morin, P. (eds) Algorithms and Computation. ISAAC 2002. Lecture Notes in Computer Science, vol 2518. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36136-7_12
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DOI: https://doi.org/10.1007/3-540-36136-7_12
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