Modular forms and effective Diophantine approximation |
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Authors: | M Ram Murty Hector Pasten |
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Institution: | Department of Mathematics and Statistics, Queen?s University, Jeffery Hall, University ave., Kingston, ON K7L 3N6, Canada |
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Abstract: | After the work of G. Frey, it is known that an appropriate bound for the Faltings height of elliptic curves in terms of the conductor (Frey?s height conjecture) would give a version of the ABC conjecture. In this paper we prove a partial result towards Frey?s height conjecture which applies to all elliptic curves over , not only Frey curves. Our bound is completely effective and the technique is based in the theory of modular forms. As a consequence, we prove effective explicit bounds towards the ABC conjecture of similar strength to what can be obtained by linear forms in logarithms, without using the latter technique. The main application is a new effective proof of the finiteness of solutions to the S-unit equation (that is, S-integral points of ), with a completely explicit and effective bound, without using any variant of Baker?s theory or the Thue–Bombieri method. |
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Keywords: | Effective Diophantine approximation Modular forms Unit equation ABC conjecture |
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