Torsion Points on Modular Curves
Abstract
Let be a quadratic field. Let be modular curves and be Jacobian curves of modular curves defined over . In this paper, we investigate torsion points on modular curves and the Mordell-Weil group of the Jacobian curves . Let be one of the primes 17,19,23,29,31. We prove that the Mordell-Weil group of the Jacobian curves over is finite and the modular curves have no -points of order . Whereas in case , we find the Jacobian curves has a finite Mordell-Weil group over where is an imaginary quadratic field and conclude that the modular curves has no -points of order .
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