Explicit isogenies of elliptic curves

• Main Author: Tsukazaki, Kiminori
• Published: University of Warwick 2013
• Subjects: 510; QA Mathematics
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.582409

Let E be an elliptic curve defined over a field K. The main topic of this thesis is to present a method for the explicit computation of all separable K- rational l-isogenies of E and isogenous curves for small primes l. The key tool for this explicit computation is that the modular curve X0(l) parametrises l- isogenies of elliptic curves. In [3], Cremona and Watkins give explicit isogeny formulae for l 2 f2; 3; 5; 7; 13g, where the modular curve X0(l) has genus 0. Their formula allow us to compute l-isogenies of E by simply substituting its j-invariant and twisting parameter into the formulae. We extend the work of Cremona and Watkins to the cases l 2 f11; 17; 19; 23; 29; 31; 41; 47; 59; 71g, where the genus of X0(l) is greater than 0 but the modular curve X+ 0 (l) has genus 0.