Tsukazaki, Kiminori (2013) Explicit isogenies of elliptic curves. PhD thesis, University of Warwick.

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Abstract

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.

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Curves, Elliptic, Modular curves
Official Date:	July 2013
Dates:	Date Event July 2013 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Cremona, J. E.
Sponsors:	University of Warwick. Mathematics Institute
Extent:	v, 112 leaves.
Language:	eng

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