Mishra, shubham and Kumar, Narasimha
(2018)
Rational Points on Elliptic Curves.
Masters thesis, Indian Institute of Technology Hyderabad.
Abstract
The aim of this thesis is to de�ne the Elliptic Curves and some of intresting properties of a special
class of terms,namely,rational point of elliptic curve ,The properties of the rational pooint of curve
is airthmetic ,and rational point on elliptic curve forms a �nitely generated group structure group it
will be done by using the chordtangent group law of composition.After completing the the abelian
group structure,so we look some elementary properties like for a given elliptic curve we �nd the
torsion subgroup of that elliptic curves.and �nally done the elliptic curve and their isomorphism.
Further, we study different families of elliptic curves which depend on different parameters.
Moreover, we look at the reduction modulo p of an elliptic curve and infer the meaning of good and
bad reduction of an elliptic curve. Lastly, the statement and proof of MordellWeil theorem is given.
The topics of the thesis are based on the book Elliptic Curves by Dale Husemoller. I have
not added anything new, except making a few observations of my own.
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