| Abstract:
Let E be an elliptic curve over Q, the field of rational numbers. For
all practical purposes, this simply means that E is a non-singular
cubic curve with rational coefficients and having a rational point.
The celebrated theorem of J. L. Mordell says that the group E(Q) of
all rational points of E is a finitely generated abelian group. As a
finitely generated abelian group, E(Q) admits the following
decomposition:
E(Q) = Z^r(E) x TorsE(Q)
where rE is the rank of the curve E and TorsE(Q) is the subgroup of
elements of finite order in E. By a theorem of Barry Mazur, not all
finite groups are admissible. In this talk, we give examples of elliptic
curves for which various admissible torsion groups are indeed
realized. Once again, we will witness here a harmonious blend of
geometry, algebra and number theory.
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