Department of Mathematics
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The concept of modular curves is an essential ingredient in the celebrated proof of FLT. Roughly speaking, modular curves X(N) are explicit equations that parametrize the family (E, N) of elliptic curves together with a torsion point of a given order N. In this talk, we show how to construct elliptic curves with given torsion, and how to prove that certain torsion groups do not occur as rational subgroups. In particular, we construct an explicit equation for X(11) thereby demonstrating that Z[11] cannot occur as a rational subgroup over E(Q) for any E. This is just a very special case of a famous theorem of Barry Mazur which says that there are only a few possibilities for the torsion subgroup of E(Q) and 11 is the first such prohibited order.

Mazur, B. Rational points on modular curves (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Lecture Notes in Math., Vol. 601, Springer, Berlin, 1977.