18.782 Arithmetic Geometry Lecture Note 25
نویسنده
چکیده
In the last lecture we proved that the torsion subgroup of the rational points on an elliptic curve E/Q is finite. In this lecture we will prove a special case of Mordell’s theorem, which states that E(Q) is finitely generated. By the structure theorem for finitely generated abelian groups, this implies E(Q) ' Z ⊕ T, where Zr is a free abelian group of rank r, and T is the (necessarily finite) torsion subgroup.1 Thus Mordell’s theorem provides an alternative proof that T is finite, but unlike our earlier proof, it does not provide an explicit method for computing T . Indeed, Mordell’s theorem is notably ineffective; it does not give us a way to compute a set of generators for E(Q), or even to determine the rank r. It is a major open question as to whether there exists an algorithm to compute r; it is also not known whether r can be uniformly bounded.2 Mordell’s theorem was generalized to number fields (finite extensions of Q) and to abelian varieties (recall that elliptic curves are abelian varieties of dimension one) by André Weil and is often called the Mordell-Weil theorem. All known proofs of Mordell’s theorem (and its generalizations) essentially amount to two proving two things:
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تاریخ انتشار 2013