Heegner points and Mordell-Weil groups of elliptic curves over large fields

• Autores: Boe-Hae Im
• Localización: Transactions of the American Mathematical Society, ISSN 0002-9947, Vol. 359, Nº 12, 2007, págs. 6143-6154
• Idioma: inglés
• DOI: 10.1090/s0002-9947-07-04364-4
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• Resumen

  • Let be an elliptic curve defined over of conductor and let be the absolute Galois group of an algebraic closure of . For an automorphism , we let be the fixed subfield of under . We prove that for every , the Mordell-Weil group of over the maximal Galois extension of contained in has infinite rank, so the rank of is infinite. Our approach uses the modularity of and a collection of algebraic points on - the so-called Heegner points - arising from the theory of complex multiplication. In particular, we show that for some integer and for a prime prime to , the rank of over all the ring class fields of a conductor of the form is unbounded, as goes to infinity.