Birch and Swinnerton-Dyer conjecture: old and new

• Oscar Rivero-Salgado ^[1]
  1. [1] Universitat Politècnica de Catalunya

     Universitat Politècnica de Catalunya

     Barcelona, España

     
• Localización: Reports@SCM: an electronic journal of the Societat Catalana de Matemàtiques, ISSN-e 2385-4227, Vol. 4, Nº. 1, 2018, págs. 9-20
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen
  • English

    The Birch and Swinnerton-Dyer (BSD) conjecture is one of the millennium problems that has not been solved yet. Although it was formulated after different numerical experiments, there are several theoretical reasons and analogies with simpler mathematical objects that lead us to believe that it is true. We go through some of these analogies, and at the same time, we explain the most relevant results and generalizations that are currently known. At the end, we move to the rank two situation, recovering the elliptic Stark conjecture, closely related to BSD.Keywords: BSD, elliptic curve, L-series, modular forms, Gross-Zagier Stark conjecture.

  • català

    La conjectura de Birch i Swinnerton-Dyer (BSD) és un dels sis problemes delmil·lenni que encara no s’ha resolt. Tot i que es va formular després de diferents experiments numèrics, hi ha diverses raons teòriques i analogies amb objectes matemàtics més senzills que ens fan pensar que l’enunciat és cert. Repassarem primer algunes d’aquestes motivacions i explicarem els resultats i generalitzacions més rellevants que es coneixen. A la darrera part ens apropem al mon en el que el rang analıtic és dos, una situació poc treballada, i ens trobem aix í amb la conjectura el·líıptica de Stark, molt relacionada amb BSD.

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