Bhargava cubes and elliptic curves

• Martí Oller Riera ^[1]
  1. [1] University of Cambridge

     University of Cambridge

     Cambridge District, Reino Unido

     
• Localización: Reports@SCM: an electronic journal of the Societat Catalana de Matemàtiques, ISSN-e 2385-4227, Vol. 7, Nº. 1, 2022, págs. 27-39
• Idioma: inglés
• DOI: 10.2436/20.2002.02.30
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• Resumen
  • català

    En les seves c`elebres Disquisitiones Arithmeticae, Gauss va descobrir una llei de composici´o que confereix una estructura de grup al conjunt de classes de formes quadr`atiques bin`aries amb discriminant fixat. Dos segles m´es tard, Bhargava va donar una reinterpretaci´o d’aquesta llei a trav´es de cubs 2 × 2 × 2 d’enters, ara coneguts com a cubs de Bhargava. El plantejament d’aquest article rau en utilitzar la mateixa idea dels cubs de Bhargava per`o en cubs 3×3×3, que donen lloc a corbes planes projectives de grau 3. L’objectiu ´es determinar lleis de composici´o an`alogues que involucrin aquestes corbes. A tal fi, es desenvoluparan els coneixements matem`atics pertinents, incloent cohomologia de Galois i geometria algebraica, fent `emfasi en corbes ellptiques i, m´es en general, en les propietats de corbes de g`enere 1

  • English

    In his celebrated Disquisitiones Arithmeticae, Gauss discovered a composition law that gives a group structure to the set of classes of binary quadratic forms of a given discriminant. Two centuries later, Bhargava gave a reinterpretation of this law through 2 × 2 × 2 cubes of integers, now known as Bhargava cubes. In this article, we aim to use the same idea of Bhargava cubes but in 3 × 3 × 3 cubes, that yield projective plane curves of degree 3. Our aim is to determine analogous composition laws involving these curves. To this end, we will review the needed mathematical knowledge, including Galois cohomology and algebraic geometry, with an emphasis on elliptic curves and, more generally, in the properties of genus one curves