Bhargava cubes and elliptic curves

Autors/ores

  • Martí Oller Riera University of Cambridge

Paraules clau:

Bhargava cubes, genus one curves, elliptic curves, Galois co- homology, arithmetic geometry, num- ber theory.

Resum

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 x 2 x 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 x 3 x 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.

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Com citar

Oller Riera, M. (2022). Bhargava cubes and elliptic curves. Reports@SCM, 7(1), 27–39. Retrieved from https://revistes.iec.cat/index.php/reports/article/view/150102

Número

Vol. 7 Núm. 1 (2022)

Secció

Articles

Llicència

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