Bhargava cubes and elliptic curves

Authors

  • Martí Oller Riera University of Cambridge

Keywords:

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

Abstract

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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How to Cite

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

Issue

Vol. 7 No. 1 (2022)

Section

Articles

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