Colloquium - John Voight - A heuristic for boundedness of ranks of elliptic curves
Title: A heuristic for boundedness of ranks of elliptic curves
Abstract: An elliptic curve over the rational numbers is smooth curve in the plane defined by an equation of the form y^2 = x^3 + Ax + B with A,B rational numbers. The chord-tangent law allows us to "add" points on an elliptic curve, and in fact the set of rational points has the structure of a finitely generated abelian group. In this talk, we suggest that the minimal number of generators required for this group is bounded for all rational elliptic curves! Our prediction is based on a probabilistic model for certain aspects of the arithmetic of an elliptic curve; this model comes from linear algebra and it seems to match with computational experiment. This is joint work with Jennifer Park, Bjorn Poonen, and Melanie Matchett Wood.
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