2020/09/18 by Alex Cowan, Cowan, Alex
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Coding theory and cryptography #Limits and Structures in Graph Theory #math.NT #msc:11G05 #msc:14J27
paper · pdf · doi:10.48550/arxiv.2009.08622
arxiv created 2020/09/21 · arxiv updated 2020/09/23
Based on an equation for the rank of an elliptic surface over ℚ which appears in the work of Nagao, Rosen, and Silverman, we conjecture that 100% of elliptic surfaces have rank 0 when ordered by the size of the coefficients of their Weierstrass equations, and present a probabilistic heuristic to justify this conjecture. We then discuss how it would follow from either understanding of certain L-functions, or from understanding of the local behaviour of the surfaces. Finally, we make a conjecture about ranks of elliptic surfaces over finite fields, and highlight some experimental evidence supporting it.