Diophantus Revisited: On rational surfaces and K3 surfaces in the Arithmetica

2015/09/21 by René Pannekoek, Pannekoek, René
Computer Science · Mathematics · #History and Theory of Mathematics #Mathematics and Applications #Polynomial and algebraic computation #math.HO #math.NT #msc:01A20 #msc:11DXX

paper · pdf · doi:10.48550/arxiv.1509.06138

41 pages; comments welcome

arxiv created 2015/09/21 · arxiv updated 2015/09/22

Abstract

This article wants to show two things: first, that certain problems in Diophantus' Arithmetica lead to equations defining del Pezzo surfaces or other rational surfaces, while certain others lead to K3 surfaces; second, that Diophantus' own solutions to these problems, when viewed through a modern lens, lead to parametrizations of these surfaces, or of parametrizations of rational curves lying on them.

Related

• On the arithmetic of one del Pezzo surface over the field with three elements
• Counting rational curves on K3 surfaces
• Two lectures on the arithmetic of K3 surfaces
• Non-Euclidean Pythagorean triples, a problem of Euler, and rational points on K3 surfaces
• Arithmetic of K3 surfaces