Simon Myerson, Undergraduate Mathematics Seminar
In 1659 Pierre de Fermat claimed to have “astonished the greatest experts” with his methods of ascent and descent. They were the first attempt at a systematic approach to solving polynomial equations in whole numbers (Diophantine equations). Descent later became part of modern number theory after being used to prove the Mordell-Weil theorem in the 1920s. Descents – in a form which would have greatly astonished Fermat – are used today in research on the ‘hot topic’ of elliptic curves.
This is the first of two seminars on this topic. We will see precedents for ascent and descent in the work of Diophantos and Bhascara and discuss Fermat’s innovations. We will see how these ideas became part of modern algebraic geometry, using the Riemann-Roch theorem to define group laws on curves.