Valuations of function fields form the geometrical point of view.
We study the variation of the reduction type of elliptic curves under base change. A complete description of the variation is given when the base field is the p-adic field and the base change is of small degree.
In this article, we develop a geometric method to construct solutions of the classical Yang–Baxter equation, attaching a family of classical -matrices to the Weierstrass family of plane cubic curves and a pair of coprime positive integers. It turns out that all elliptic -matrices arise in this way from smooth cubic curves. For the cuspidal cubic curve, we prove that the obtained solutions are rational and compute them explicitly. We also describe them in terms of Stolin’s classication and prove...
The universal vectorial extension of a curve is described in terms of the geometry of the curve.