Counting rational points with Stöhr-Voloch theory and Tate-Shafarevich results - Sousa, João Paulo Guardieiro (2025)
This thesis present some results in the number of rational points of curves defined over finite fields. The first part of this work involves the Stöhr-Voloch theory. We classified the trinomial curves that are Frobenius nonclassical with respect to the morphism of lines, and obtained a formula for the number of its rational points. In addition, we obtained the intersection numbers of the Frobenius nonclassical curves with tangent lines, and some partial results on the Weierstrass semigroups at the places of their function fields. The second part of this work concerns elliptic surfaces which can be seen as elliptic curves over function fields of characteristics 2 and 3. Starting from fixed supersingular elliptic curves over F16 and F9, we obtained formulas for the Mordell-Weil rank of suitable twists of them in Artin-Schreier and Kummer extensions of the function field of a curve. We also studied the singular fibers of the associated elliptic fibration and derived a formula for the geometric Mordell-Weil rank of such twists.