Almost André in Positive Characteristic

Abstract: Abstract: The André–Oort theorem (Jacob Tsimerman together with Pila and Shankar, 2021) characterizes algebraic subvarieties of Shimura varieties that contain a Zariski-dense set of special points: such subvarieties must themselves be special.

In the simplest nontrivial case, $Y(1)^2_{\mathbb{C}}$ , this classification is particularly explicit. The special curves are the vertical and horizontal fibers above CM points, together with the modular curves defined by the modular polynomials $\Phi_N(X,Y)$. André's theorem asserts that any irreducible algebraic curve containing infinitely many special points must belong to this list.

In this talk, I will present a proof of a positive-characteristic variant of André's theorem for the product of two modular curves.

algebraic geometrynumber theory

Audience: researchers in the discipline

Bilkent University Arithmetic Geometry Seminar