p-adic heights of special cycles

The proposed research deals witli the theoretical problem of finding rational solutions to polynomial equations. These problems were already of interest to ancient Greek math- ematicians and are still largely unsolved today. Our project will significantly advance our understanding of new cases in which the polynomial equations define not only curves or surfaces, but algebraic varieties of arbitrarily high dimension.

The idea is inspired from the Langlands programme, a series of partially—proved conjec- tures predicting a deep link between the symmetries of algebraic numbers and functions on certain symmetric spaces. We will further develop a method based on functions that take values iii a strange but useful system of numbers encoding congruences modulo powers of a prime (e.g. if the prime is 3, iii this system of numbers 81 : 34 is very close to 0, like 0.0001 : 0.14 is in the usual decimals). This will allow us to study certain systematic methods of constructing solutions to the equations of interest based on more symmetric equations. It will also help us understand the connection between our problem and the study of solutions in simpler systems of numbers, such as the finite system of numbers from 0(: 12) to 11 used for the hours of the day.

The main point of interest of this project is thus to uncover modern relations among three ancient branches of mathematics — arithmetic, symmetry, and geometry —, and put them to good use. Date: November 20. 2022.