Special cycles and p-adic Relative Trace Formulas

SPECIAL CYCLES AND p-ADIC RELATIVE TRACE FORMULAS (BSF PROPOSAL n 2018250) DANIEL DISEGNI (BGU), WEI ZHANG (MIT) Abstract (in lay terms) The proposed research deals with the theoretical problem of finding rational solutions to polynomial equations. These problems were of interest to ancient Greek mathematicians and are still unsolved today. Our project will significantly advance our understanding of new cases in which the polynomial equations define algebraic varieties of arbitrarily high dimension.

The main idea is inspired from the Langlands programme, a series of partially-proved conjectures predicting a deep link between the symmetries of algebraic numbers and func- tions on certain symmetric spaces. We will develop a new method based on functions which take values in a strange but useful system of numbers encoding congruences modulo powers 4 of a prime (e.g. if the prime is 3, in this system of numbers 81 = 3 is very close to 0, like 0:0001 = 0:1 in the usual decimals). This will allow us to study a systematic method of constructing solutions to the equations of interest. 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 interest of this project is currently theoretical. Nevertheless the problem is a natural development of decades of work on the study of elliptic curves: these are equations in two variable that have found ample applications in cryptographic security, and were recently in the news for evidence of tampering by the NSA.

Date: October 20, 2018.