Pol van Hoften recently published his article "Mod p points on Shimura varieties of parahoric level" in Forum of Mathematics, Pi. This article, which is based on his PhD thesis, makes significant progress on a long-standing conjecture of Langlands and Rapoport. The conjecture in question describes the set of mod p points of Shimura varieties—essentially, the solutions modulo p to specific systems of polynomial equations. It plays an important role in the Langlands programme, which is an influential and very active field of modern pure mathematics that concerns a web of far-reaching conjectural connections between number theory, harmonic analysis and algebraic geometry. Pol was recently awarded a Veni grant to further pursue this line of research.
The Langlands program summarized in one paragraph: Zeta functions of algebraic varieties—loosely speaking, systems of polynomial equations with integer coefficients—generalize the famous Riemann zeta function and play a central role in modern algebraic number theory. Conjecturally, these zeta functions have good analytic properties, such as meromorphic continuation and a functional equation, but this is known in only a small number of examples. Langlands proposed to prove these properties for zeta functions of Shimura varieties, a special class of highly symmetric algebraic varieties, by expressing their zeta functions in terms of zeta functions of analytic objects known as automorphic representations.
The conjecture of Langlands and Rapoport is a crucial part of Langlands's approach. Zeta functions of algebraic varieties have, like the Riemann zeta function, an Euler product formula as a product of factors, one for each prime number p. These factors are called local zeta functions; they are determined by the number of modulo p solutions to the system of polynomial equations. Therefore, point-counting on Shimura varieties is closely related to understanding their zeta functions.