Strategic Research to Solve Conjectures in Arithmetic Geometry

The KiPAS Arithmetic Geometry and Number Theory Group is researching various conjectues concerning special values of L-functions, especially by using a geometric object called the polylog. “Algebraic varieties are geometric objects given as the zeros of polynomials. Various conjectures exist regarding special values of L-functions for such algebraic varieties. Our primary objective is to explore such conjectures.” All positive integers can be uniquely expressed as a product of prime numbers. Prime numbers are like the “atoms” constituting an integer with respect to the product, and the properties of prime numbers are captured by L-functions. The simplest example of an L-function is a function called the Riemann zeta function, which is expressed by this type of infinite sum with respect to a real numbers that are greater than 1. This infinite sum can be re-expressed as an infinite product parameterized by prime numbers. “Since the time of Euclid, it has been known that there are an infinite number of prime. It was discovered by Euler that this fact corresponds to the analytic fact that the Riemann zeta function diverges when the variable s approaches 1. This type of phenomenon is now being researched in even greater depth.” Important invariants in number theory and special values of L-function are of totally different worlds, so that it is surprising to expect a connection. The idea behind arithmetic geometry is to introduce geometric objects in order to make a connection between the two. The KiPAS Arithmetic Geometry and Number Theory Group is focused on a geometric object called the “polylog”. “Arithmetic conjectures of L-functions try to connect two entirely different quantities of algebraic varieties, namely special values of L-functions which are analytic, and arithmetic information of the algebraic variety. One method to achieve this goal is to introduce a certain geometric object that intrinsically contains both quantities. One simple example of a geometric is a polygon, which intrinsically contains quantities such as the number of vertices and the number of sides. For our conjecture, the polylog is an extremely promising but complicated geometric object. Grothendieck refers to this type of geometrical object as a “motif”. Motif is a word that originates in the description of the work of Cézanne, and it has the same type of connotation as a motif of an artwork. If important but seemingly very different invariants can be interpreted as intrinsic features of a common motif, then one may be able to deduce relation between the different invariants.” The members of the KiPAS Arithmetic Geometry and Number Theory Group are conducting research as a team, and by studying the polylog, the team is endeavoring to solve important conjectures in arithmetic geometry.