Mapping P-adic Spaces with Height Pairings Professor Amnon Besser MAPPING P-ADIC SPACES WITH HEIGHT PAIRINGS Professor Amnon Besser of Ben-Gurion University of the Negev and his colleagues are exploring p-adic numbers – one of the most difficult areas of number theory – in order to solve long-standing open problems bridging several fields of mathematics. Solving Unsolvable Equations with a New cryptography. At that time, such an Number System algorithm had already been produced, but Professor Besser felt he could create a If you are somewhat familiar with advanced better, more general one. Later on, he joined mathematics, specifically number theory, it forces with Jennifer Balakrishnan, who was is likely you have heard of p-adic numbers. working on finding certain solutions specific Often spoken of with reverence, and to computing Coleman integrals. Their work described as one of the most difficult areas produced a new method for computing a in pure mathematics, p-adics are regarded function called a p-adic height pairing, as inaccessible to the public. However, and facilitated the generalisation of they have many real-world applications, Professor Besser’s previous results. Just like complex and real numbers, p-adic such as encryption algorithms, and have numbers extend the rational number system the potential to solve several problems Now, a large part of Professor Besser’s ℚ and its associated arithmetic operations. If in fundamental physics through theories research focuses on using numerical rational numbers can be expressed as a ratio such as p-adic quantum mechanics. The methods to find rational solutions of between two integers p/q, real numbers are modular way in which p-adics retain algebraic equations. Developing numerical expressed as an endless decimal expansion. information about the equivalence between methods can address situations where One such number is π and another is √2, and different numbers is fundamental to the no exact solutions are known for a class even 1 can be written as 1.000… or 0.999… for demonstration of Fermat’s Last Theorem, as of equations, by reducing the problem to example, which is a sum of an infinite number discovered by Andrew Wiles no less than 350 number crunching. In general, these types of of powers of 1/10. Unlike real numbers, p-adics years since Fermat first stated it. In fact, the numerical solutions can help scientists and are expressed as sums of powers of a prime consequences of the p-adic number system engineers when dealing with an ill-behaved number, usually denoted by p. Since the stretch far beyond that. Although they started function that models an aspect of applied letter p is simply a placeholder for the base, as a natural quirk derived from the need to research. replacing it with some number will yield base- solve Diophantine equations (polynomial dependent names like 2-adics or 17-adics and equations in which only integer solutions In order to have a better understanding of so on. are allowed), it is believed that they have these problems, let’s go back to the basics of deep implications touching upon the Birch p-adic number theory. In a way, p-adic numbers are the opposite and Swinnerton-Dyer conjecture and the of real numbers. The difference derives from Riemann Hypothesis – two open Millennium What Are P-adics? the ordering or closeness within the number Prize Problems. system. If two real numbers that differ in the P-adics were created from a desire to apply 10ᵗ decimal digit are closer than two other For Professor Amnon Besser of Ben-Gurion the techniques of power series methods to numbers differing in the 2 ᵈ decimal digits, for University of the Negev, p-adics have been a number theory, as a way to express certain p-adics the concept of closeness is a much source of hope for finding a specific answer numbers or mathematical problems in a more interesting one. Albeit counterintuitive, to a computational problem. Part of his tractable way. After all, number theory, or this is thoroughly consistent within the p-adic problem involved an integration method the queen of mathematics as Gauss called number system. Thus, p-adics are close when developed by Robert Coleman, a method it, is famous for posing some of the most their difference can be divided by a high power which is the p-adic equivalent of integration difficult conjectures in the entire field of of p. The higher this power is, the closer the along a curve between two points. After his mathematics. Thus, the idea to create tools two numbers are said to be. In our example PhD, Professor Besser often used Coleman that would transform intractable problems where p is a fixed prime and e a variable power, integration while delving into p-adic number into approachable ones is a very natural the difference between p-adics is divisible theory. Later on, he heard that algorithms strategy. by pe and the numbers are close when e that compute a function called p-adic is very large. In other words, two decimal height could have important applications in WWW.SCIENTIA.GLOBAL ‘I look for new methods for solving equations. I put the usual well-known numbers inside some bigger space from which one can cut out the pieces containing the solutions.’ representations of a real number are close arithmetical operations behave differently when the difference between them is a large from those of usual arithmetic. One reason negative power of 10, whereas two p-adic why p-adics are a particularly neat number expansions are close when their difference is system is that the base must be a prime a large positive power of p. For p-adics, the number, otherwise the set of all numbers concept of closeness opens the possibility obtained from arithmetic operations is not to encode information about congruence of closed under these operations. In other absolute value, which could be called their words, you can divide p-adics with a non- measure, in a new and interesting way. prime base and get a number that was not part of the initial set. In addition, p-adics can be considered opposites of real numbers in terms of their Mapping the Path to Unique Solutions representations. Due to change of base, Elliptic curve p-adic numbers are written from right to Although p-adics in particular and number left, each digit added to the left increasing theory in general seem to be purely abstract form, the Diophantine equations that led to the precision of the number representation. mathematics, they are both born of the the development of p-adics can be written Recall that expansions of real numbers are very simple goal of solving equations. as ax + by = c, where x and y are variables written the other way around, from left to As Professor Besser explains, ‘I look for new and a, b and c are constants. In addition, right, with a finite number of digits before methods for solving equations. I put the p-adics methods can be used to study the decimal point and an infinite number usual well-known numbers inside some superelliptic equations, where yᵐ = f(x), and of digits after it. P-adics, on the other hand, bigger space from which one can cut out the hyperelliptic equations of the form y2 = f(x). can have an infinite number of digits to pieces containing the solutions.’ The representations of these equations are the left. Moreover, the p-adic number often called curves in the formal language of system is based on the modular number These equations are important for all types mathematics due to their shape. Sometimes, system, like a clock that resets to zero when of applied problems in physics, engineering, these curves self-intersect to form a closed reaching 12-midnight, which means that the finance, and other fields. In their simplest boundary. WWW.SCIENTIA.GLOBAL Meanwhile, Professor Besser’s colleagues, including Minhyong Kim from Oxford, used a non-abelian generalisation of a method The only solutions in integers to devised by Claude Chabauty in 1941 to find rational-valued points on curves with special properties. The Chabauty method allowed y = x(x 1) + 1. them to progress in theoretical directions, are (2, 3); (1, 1), (0, 1). by showing that some Coleman integrals become equal to zero on rational solutions of certain equations. This important finding proved that these integrals have a finite number of solutions, because a Coleman integral can become 0 only a finite number of times. During this time, Professor Besser focused on the practical question of actually finding the rational and integral solutions to equations using the Chabauty method. His approach involved using p-adic height pairings. The approach is based on the idea that rational solutions to equations can be added 1 together in some sense. The advantage of the p-adic height pairing is that it transforms in a very simple way when solutions are added, and this leads to equations on the values of the height pairing at integral points. Based on Professor Besser’s previous work, these equations can be expressed as a Coleman integral becoming zero. Although this One of the peculiarities that follow from the Professor Besser found a way to overcome seems like a simple enough exercise, there rules of p-adics is that they cause unusual the problem of multi-valued results for the are in fact many categories of unsolvable behaviour. In a geometric interpretation, same function by continuing Coleman’s work equations. Solutions to these equations, as p-adic numbers form a space much like on integration.