Contemporary Methods for Solving Diophantine Equations

The topic of this summer school was Diophantine equations, which are among the oldest studied mathematical objects. A Diophantine equation is an equation where admissible solutions are restricted to the rationals or the integers, or appropriate mathematical generalizations of such objects. The equations themselves tend to be polynomial, exponential, or a mixture of both, where variables in the exponents are usually restricted to (positive) integers. A characteristic example is the equation that is central to Fermat’s Last Theorem, x + y = z, with x, y, z ∈ Z and n ∈ {3, 4, . . .}. Because Diophantine equations concern themselves with objects so fundamental to mathematics, they tend to arise whenever one uses the mathematical language to formulate problems or theories. This supplies a dual motivation to the field. On the one hand, there is an interest to understand theoretically the set of solutions to the equations and its relationship to the geometric objects defined by the equations. On the other hand, there is a demand for practical methods that, given an explicit equation, provide a complete and explicit description of the set of solutions. In recent years, a combination of development of general theory, computational tools and computational techniques has greatly improved our ability to explicitly solve Diophantine equations. Some of the methods that have proved to be particularly relevant recently are the following. • The modular method (see Section 4). The application of the ideas of Frey, Ribet, Wiles, and others, that led to the proof of Fermat’s last theorem. The method only applies to a limited class of equations, but it is currently the only method that is able to handle statements about rational solutions to families of equations with varying exponents. • Linear Forms in Logarithms (See Section 5). This method applies to a wide class of equations for which one wants to determine integral solutions. It provides explicit versions of results along the lines of Roth’s theorem, which has been used to prove finiteness of integral points on affine hyperbolic curves. • Hypergeometric method (See Section 6). This method applies to a subset of problems where linear forms in logarithms apply. When it does apply, it usually yields sharp results. • Cohomological obstructions (See Section 7). One of the oldest and simplest methods to show that a Diophantine equation has no solutions is by showing that there is some local obstruction to having solutions. Equations that have no local obstructions to having solutions are said to have solutions everywhere locally. It is well known that local obstructions are not the only obstructions to having solutions. Various other obstructions have been identified, many of which were eventually shown to