Algebraic Number Theory and Diophantine Geometry
Number theory is the branch of mathematics whose motivation comes from properties of the integers. There are many sub-fields of number theory and a good, but brief, description of some can be found here. I am interested in answering various questions about Diophantine equations - an equation where we look for solutions in a restricted class (usually the set of integers or the rational numbers). The two primary approaches I take involve methods from algebraic number theory and Diophantine geometry.
Algebraic number theory generalizes the notion of an integer to that of a Dedekind domain. We then use techniques of commutative algebra to answer some questions about Diophantine equations. For example, solving certain Diophantine equations amounts to asking whether or not we have unique factorization in some Dedekind domain (see SUMS talk). In fact, this approach was mainly developed in pursuit of a solution to the famous "last theorem" of Fermat.
In Diophantine geometry, we primarily combine the techniques of algebraic number theory and algebraic geometry to obtain generally finiteness conditions on Diophantine equations. Our equations are viewed as geometric figures in projective space and this provides a global understanding of our solutions. We might not be able to say specifically what the solutions are, but we are sometimes able to describe finiteness conditions on the solutions. When our geometric figures are curves, we obtain three important finiteness theorems: the Mordell-Weil theorem, Siegel's theorem and Faltings's theorem.
It is useful to embed our curve in a corresponding geometric structure called its Jacobian: This is is an abelian group whose dimension, called the genus, gives the best way for classifying curves. The Mordell-Weil theorem says that the subgroup of rational points for a smooth curve of genus g &ge 1 is finitely generated. This means that all the rational solutions of a Diophantine equation describing a curve of positive genus can be obtained from a finite set of solutions. Siegel's theorem says that a smooth curve with genus g &ge 1 has finitely many integral points, and Faltings's theorem, which is a proof of the Mordell conjecture, says that a smooth curve of genus g &ge 2 has finitely many rational points. Faltings's theorem, which is the main reason Faltings earned a fields medal in 1986, generalizes Siegel's theorem when the genus g &ge 2.
Another important theorem worth mentioning is Roth's theorem from Diophantine approximation, which is the surprising generalization of a theorem by Liouville. Intuitively, it says that all but finitely many rational approximations to an irrational algebraic number must have large denominators. Even though Roth's theorem is not immediately a finiteness condition on curves, it is crucial in proving some of the finiteness theorems mentioned before. For example, the proof of Siegel's theorem involves contradicting Roth's theorem by showing infinitely many integer solutions to a curve of genus g &ge 1 gives infinitely many small denominator rational approximations to an algebraic number.
Arithmetic Dynamics
Arithmetic dynamics asks about various arithmetic properties of the iterations of a map &phi: S &rarr S from a set S into itself. For example, let S be the set of rational numbers Q and &phi(z) a rational map with coefficients in Q. Pick a rational number p and define its forward orbit Orb(p) as the set iterations {p, &phi(p), &phi(&phi(p)), &phi (&phi(&phi(p))), . . .}. We could ask, under what conditions does Orb(p) contain finitely many integer points? This analogous Siegel type theorem for forward orbits was answered by J. Silverman in 1993: If &phi(&phi(z)) is not a polynomial then Orb(p) will contain finitely many integers. We could also ask, when is Orb(p) finite and how does the size vary with p? Defining the backward orbit of p as the collection of pre-images under iterations of &phi, we may ask similar questions as with the forward orbits. For the most part, these orbits seem analogous to the curves in diophantine geometry, and this serves as a guide to many of the current results in the field.