The theory of ordinary differential equations is a fundamental instrument of continuous mathematics, in which the central objects of study are functions involving real numbers. It is not immediately apparent that this theory has anything useful to say about discrete mathematics in general or number theory in particular.

In this book we consider ordinary differential equations in which the role of the real numbers is instead played by the field of p-adic numbers, for some prime number p. The p-adics form a number system with enough formal similarities to the real numbers to permit meaningful analogues of notions from calculus, such as continuity and differentiability. However, the p-adics incorporate data from arithmetic in a fundamental way; two numbers are p-adically close together if their difference is divisible by a large power of p.

In this chapter, we first indicate briefly some ways in which p-adic differential equations appear in number theory. We then focus on an example of Dwork, in which the p-adic behavior of Gauss's hypergeometric differential equation relates to the manifestly number-theoretic topic of the number of points on an elliptic curve over a finite field.

Since this chapter is meant only as an introduction, it is full of statements for which we give references instead of proofs. This practice is not typical of the rest of this book, except for the discussions in Part VI.

Whyp-adic differential equations?

Although the very existence of a highly developed theory of p-adic ordinary differential equations is not entirely well known even within number theory, the subject is actually almost 50 years old.