In p-adic Analysis (Spring 2018), we will look at a family of number systems called the $p$-adic numbers, one for each prime. The real numbers are a familiar completion of the rational numbers: any real number can be written as a limit of rational numbers. But using a different definition of the absolute value or distance, we can define new number systems known as the $p$-adic numbers. Like the real numbers, they are well-suited to analysis, so we can talk about sequences and series of $p$-adic numbers. But the $p$-adic versions typically behave better than the real ones. For instance, there are sequences $a_n$ of real numbers so that $a_n\to 0$ as $n\to\infty$, but the corresponding series $\sum_{n=0}^\infty a_n$ does not converge. Not so for the $p$-adic versions: the series converges if and only if the terms tend to 0. It is also easy to determine whether a $p$-adic polynomial has a root, and how many of them there are. In fact, there is even a simple method of calculating the number of roots of a polynomial with a given absolute value, something that has no analogue in the real numbers.
Many familiar functions on the natural numbers, including the Fibonacci numbers, can be extended to $p$-adic functions, or, more precisely functions on the product of all the $p$-adic numbers. We will look at how that works and what conditions are needed for a function to be extensible to the $p$-adic numbers. There are also $p$-adic exponential, logarithm, $\Gamma$, and $\zeta$ functions, among many others. Along the way, we will introduce topics from analysis, topology, and abstract algebra as needed so that we can understand the $p$-adic version, which are generally quite a bit simpler. For instance, there are only finitely many field extensions of the $p$-adic field $\mathbb{Q}_p$ of any given degree, in contrast to the case of $\mathbb{Q}$.
The $p$-adic numbers are important in number theory. Since $\mathbb{Q}_p$ contains $\mathbb{Q}$, a polynomial that has a root in $\mathbb{Q}$ must also have a root in each $\mathbb{Q}_p$. But sometimes, it also turns out that we can detect the presence of a root in $\mathbb{Q}$ from a root in each $\mathbb{Q}_p$, and also in $\mathbb{R}$. In particular, this is true for quadratic forms; this is the celebrated Hasse-Minkowski Theorem. We will also look at other applications of $p$-adic numbers to number theory. For instance, in an integer sequence satisfying a linear recurrence relation, the zeros form a union of arithmetic progressions, together with finitely many exceptions. All known proofs of this theorem rely on the $p$-adic numbers.
There are many problems related to $p$-adic numbers that have not been explored much, if at all. Thus there are plenty of opportunities to start on research projects based on material in the class, and we will include some possible problems on the problem sets.