Description
We are all familiar with the usual notion of distance between two rational numbers: \(d(x,y) = |x-y|\). The real numbers may then be constructed by completing with respect to this metric, or `filling in the gaps'. The \(p\)-adic numbers arise when a very different metric is used, depending on the choice of a prime \(p\). We will define this in the project but, loosely, we consider rational numbers \(p\)-adically close when their difference is divisible by a large power of \(p\). We can then construct the field of \(p\)-adic numbers, \(\mathbb{Q}_p\), as the completion of \(\mathbb{Q}\) with respect to this metric.
This has strange consequences; for example, \[1 + p + p^2 + \ldots = \frac{1}{1-p}\] is a valid convergent series in the \(p\)-adic numbers! In fact, in some ways the \(p\)-adic numbers are much nicer to work with: all series whose terms tend to zero are convergent! Had this been true for \(\mathbb{R}\), Analysis I would have been much easier.
The \(p\)-adic numbers are not just an interesting construction: they are fundamental in number theory. One reason is that they can rule out the existence of rational solutions to equations: if an equation such as \(aX^2 + bY^2 = cZ^2\) has no (nonzero) solutions \(X, Y, Z \in \mathbb{Q}_p\), then it can't have any rational solutions either.
In this project we will explore the construction and basic theory of \(p\)-adic numbers, learn how to compute with them, and see how they give new ways to think about polynomial equations and number-theoretic problems.
Group project
The group project will revolve around learning the basic theory of \(p\)-adic numbers from Gouvea's p-adic Numbers: An Introduction ([G]). We will cover most of the first five chapters. By the end of the group project we will have learned:
- The \(p\)-adic absolute value and the construction of \(\mathbb{Q}_p\).
- The \(p\)-adic integers \(\mathbb{Z}_p\) and \(p\)-adic expansions.
- Hensel's lemma and applications.
- The local-global principle and Hasse-Minkowski theorem (statement).
- Convergence of \(p\)-adic power series and examples: \(\exp, \log\) and the binomial series.
Mode of Operation and Evidence of Learning for the group project
The project will revolve around learning through reading, examples, and problem solving, with focus on mathematical rigour and the development of conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and applications of the material, and clearly communicating it in both written and oral formats.
Individual project
The \(p\)-adic numbers are so fundamental in number theory there are near-infinite possibilities. Some ideas below:
- Quadratic forms and the proof of the Hasse-Minkowski theorem (see [S1]).
- The \(p\)-adic complex numbers, \(\mathbb{C}_p\) (see [G] chapters 6 and 7, [K] Chapter III).
- \(p\)-adic special functions: e.g. the \(\Gamma\)-function, Artin-Hasse exponential. (see [K Chapter IV]).
- Further \(p\)-adic analysis; for example, Newton polygons, Mahler's theorem, the Weierstrass preparation theorem (see [G] Chapter 7, [K] Chapter IV).
- Extensions of \(\mathbb{Q}_p\) and ramification, and Galois groups. (see [G] chapter 6 and then [S2] or [C].) This strand would benefit from Galois Theory, Group and Geometry III and/or Number Theory III.
- The \(p\)-adic Riemann \(\zeta\)-function. (see [K] Chapter II)
Mode of Operation and Evidence of Learning for the individual project
The project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.
Prerequisites and Co-requisites
Prerequisites : Algebra II and Complex Analysis II.
Co-requisites : None.
Galois Theory, Groups and Geometry III and Number Theory III (Algebraic Number Theory) would be helpful for some individual project directions
Feel free to contact me with any questions at jack.g.shotton@durham.ac.uk.
Resources
- [G] F. Q. Gouvea: p-adic Numbers: An Introduction. The main text for the group project.
- [K] N. Koblitz: p-adic Numbers, p-adic Analysis, and Zeta-Functions.
- [R] A. M. Robert: A Course in p-adic Analysis. The most comprehensive reference, contains material on most of the listed directions.
- [S1] J.-P. Serre: A Course in Arithmetic.
- [S2] J.-P. Serre: Local Fields.
- [C] J. W. S. Cassels: Local Fields.