Wednesdays, 7:30pm; Room 622, Mathematics
Topic: p-adic Numbers
Text: p-adic Numbers, p-adic Analysis, and Zeta-Functions by Niel Koblitz
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ums [at] math.columbia.edu
| Date | Speaker | Title | Abstract |
| June 8 | Linus Hamann | Introduction | Tonight we are having a meeting to discuss topics for the summer. Time permitting, I will review some topics in Complex Analysis and Riemann Surfaces. If you are at all interested in participating in the summer talks, please come to this meeting. We are meeting in room 628 as the lounge and room 507 will be undergoing construction this summer. |
| June 15 | Linus Hamann | p-adics Introduction | We will begin our summer seminar on p-adic numbers and their generalization in local fields. I will be introducing various viewpoints of looking at p-adic numbers, analytic, algebraic, and topological, outlining the core properties. We will for a short time be using "p-adic analysis" by Niel Koblitz, but I will give a detailed outline of the course material today. |
| June 22 | Kat Christianson | Hensel's Lemma |
We will focus on the question: to what extent does the behavior of the integers determine the behavior of the p-adic integers? There will be a brief recap what we learned last week. Then, we'll discuss Hensel's lemma, which allows one to generate p-adic roots of polynomials by solving modular congruences. We'll also start talking about p-adic interpolation, which allows one to extend function on the integers to functions on the p-adic integers. |
| June 29 | Willie Dong | p-adic Distributions | In this talk, I will introduce what a p-adic distribution is. I will prove a criteria for which a map can be extended to a p-adic distribution, and then go over some examples, i.e. the Haar and Dirac distributions. Time permitting, I'll go over a lengthier example of the Bernoulli distribution. |
| July 6 | No talk | ||
| July 13 | Jun-ho Won | p-adic Interpolation of the Zeta function | We will discuss the p-adic interpolation of the Riemann zeta function. We will begin by looking at the familiar example of the function f(x)=a^x, where a is a fixed positive real number and x is a real variable. In order to define f we may first define it on rational x, then "extend by continuity" to real x. We will discuss how a similar process can be carried out in the p-adic case. Next, we will expand on Willie's talk on p-adic distributions, and develop some essential measure and integration theory for p-adic numbers. If time remains we will define the p-adic zeta function and its "branches," and see how p-adic interpolation comes into play. |
| July 20 | Theo Coyne | p-adic Interpolation of the Zeta function | We will discuss p-adic interpolation. Given a function on the integers, when can it be extended to a continuous function defined on the p-adic integers? We will show that f(m)=a^m for fixed natural a can be extended this way to a domain of Z_p, with the goal of later doing the same for the Riemann zeta function. Along the way, we will find the need to further discuss p-adic integration, and in particular the Bernoulli measures introduced by Junho last week. |
| July 27 | Jun-Ho Won | Theory of Local Fields | We revisit our study of p-adic numbers, and develop the general theory for valuations on fields. We will go over how Ostrowski's theorem, Newton's Lemma, and Hensel's Lemma generalizes in this setting. After developing some dictionary in algebra including dedekind domains, fractional ideals, and norms of ideals, we will see how to extend nonarchimedian valuations to complete algebraically closed valued fields, and introduce the notion of local fields. |
| August 3 | Theo Coyne | p-adic Interpolation and Absolute Values | First, we will quickly revisit the p-adic interpolation of the zeta function discussed two weeks ago and use it to prove some important congruences involving the Bernoulli numbers (Kummer, Clausen and von Staudt congruences). Then we will continue our discussion of absolute values on algebraic number fields. In particular, we will see how our theory of valuations and absolute values can be used to generalize the p-adics. |