Higher reciprocity laws

Today I wanted to share with you a write-up of a talk I have given to other PROMYS Europe 2018 counsellors about two weeks ago. The topic of the talk was higher reciprocity laws, which is a general term used to describe a great variety of results which, in one way or another, can be seen as generalizations of the classical law of quadratic reciprocity. I have tried to give a somewhat motivated account of how each and every one of those generalizes the previous ones and, ultimately, how all of them are just special cases of one theorem (or of a single, even more general conjecture!)

This is a topic I have been fascinated by for years now and I have been seeking an opportunity/excuse to read more into it. Around a year ago I have given a 30-minute talk on this topic during a student conference I was attending, but due to time constraints I couldn’t get far into the topic nor did I get deep into it myself. My talk during PROMYS Europe was different – I would like to thank all counsellors for having patience to sit through my 2.5 hour long (!) talk, despite my promises I will keep it reasonably short. I have greatly enjoyed reading up on the topic and then preparing to present it to others.

I must admit I am quite proud of the write-up for this talk, which is why I would like to share it with everyone. Except for some minor fixes, this is a version which ended up in our yearbook, but I hope some people find some benefit in this, perhaps even find this topic as interesting as I do!

Direct download link to the PDF

Long well-ordered chains of functions under eventual domination

\(\newcommand{\N}{\mathbb N}\)Consider the set \(\N^\N\) of all functions from \(\N\) to itself. We impose a partial order of eventual domination on this set: we say that \(f\) eventually dominates \(g\) if for large enough \(n\) we have \(f(n)< g(n)\). In this poset, we consider chains, in particular the well-ordered ones. One question we can ask is, how long can such chains be? Here we establish a lower bound \(\omega_2\) for the supremum of possible lengths.

All background necessary for this post is some basic knowledge about ordinals, in particular cofinalities.

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PNT for polynomials using the zeta function

In this blog post of mine I talk about the “polynomial zeta function” \(\zeta_q(s)\) and prove an analogue of Riemann hypothesis for it (indeed, this zeta has no zeroes at all). At the end I mention that this zeta function is, in fact, the “correct” zeta function for this purpose. In this blog post I am going to derive a formula analogous to the von Mangoldt’s explicit formula. This is a little project for me to see whether I’m able to work out all the details.

The proof is based on the exposition in Davenport’s Multiplicative Number Theory and requires some understanding of complex analysis.

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Finiteness of subsets of arbitrary finite sets

The goal of this blog post is to provide a novel proof of the following result:

Proposition: Let \(A\) be a finite set and let \(B\) be a subset of \(A\). Then \(B\) is finite.

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Proof sketch of Thue’s theorem

In this post a proof of the following theorem is going to be sketched, following the treatment in Borevich and Shafarevich’s Number Theory. This sketch is by no means meant to be highly detailed and I am writing it mostly for my own purposes, so I avoid proving some things, even if they aren’t that straightforward.

Thue’s theorem: Suppose \(f(x,y)=a_0x^n+a_1x^{n-1}y+\dots+a_ny^n\) is a binary form which has degree \(n\geq 3\), is irreducible (i.e. \(f(x,1)\) is an irreducible polynomial in \(x\)) and \(f(x,1)\) has at least one nonreal root in \(\mathbb C\). Then for any nonzero integer \(c\) the equation \(f(x,y)=c\) has only finitely many integral solutions.

Proof: Suppose otherwise…

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Abstract divisors

The goal of this blog post is to provide an overview of the general theory of divisors, as described in a book Number Theory by Borevich and Shafarevich. The point of this post is for it to be somewhat expository, so it will avoid the longer proofs, sometimes just sketching the ideas.

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Complexity theory is a mathematical study of algorithmic problems with regard to how cost-effective solutions they have. Most prominent are the decision problems, which are yes-no questions for which answer depends on some further input, for example, “Is \( n\) a prime number?” is a decision problem depending on the input \( n\). I assume that a reader of this post has some familiarity with the basic concepts of this theory, in particular, what PSPACE complexity class is and what does it mean that a problem is PSPACE-complete. A well-known theorem due to Shamir states that the class PSPACE and another class, IP (which I explain below, since I wouldn’t count it as a “basic” concept), are equal. The goal of this post is to prove this theorem. It is not based on a single source, but there is a very detailed exposition due to B.J. Mares, which I shall leave as a reference.

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Starting a new project – Solutions to exercises in Marcus’ book (+some info)

The book “Number Fields” by D. Marcus is a very well-known introductory book on algebraic number theory. Its most memorable aspect is, without a doubt, the great number of exercises it contains. They vary from short(ish) computational exercises, through various technical results used later in the book, to series of exercises aimed to establish (sometimes very deep) results in number theory. They are structured in a way which allows even an unexperienced reader be able to solve most, if not all, exercises even on their first reading, thanks to (often very elaborate) hints provided.

However, even then a reader might want to refer some external source in order to see how the exercise can be solved, because otherwise it might be difficult to proceed any further (I myself would appreciate such a source at times). And, as they say, if you want something done right, do that yourself.

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Higher ramification groups

This post is based on Marcus’s Number Fields. More specifically, it is based on a series of exercises following chapter 4.

Recall the definition of the intertia group of a prime \(\frak P\) in \(\mathcal O_L\) lying over a prime \(\frak p\) in \(\mathcal O_K\) (\(L/K\) is a Galois extension of number fields) – it’s the set of all \(\sigma\in G=\mathrm{Gal}(L/K)\) such that, for all \(\alpha\in L\), we have \(\sigma(\alpha)\equiv\alpha\pmod{\frak P}\). We now generalize this group.

Definition: In setting as above, we define the \(n\)th ramification group \(E_n\) to be the set of all \(\sigma\in G\) such that \(\sigma(\alpha)\equiv\alpha\pmod{\frak P^{n+1}}\). The groups \(E_n,n>1\) are called the higher ramification groups.

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Discriminant and different

Discriminant of a number field is arguably its most important numerical invariant. Quite closely connected to it is the different ideal. Here we discuss the most basic properties of these two concepts. The discussion mostly follows this expository paper by K. Conrad, but also takes from a series of exercises in chapter 3 of Number Fields.

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