I am an undergraduate. I am interested in algebraic number theory.

My Background:

(1) I have read the first 5 chapters of the book Number Fields by Daniel A Marcus.

(2) I have read the first and third chapter of Koblitz's book on $p$-adic numbers.

(3) I have read the first two chapters of Janusz's Algebraic Number Fields and also some parts of the fifth chapter on class field theory.

Very often what happens with me is, I learn something properly only when I have to use it frequently while studying something more advanced. (For example, I got a better understanding of metric spaces when I took a course on functional analysis.) Keeping this in mind, I think it will be a good idea to read some papers in algebraic number theory. This will help me to consolidate whatever I have learnt so far. Also, it will help me to learn some new material.

Looking at my background, can you point me to some research papers which are accessible to undergraduates ?

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    $\begingroup$ It's worth noting that (perhaps) more so than in other areas of mathematics, the jump from undergraduate to research level algebraic number theory is particularly large. In particular, modern algebraic number theory often has as much to do with representation theory, complex analysis, algebraic topology and algebraic geometry as it does with "undergraduate" number theory. As such, most modern papers will be completely inaccessible to you. ... $\endgroup$ – Mathmo123 Mar 4 '17 at 20:33
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    $\begingroup$ ... There are, however, many papers from the 60s-70s which have been hugely influentual in shaping modern number theory, and could be slightly more manageable. The ones I'm familiar with would require you to learn a good deal about modular forms and Galois representations. $\endgroup$ – Mathmo123 Mar 4 '17 at 20:33

I think that the most accessible research-level number theory for an undergraduate with your background is where number theory intersects combinatorics. Two topics that I know professors who use number theory in their combinatorics work work on the arithmetic properties of finite fields and questions about Latin squares. There's been some recent breakthroughs in studying the combinatorial structure of the generalized game of Set too. Although these combinatorial problems don't look particularly algebraic, there is a lot of algebraic machinery (particularly Galois theory and group theory) going on. This isn't "algebraic number theory" in the mainstream sense of the field, but it's an algebraic approach to number theory that is going to be accessible.


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