How does "modern" number theory contribute to further understanding of $\mathbb{N}$?

Question

I hope this question is appropriate for MO. It comes from a genuine desire to understand the big picture and ground my own studies "morally".

I'm a graduate student with interest in number theory. I feel like I'm in danger of losing the big picture as I venture a bit deeper and reflect on where I am at. My fundamental is this: I care about the natural numbers - and thus naturally care about the Riemann Zeta function. Number theorists have embarked on various adventures in studying generalized integers (rings of integers of Q-extensions), and their associated zeta functions, and beyond (e.g. Langlands program). Some mathematicians seem to be interested in these generalized integers and zeta functions for their own sake. I am not.

Given my passion for $\mathbb{N}$ and zeta, why should I study these other objects? I understand that philosophically to understand an object it's good to understand its context, and its similarities and differences to its brothers and cousins. This principle makes a lot of sense.

But specifically, what new understandings of $\mathbb{N}$ and zeta have we gained thus far by studying these more general systems? Are there clearly articulated reasons why we can hope to bring back more "treasure" from these more general searches that may shed light on $\mathbb{N}$ in particular? I worry sometimes that number theory is becoming divorced from its original "ground", though I believe (and hope) this feeling derives mainly from ignorance.

EDIT: My question was probably not written very well. I am aware of some of the benefits of studying solutions of polynomials in ring extensions (e.g. solving cases of FLT). My concern is with the broad scope of number theory research today, particularly in the land of generalized zeta-functions and Langlands program. I am uncomfortable (in my ignorance, I admit!) with the apparent lack of a clear connection to the "natural" concerns of number theorists prior to the mid 20th century.

I hope that my question is taken in the spirit of a naive apprentice asking masters for motivation, and a layout of the land of modern research as it connects to concerns that used to be universal.

Answer 1

You can find the answer in the history of the subject. For brevity let us consider the following two genuinely number theoretic questions that were of great interest already to Gauss (and Fermat, Euler, Lagrange, Legendre, Jacobi, Dirichlet, Eisenstein):

(1) For which primes is a given integer a quadratic residue?

(2) Which numbers can be written as a sum of three squares and in how many ways?

These questions were pretty well understood by Gauss and his contemporaries, but they admit equally natural generalizations which turned out to be much much harder (and they are being studied until the present day):

(3) Given an irreducible polynomial over the integers, over which primes decomposes the polynomial in a particular way (e.g. splits into linear factors)?

(4) Which numbers are represented by a positive integral ternary quadratic form and in how many ways? How do the representations distribute on the corresponding ellipsoid?

The best answers to these questions rely heavily on the theory of automorphic forms and their $L$-functions. Question (3) leads naturally to Artin $L$-functions, the question if there is an alternate way to describe their coefficients, for which the best answers are produced by the Langlands program. Question (4) leads naturally to the theory of genera and spinor genera, theta series and cusp forms, Siegel's maass formula and Eisenstein series, Siegel's bound for the class number, the Shimura lift and Waldspurger's formula, bounds for automorphic $L$-functions, all which necessitate a global automorphic thinking. Question (4) also leads to more general questions such as representing a quadratic form by another one, or their counterparts over number fields, which brings to surface a wider class of automorphic forms (e.g. Siegel and Hilbert modular forms).

Automorphic forms and their $L$-functions is not a digression from the natural numbers but the most fitted tool to formulate and study their properties. I often wonder which comes first: the natural numbers, or automorphic forms?

I understand I have not answered your question completely. My aim was to indicate two hot spots (Galois representations, and quadratic forms) where automorphic forms have had a tremendous influence, including contemporary research.

Answer 2

Where to begin? As a tiny example, suppose that you are interested in the solutions in rational numbers to an equation $Y^2=X^3+AX+B$, i.e., the rational points on an elliptic curve $E$. One naturally looks at the algebraic solutions $E(\overline{\mathbb{Q}})$, since one can do geometry over the algebraically closed field $\overline{\mathbb{Q}}$, and then picks out the rational points $E(\mathbb{Q})$ by studying the action of the Galois group $G(\overline{\mathbb{Q}}/\mathbb{Q})$ and picking out the points that are invariant for the group action. Hopefully you're also interested in rational numbers, since they're just ratios of integers, but if you insist on problems with integers, then one can look at the integer solutions $E(\mathbb{Z})$, which forms a finite set (Siegel's theorem, made effective by Baker). But even there, one way to study $E(\mathbb{Z})$ is by analyzing it as a subset of $E(\mathbb{Q})$, so you're back to rational solutions.

So that's a bit long-winded, but it illustrates a general principle. If a problem involving a set is hard, for example a problem involving integers, it may be easier to solve a problem involving a larger set and then pick out the subset that you're really interested in.

Having said all of this, it is also true that there are mathematicians who find studying "new" objects to be intrinsically interesting, regardless of the original applications that they were devised for. As an example, there are lots of people who study automorphic forms for their own sake. If there are applications to classical sorts of problems involving integers, that's great, but it's not their motivation or primary interest. Luckily, there's room for everyone in the big tent that makes up the mathematical research community.

Answer 3

If you are not already aware of it, I'd recommend reading Representation theory: Its rise and its role in number theory by Langlands himself. He motivates the Langlands program in terms of one of the most concrete and classical number-theoretic questions, namely finding integer solutions of polynomial equations modulo a prime.

Let $l$ be a prime, congruent to 1 mod 4 for simplicity, and let's say you are interested in knowing for which primes $p$ the equation $$x^2 - l \equiv 0 \pmod p$$ has an integer solution, or to put it another way, you want to know for which primes $p$ the polynomial $x^2-l$ factors modulo $p$. The answer, of course, is given by quadratic reciprocity, and depends only on the value of $p \bmod l$.

This is such a nice fact that it cries for generalization. We'd like the factorization behavior of the polynomial modulo $p$ to depend solely on "$p$ modulo something." For polynomials with abelian Galois group, class field theory yields a very beautiful and satisfying answer, telling us that "$p$ modulo something" should be interpreted in terms of a certain class group. (In the quadratic reciprocity case, the relevant class group is isomorphic to $\mathbb{Z}/l\mathbb{Z}$.) Pursuing the answer in general leads more-or-less directly to the Langlands program. As Langlands writes:

Another example, the reasons for whose choice will be explained later, is $$x^5 + 10x^3 − 10x^2 + 35x − 18.$$ It is irreducible modulo $p$ for $p = 7, 13, 19, 29, 43, 47, 59, \ldots$ and factors into linear factors modulo $p$ for $p = 2063, 2213, 2953, 3631, \ldots\,$. These lists can be continued indefinitely, but it is doubtful that even the most perspicacious and experienced mathematician would detect any regularity. It is none the less there.

For more of the story, read the paper.

Answer 4

Let me illustrate the point with a specific and well-known example: the congruent number problem, perhaps the oldest open problem in mathematics. An integers $n>0$ is said to be a congruent number if it can be written as the area of right-angled triangle with rational sides. Which integers are congruent in this sense ?

Using his method of infinite descent, Fermat proved that $1$ is not congruent, thereby settling a conjecture of Fibonacci.

The best current answer to the problem, which depends heavily on the mathematics of the 20th century, is Tunnell's conjectural characterisation of congruent numbers. You can find many expository articles on the web (for example by Karl Rubin, Pierre Colmez, Franz Lemmermeyer's translation of an article by Guy Henniart, and John Coates). My own attempt can be found on the arXiv. Koblitz has written a whole book about it.

More 20th century mathematics (and perhaps even some 21st century mathematics) will be needed for the proof that this conjectural characterisation is indeed correct.

Very substantial progress has been made recently by Ye Tian. See for example his preprint or the video of his Moscow talk.

Try doing this with just $\mathbf{N}$ (and $\zeta$) !

Addendum. You might also be interested in Lang's article Mordell's review, Siegel's letter to Mordell, Diophantine Geometry, and 20th Century Mathematics in the Gazette (1995) 63.

Addendum 2. To get a broad overview of Number Theory today, you may want to consult Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories by Yu. I. Manin and Alexei A. Panchishkin.

Addendum 3 (2013/01/19). John Coates has written a short account of Tian's result in the PNAS 109 (52), 21182–21183.

Answer 5

As yet-another comment/question, in addition to the several excellent responses: the question in itself is entirely reasonable, indeed, and is not obviously answered by contemporary professional literature, nor by our current "standard texts". One should rightly suspect/worry that there's been some sort of swindle perpetrated...

And, indeed, the monuments and icons of past times do seem to assume a life and mythology of their own, which all too often has dropped explicit or pointed references to previous history.

So, indeed, one should ask oneself these questions...

And, simultaneously, it would seem fair to ask what all this chatter is about while we've not only not proven RH, but not even proven a zero-free strip!?! Cart before the horse? :)

Probably the genuine explanation in human terms is that the immediate description of things misled us as to what was a cart, and what was a horse, and... maybe, what was "before".

So, yes, while we seem to have made frustratingly little progress on "old issues", and we make only tenuous analogies in further generalizations... what is the alternative?

That is, we do what we can. And we do have some pressure to "hype" it to funding agencies, to Deans, and to each other. Yes, this seems to claim remarkable progress... while we've not really made so much, by more sober standards. :) But it is not easy to construct a better model of reporting progress.

At some point, then, one observes that it'd be extremely unwise to decide to prove RH for a PhD thesis, or postdoc project, ... or maybe anything else... despite the extreme desirability of progress on exactly such a thing.

One true operational point is that people have to "make a living", which currently requires reporting incredible progress to department heads and deans. Thus, ... noise.

The notion of "deconstruction" of any human activity (that is, acknowledging that it is a human activity, rather than the human-independent, context-independent "absolute" that has a sort of pernicious charm ... for humans) greatly clarifies the seeming paradoxes of human activities, in many cases. :)

Answer 6

A short answer: the kind of "structure" recognised in the analytic number theory of the period 1900 to 1930, successful as that theory was, doesn't go far enough. You need at least functions of several complex variables, harmonic analysis that is non-commutative, algebraic geometry that isn't just cartesian coordinates, and a few more things. Debates on aesthetics, such as Felipe alludes to, have tended to get in the way of these insights.

Answer 7

Consider an abelian variety with integer coefficients with a group map. The points of order $p$ for $p$ a prime are acted on by the absolute Galois group of $\mathbb{Q}$, so we have a naturally associated modular representation of the absolute Galois group. If this representation satisfies the condition of the Langlands conjectures, that translates into information about the arithmetic of points.