4
votes
0answers
214 views

Advice for number theory library

Hi I just got a faculty position and it comes with a generous start up funds for "office supplies", which I must use or lose. What does a pure mathematician need? I have good computers already. I ...
4
votes
1answer
312 views

Basics on anabelian geometry and Grothendieck's section conjecture

Even I can find similar questions and some answers on that questions, most of them are not quite unsatisfactory to me. Maybe this is a very stupid question, but there is no other place that I can ask ...
4
votes
1answer
372 views

Examples of “nice” properties of algebraic extensions of $\mathbb{Q}$

I am writing a short survey of some "nice'' properties of algebraic extensions of $\mathbb{Q}$. Let's say a property (P) is nice if every finite extension of $\mathbb{Q}$ satisfies (P), and if $K ...
13
votes
6answers
994 views

Text for Algebraic Number Theory

I have the privilege of teaching an algebraic number theory course next fall, a rare treat for an algebraic topologist, and have been pondering the choice of text. The students will know some ...
2
votes
2answers
537 views

L-functions and algebraic geometry

Robert Langlands commented in a letter to Deligne that perhaps some of the deepest problems of algebraic geometry lie in L-functions. I want to understand the general philosophy and the connection ...
4
votes
1answer
391 views

a conjecture in sum-free sets

Let $ A $ be a set of non-zero integers. Then $A$ contains a sum-free subset $B$ of size $ |B|> \frac{|A|}{3} $ (a result of Erdős). It is conjectured that RHS can be improved to $\frac{|A|}{3} ...
1
vote
0answers
143 views

English version of “Quasi-Hopf Algebras”

I was wondering where I can find a pdf of Drinfeld's paper "Quasi-Hopf Algebras," which formulated the Grothendieck-Teichmuller group. The Russian version is in Algebra i Analiz, 1:6 (1989), 114–148, ...
5
votes
1answer
418 views

Naive question on adelic groups

The ever-reliable Wikipedia says: ... an adelic algebraic group is a semitopological group defined by... No more details are given, and I was wondering if the multiplication only being ...
7
votes
1answer
343 views

Number theory underlying Euler's theory of music

I've recently been studying Euler's theories on music, and I came across Euler's concept of gradus suavitatis or 'degree of pleasure' of a rational number representing the ratio of two tones. (I found ...
8
votes
1answer
878 views

How many proofs of the Weil conjectures are there?

I hope this this is not seen as too much as jumping on the band-wagon, but here goes. Deligne's proof of the last of the Weil conjectures is well-known and just part of a huge body of work that has ...
6
votes
4answers
841 views

fourier analytic proofs

While searching through Mathoverflow, I found out a fourier analytic proof of the Isoperimetric Inequality.Also, by google search I found a fourier analytic proof of Quadratic Reciprocity theorem.I ...
22
votes
1answer
1k views

How should a number theorist learn a modest amount of algebraic geometry?

A little bit vague, but I hope useful for the entire community. I am, by training, an analytic number theorist. I have managed to learn some algebraic geometry, by reading parts of Silverman's ...
0
votes
2answers
204 views

Algebraic maximal extension and algebraic closure

Let $K$ be a valued field. We say that $K$ is algebraic maximal if any algebraic extension of $K$ has either a bigger value group or a bigger residue field. Under which condition is an algebraic ...
3
votes
1answer
444 views

Request: Kato's article “Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions.” Part II

Hello, The question (similar to MO.96531) is about the article by Professor Kazuya Kato in this book. In this article, Professor Kato indicates the contents of the second part. MathSciNet does not ...
7
votes
3answers
497 views

Intuition Behind a Decimal Representation with Catalan Numbers

From $0 = 0.5 - 0.5 = 0.5 - \sqrt{0.25}$, we can adjust the subtrahend slightly to obtain $$0.5 - \sqrt{0.249} = 0.001\ 001\ 002\ 005\ 014\ 042\ldots$$ where the decimal representation contains the ...
15
votes
4answers
3k views

number of zeroes in 100 factorial.

I was on math.stackexchange the other day and i found a question that said How many zeroes are there in 100!. I quickly factored it out and said that there where 24 zeroes. However thats only the ...
50
votes
35answers
8k views

Where is number theory used in the rest of mathematics?

Where is number theory used in the rest of mathematics? To put it another way: what interesting questions are there that don't appear to be about number theory, but need number theory in order to ...
34
votes
10answers
5k views

Elementary results with p-adic numbers

I'm giving a talk for the seminar of the PhD students of my math departement. I actually work on Berkovich spaces and arithmetic geometry but, of course, I cannot really talk about that to an audience ...
2
votes
1answer
498 views

Primes and Ackermann's function

If $A(m,n)$ is Ackermann's function, and $c$ is a fixed integer, are there any heuristics/conjectures/obvious things that can be said about primes of the form $A(m,n)+c$, $m \geq 4$,at all? EDIT: I ...
4
votes
2answers
453 views

Is there a reasonable definition of the height of a transcendental number

For an algebraic number $\alpha$ one can define its "height" in many ways. Informally, you could use its minimal polynomial over $\mathbf{Q}$ and consider the maximum of the heights of its ...
3
votes
1answer
661 views

motive of a modular form

What is the idea behind a "motive of a modular form"? (I know what a motive is and what a Weil cohomology is. I want to know how to get the motive [what is the idea of Scholl?], and why this is ...
11
votes
4answers
801 views

Applications of full integral weight modular forms in elementary number theory

Except for Eisenstein series having the divisor functions as their Fourier coefficients, is there any other full integral weight modular form (of some level, preferably full) having arithmetic ...
4
votes
1answer
433 views

Characterization of algebraic points on Shimura varieties?

Is there any (conjectural) characterization of $\overline{\bf{Q}}$-points on Shimura varieties? The question of course does not always make sense for ${\bf{Q}}$-points: a theorem of Shimura shows ...
14
votes
6answers
2k views

A non-technical account of the Birch—Swinnerton-Dyer Conjecture

I was wondering whether anyone knows of any good non-technical or even popular expositions of the Birch—Swinnerton-Dyer conjecture, for someone with minimal background in elliptic curves. I was ...
13
votes
9answers
2k views

New proofs to major theorems leading to new insights and results?

I am wondering, historically, when has a new proof of an old theorem been particularly fruitful. A few examples I have in mind (all number theoretic) are: First example is classical... which is ...
1
vote
1answer
687 views

Good Minkowski Theory and Commutative Algebra Books

I am not so familiar with the theory of measures which Andre Weil uses to develope the Class Field Theory. However, I am interested in learning algebraic number theory and I recently found that the ...
8
votes
1answer
2k views

Is there any book explaining in detail the book “Basic Number Theory” by Andre Weil as Dirichlet did to “Disquisitiones Arithmetica” by Gauss?

Is there any book explaining in detail the book "Basic Number Theory" by Andre Weil as Dirichlet did to "Disquisitiones Arithmeticae"? This is because I have read the two books mentioned above and I ...
6
votes
1answer
318 views

Would an oracle for integral points on elliptic curves be a factoring oracle?

Oracle finding all integral points on genus 0 curves is a factoring oracle (e.g. $xy=n$ and $x^2-y^2=n$ I asked Can the number of solutions $xy(x−y−1)=n$ for x,y,n∈Z be unbounded as n varies? and ...
5
votes
1answer
527 views

Why are absolute values more natural than discrete valuations?

It is true that considering the archimedean places as well is more general, but that still doesn't explain why it is more natural. If we consider both the definitions of an absolute value and that of ...
9
votes
5answers
2k views

Complex Analysis applications toward Number Theory

I'm an undergrad who is taking a Complex Analysis Course mainly for its applications in number theory. So I would like to ask some guidelines about which theorems/concepts should I focus on in order ...
2
votes
4answers
2k views

Best way to introduce the Chinese Remainder Theorem (to a high school student)

What do you think to be the most effective way to teach the Chinese remainder theorem to a smart high school student, which is supposed to only have a soft idea about how modular arithmetic works, and ...
49
votes
11answers
4k views

Why certain diophantine equations are interesting (and others are not) ?

It is quite clear why certain differential equations, among the jungle of possible diff equations that is possible to conceive, are studied: some come from physical problems, or from "spontaneous" ...
1
vote
2answers
866 views

Arithmetic geometry from a bird's-eye view

Is ist true that Arithmetic Geometry can roughly be separated into two areas: 1) Showing that motivic $L$-functions are automorphic. 2) Calculating special values of these $L$-functions.
12
votes
3answers
1k views

Are there any (interesting) consequences of the irrationality of π? [closed]

I am not sure how appropriate this question is for MO. If it is not, I apologize in advance but I could not resist asking it and if by any chance I get some interesting answers, it will for sure be ...
88
votes
12answers
12k views

Have any long-suspected irrational numbers turned out to be rational?

The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surprise demonstration ...
7
votes
3answers
2k views

How to find/guess a polynomial sequence?

My question is motivated by the recent question and more recent appearance of its author Bruce Westbury. Most of you know that the best way to find a sequence of integers is looking for it on The ...
6
votes
1answer
985 views

Who is the Youngster in the Automorphic Room?

Iwaniec and Friedlander wrote a short survey article for the notices of the AMS, entitled "What is the Parity Phenomenon?" http://www.ams.org/notices/200907/rtx090700817p.pdf At the end of the ...
9
votes
2answers
2k views

Separable and algebraic closures?

I have no intuition for field theory, so here goes. I know what the algebraic and separable closures of a field are, but I have no feeling of how different (or same!) they could be. So, what are the ...
20
votes
0answers
1k views

Name of amateur who gave a new proof of the Ramanujan-Nagell theorem?

In an article by George Johnson in the New York Times back in 1999, it says that an amateur mathematician from India once sent Ian Stewart a proof of the Ramanujan-Nagell theorem that the Diophantine ...
12
votes
1answer
537 views

Geometry for Anderson's motives?

Anderson's $t$-motives satisfy most of what is expected of a reasonable category of mixed motives, except of course that everything is in positive characteristic. For instance, it is a linear category ...
5
votes
3answers
1k views

Why are they called L-functions?

I was hoping to see this pop up on the recent big list question about etymology or terms and symbols. Since it has not, and I can't find an answer, I will ask: What is the reason for the $L$ in ...
6
votes
3answers
1k views

Terminology occuring in automorphic representation and relationship between them

When one tries to read about automorphic representation few terms come up more than others namely, 1.Cuspidal 2.Square Integrable 3.Absolutely Cuspidal 4.Super Cuspidal My understanding about ...
2
votes
1answer
226 views

Uniformly computable classes of graphs

[Follow-up to Can every finite graph be represented by one prescribed sequence of natural numbers?, reformulated thanks to a hint from Jacques Carette] Let $V(n,\nu)$ and $E(n,m,\mu)$ be ...
0
votes
1answer
621 views

Can every finite graph be represented by one prescribed sequence of natural numbers?

(This is a follow-up to my previous question Can every finite graph be represented by an arithmetic sequence of natural numbers?) Since it is obviously false that every finite graph can be ...
1
vote
8answers
888 views

What is “rich structure”, actually? [closed]

An ubiquitous claim in mathematics is that such-and-such mathematical entity has a rich structure or more structure than another one. Most oftenly the entity is a structure - a set explicitly equipped ...
6
votes
1answer
655 views

An ubiquitous pattern of questions

There is an ubiquitous pattern of questions concerning assumedly any kind of mathematical object or structure: groups, graphs, numbers, categories, and so on. It goes like this (informally): Can a ...
3
votes
3answers
1k views

Why is 2 so odd? [duplicate]

Possible Duplicate: Is there a high-concept explanation for why characteristic 2 is special? There are so many results on primes that either fail for $p=2$ or are not known to be true for ...
3
votes
1answer
1k views

moduli space and modularity

I recently realized some kind of analogy when considering modularity results (such as the modularity of elliptic curves over Q). The analogy comes from algebraic groups. Take one point (say, the ...
30
votes
5answers
4k views

Heuristically false conjectures

I was very surprised when I first encountered the Mertens conjecture. Define $$ M(n) = \sum_{k=1}^n \mu(k) $$ The Mertens conjecture was that $|M(n)| < \sqrt{n}$ for $n>1$, in contrast to the ...
29
votes
6answers
4k views

The current status of the Birch & Swinnerton-Dyer Conjecture

[Une traduction française suit la version anglaise.] The question is only about elliptic curves $E$ over $\mathbb{Q}$ and concerns only the aspect (order of vanishing of $L(E,s)$ at $s=1$)$\ =\ ...