Questions tagged [nt.number-theory]

Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

26
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5answers
2k views

Are there two non-isomorphic number fields with the same degree, class number and discriminant?

If so, do people expect certain invariants (regulator, # of complex embeddings, etc) to fully 'discriminate' between number fields?
13
votes
5answers
1k views

Very strong multiplicity one for Hecke eigenforms

In Invent. math. 116, 645-649 (1994) Dinakar Ramakrishnan proves a theorem which I understand to imply that the following statement (in light of the fact that elliptic curves over $\mathbb{Q}$ are ...
31
votes
1answer
3k views

Is the set of primes “translation-finite”?

The definition in the title probably needs explaining. I should say that the question itself was an idea I had for someone else's undergraduate research project, but we decided early on it would be ...
4
votes
2answers
563 views

“half arithmetic progressions” in dense sets

Fix a positive real number d>0. Szemeredi's theorem implies that for every integer k, there exists an integer N(k,d) such that if A is a subset of the interval [1,N] with density greater than d >0, ...
20
votes
1answer
15k views

How hard is it to compute the Euler totient function?

Are there any efficient algorithms for computing the Euler totient function? (It's easy if you can factor, but factoring is hard.) Is it the case that computing this is as hard as factoring? EDIT: ...
44
votes
6answers
5k views

Intuition for the last step in Serre's proof of the three-squares theorem

Serre's A Course in Arithmetic gives essentially the following proof of the three-squares theorem, which says that an integer $a$ is the sum of three squares if and only if it is not of the form $4^m (...
5
votes
1answer
460 views

Finiteness of Obstruction to a Local-Global Principle

Say that a projective variety V over Q satisfies the local-global principle up to finite obstruction (#) if there are only finitely many isomorphism classes of projective varieties over Q that are not ...
48
votes
3answers
3k views

What are the higher homotopy groups of Spec Z ?

The homotopy groups of the étale topos of a scheme were defined by Artin and Mazur. Are these known for Spec Z? Certainly π1 is trivial because Spec Z has no unramified étale covers,...
12
votes
2answers
591 views

Number theoretic sequences and Hecke eigenvalues

What are some number theoretic sequences that you know of that occur as (or are closely related to) the sequence of Fourier coefficients of some sort of automorphic function/form or the sequence of ...
4
votes
3answers
1k views

Modular forms reference

If f is a weight 2 newform on $\Gamma_1(N)$ then there exists an abelian variety Af whose endomorphism algebra is isomorphic to the field generated by the coefficients of f. I've seen this proven in ...
11
votes
4answers
623 views

Behaviour of Zeta-function under Finite Morphism

Let X ---> Y be a finite surjective morphism of smooth, projective, connected varieties over a finite field F_q. Can one describe the zeta function Z(X, t) in terms of the zeta-function Z(Y,t) of ...
12
votes
3answers
1k views

How should I approximate real numbers by algebraic ones?

Given a high precision real number, how should I go about guessing an algebraic integer that it's close to? Of course, this is extremely poorly defined -- every real number is close to a rational ...
6
votes
3answers
404 views

Prime numbers and strings of symbols

Suppose you have N symbols (e.g. "1","2",...,"N" or "a","b",...,"$") and a string of these symbols (say, the first trillion digits of pi). Then does there exist a prime number whose N-ary ...
19
votes
1answer
3k views

Equivalent forms of the Grand Riemann Hypothesis

I have long been curious about equivalent forms of the Riemann hypothesis for automorphic L-functions. In the case of the ordinary Riemann hypothesis, one gets a very good error term for the prime ...
86
votes
10answers
12k views

“Understanding” $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$

I have heard people say that a major goal of number theory is to understand the absolute Galois group of the rational numbers $G = \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. What do people mean when ...
20
votes
5answers
3k views

Proof of no rational point on Selmer's Curve $3x^3+4y^3+5z^3=0$

The projective curve $3x^3+4y^3+5z^3=0$ is often cited as an example (given by Selmer) of a failure of the Hasse Principle: the equation has solutions in any completion of the rationals $\mathbb Q$, ...
6
votes
2answers
395 views

Strong Bertrand postulate

Is it known that for every epsilon there is N_0 such that all intervals of the form [N, (1+\epsilon)*N], where N > N_0, contain ...
3
votes
7answers
2k views

Bertrand's postulate [closed]

I believe there was an old conjecture that there's always a prime number between N and 2N. What's the history and how is this ...
11
votes
3answers
2k views

Is there a known bound in prime gaps?

Is there known to be an $x$ such that for all positive integers $N$ there exists some $n>N$ such that $p_{n+1}-p_n \leq x$, where $p_n$ is the $n$th prime? Or, in other words, is it known that ...
1
vote
1answer
903 views

Neukirch's class field axiom and cohomology of units for unramified extension

This question may be too detailed but perhaps somebody knows the answer: Neukirch proofs in his algebraic number theory book in chapter IV, proposition 6.2, that his class field axiom implies that the ...
13
votes
3answers
3k views

What is Eisenstein series?

There are several related questions here, the latter being especially interesting. We know the classical Eisenstein series. What are the Eisenstein series on a group G and why they are interesting? ...
11
votes
4answers
859 views

Abel's equation for the dilog

Abel's identity for the dilogarithm (see the wikipedia page about polylogarithms) plays a role in web geometry as it is one of the abelian relations of the first example of exceptional web (Bol's 5-...
14
votes
6answers
1k views

Solving polynomial equations when you know in which number field the solutions live

Suppose I have a bunch of polynomial equations with coefficients in a number field, and suppose further that I'm guaranteed a priori that they have a solution in that number field. Can I leverage ...
3
votes
1answer
385 views

Any work on the Adams-Watters triangle?

Does anyone know whether any arithmetical or asymptotic results have been obtained about the Adams-Watters triangle?
16
votes
3answers
2k views

2-adic Coefficients of Modular Hecke Eigenforms

Suppose that $N$ is prime, and consider the normalized cuspidal Hecke eigenforms of weight 2 and level $\Gamma_0(N)$. For such an eigenform $f$, the coefficients generate (an order in) the ring of ...
10
votes
2answers
875 views

A comprehensive overview of finite fields

I've read numerous introductions to finite fields, but I feel like my intuition about them is fairly lacking. Considering that finite fields are the the most "inert" objects in algebraic geometry, I ...
11
votes
1answer
685 views

Covering the primes by 3-term APs ?

Hello, the Green-Tao theorem says infinitely many k-term Arithmetic Progressions exist for any integer k. My question is: can we actually partition the primes into 3-term APs only (or is there a ...
5
votes
3answers
562 views

Solving “a, b, a+b have given divisors” problem

I've read an interesting article, math.NT/0409456 where you're just trying to solve a simple problem: For a given (finite) set of primes S find all solutions to an equation ...
10
votes
3answers
646 views

A “round” lattice with low kissing number?

Historically, the lattices with high density were studied intensively, e.g. E_8 lattice or Leech Lattice. However, there are situations that lattices with low kissing number are required. Specifically,...
6
votes
1answer
1k views

Learning about Galois representations

My goal was to learn about l-adic representations on some example — I'm a newbie in these topics. Thus take pt = Spec F_q, ...
7
votes
2answers
2k views

What is the Beilinson regulator?

Trying to understand answer to this question. What is the (Beilinson) higher regulator of a number field?
5
votes
2answers
328 views

Density of a subset of the reals

The rationals are clearly dense in the real number system, i.e. for every pair a < b of real numbers there exists a rational number p/q s.t. a < p/q < b. I conjecture the same to be true with ...
35
votes
10answers
5k views

Why are functional equations important?

People who talk about things like modular forms and zeta functions put a lot of emphasis on the existence and form of functional equations, but I've never seen them used as anything other than a ...
35
votes
5answers
7k views

Definition and meaning of the conductor of an elliptic curve

I never really understood the definition of the conductor of an elliptic curve. What I understand is that for an elliptic curve E over ℚ, End(E) is going to be (isomorphic to) ℤ or an ...
7
votes
3answers
1k views

p-power roots of unity in local fields

Let $K$ be a number field and suppose $K$ contains no $p$-power roots of unity. Let $\mathcal{P}$ be a prime of $K$ above the rational prime $p$. Can someone prove or disprove the assertion that the ...
19
votes
3answers
2k views

Regulators of Number fields and Elliptic Curves

There is supposed to be a strong analogy between the arithmetic of number fields and the arithmetic of elliptic curves. One facet of this analogy is given by the class number formula for the leading ...
26
votes
4answers
2k views

Why do zeta functions contain so much information?

Is there some intuitive explanation why Dedekind zeta functions contain so much information about their number field? For example the residue at the pole $s=1$ relates several invariants of the ...
1
vote
2answers
491 views

Is an nth root of unity a square?

Suppose w^(2n)=1 (w is a complex number). For which n (if any) \sqrt(w) \in Q(w) ?
7
votes
2answers
719 views

How can I prove that a sequence of squares of graph norms is never cyclotomic?

The norm of a graph is the largest eigenvalue of the adjacency matrix. I'll write ||G|| for the norm of G. Now, fix some graph <...
7
votes
1answer
448 views

Lifting bases for (Z/pZ)^n to Z^n

The following question came up in my research. I suspect that it has a slick answer, but I can't seem to find it. Fix an integer n>=2 and a prime p. Define X(n) to be the set of primitive vectors ...
24
votes
3answers
8k views

Is 8 the largest cube in fibonacci sequence?

Can you prove that 8 is the largest cube in fibonacci sequence?
0
votes
7answers
1k views

What is the smallest integer whose primality status is not known?

Closely related: what is the smallest known composite which has not been factored? If these numbers cannot be specified, knowing their approximate size would be interesting. E.g. can current ...
2
votes
7answers
5k views

What's so special about transcendental numbers? [closed]

It's hard to prove a number is transcendental (non-algebraic) yet there are some wonderful examples amongst them like π,e and Liouville's number. What's so special about them? Are most numbers ...
90
votes
22answers
27k views

What's the “best” proof of quadratic reciprocity?

For my purposes, you may want to interpret "best" as "clearest and easiest to understand for undergrads in a first number theory course," but don't feel too constrained.
25
votes
5answers
4k views

Global fields: What exactly is the analogy between number fields and function fields?

Note: This comes up as a byproduct of Qiaochu's question "What are examples of good toy models in mathematics?" There seems to be a general philosophy that problems over function fields are easier to ...
7
votes
2answers
4k views

What does “supersingular” mean?

Are supersingular primes and supersingular elliptic curves related? (this was essentially a subquestion in my earlier question, but still looks sufficiently different to me to deserve a separate post)...
7
votes
6answers
2k views

Primes are pseudorandom?

I've been reading the wonderful slides by Terry Tao and thought about this question. Primes appear to be quite random, and the formal statement should be that there are some characteristics of primes ...
26
votes
5answers
4k views

Where stands functoriality in 2009?

Robert Langlands is famous in number theory for making famous and deep conjectures about very abstract things called automorphic forms, somewhere in the 60s. There's a very interesting article by ...
7
votes
1answer
562 views

Ways to characterize supersingular primes?

I've read the definition, and it basically says p is a supersingular prime iff the fundamental domain of a group generated by \Gamma(p) and a matrix ((0, 1), (-p, 0)) is rational. And there's a ...
11
votes
4answers
1k views

Mystery of the Monstrous Moonshine

There's a very famous group, the largest sporadic simple finite group, sometimes called a monster whose size is quoted below. What's the explanation that the primes appearing in it, ...