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

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When are all signatures represented by units?

What conditions on a number field imply that every signature is represented by a unit? Are there conditions that imply that it is not the case that every signature is represented by a unit? For ...
Christine McMeekin's user avatar
6 votes
1 answer
321 views

"Sub-logarithmic" zero-free regions from Deuring-Heilbronn/Linnik's repulsion theorem

For each $n\in\mathbb{N}$, let: $\chi_n\pmod{q_n}$ a real non-principal Dirichlet character ($q_1 < q_2 < \cdots$), $\beta_n$ the largest real zero of $L(s,\chi_n)$, $\delta_n := (1-\beta_n)\...
Alufat's user avatar
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6 votes
1 answer
846 views

Langlands Reciprocity and Fermat's Last Theorem

Question: Can Langlands Reciprocity be used to prove Fermat's Last Theorem? Background A few years ago I was reading a book on the Langlands Program and the introduction provided a list of ...
OscarCorx's user avatar
6 votes
1 answer
282 views

What is the growth rate of the products of binomial coefficients?

Question 1: Are the following empirically observed relationships true $$ {n \choose 1^a}{n \choose 2^a}{n \choose 3^a}\cdots {n \choose m^a} \sim \exp\bigg(\frac{2n^{1 + \frac{1}{a}}}{a+3}\bigg) $$ ...
Nilotpal Kanti Sinha's user avatar
6 votes
1 answer
472 views

Analogues of Hecke relations for Maass forms

If a (suitably normalised) holomorphic cusp newform has q-expansion $$f(z) = \sum_n \lambda_f(n) e(nz),$$ then we know the Hecke relations for $(mn,q)=1$, $$(\star) \qquad \lambda_f(m)\lambda_f(n) = \...
TheStudent's user avatar
6 votes
3 answers
379 views

Strings of consecutive integers divisible by 1, 2, 3, ..., N

For each n, let $a_n$ be the least integer, greater than n, such that the numbers $a_n$, $a_n$+ 1, $a_n$+ 2, ..., $a_n$+ (n – 1) are divisible, in some order, by 1, 2, 3, ..., n. For example $a_{12}$ =...
Bernardo Recamán Santos's user avatar
6 votes
2 answers
366 views

Hyperelliptic Jacobians with (or without) CM

Let $C$ be a hyperelliptic curve $y^2 = f(x) $ defined over $\mathbb{Q}$, where $f(x) \in \mathbb{Q} [x]$ is a polynomial of degree $n=5$ or $6$, and $J = Jac(C)$ its Jacobian. I know Zarhin's result [...
Kazuki  Sato's user avatar
6 votes
1 answer
187 views

Is it expected that the mod $p$ representation determines a normalized Hecke newform of fixed weight for p large enough?

Mazur's conjecture on the image of Galois representations of Elliptic curves states that for $N$ large enough there is a unique elliptic curve $E$ over $\mathbb{Q}$ giving rise to a fixed mod $N$ ...
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6 votes
1 answer
364 views

The outer automorphism of the dihedral group $D_4$ and quartic polynomials

Let $L$ be a dihedral quartic field. That is, we have $[L : \mathbb{Q}] = 4$ and the Galois closure $M$ of $L$ is a degree 8 extension of $\mathbb{Q}$ with Galois group isomorphic to the dihedral ...
Stanley Yao Xiao's user avatar
6 votes
2 answers
369 views

On the parity of $|\{(j,k):\ 1\le j<k\le\frac{p-1}2\ \&\ \ j(j+1)\ \text{mod}\ p\,>\,k(k+1)\ \text{mod}\ p\}|$ with $p$ prime

Let $p=2n+1$ be an odd prime, and let $a_1<\ldots<a_{n}$ be all the quadratic residues mod $p$ among $1,\ldots,p-1$. For $a\in\mathbb Z$ let $\{a\}_p$ be the least nonnegative residue of $a$ ...
Zhi-Wei Sun's user avatar
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6 votes
1 answer
297 views

Can a primitive root-permutation of $A=\{1, 2, \ldots, p-1 \}$ be a cycle of length $p-1$ only for finitely many $p$?

Let $p$ be an odd prime, $g$ a primitive root of $p$ and $A=\{1, 2, \ldots, p-1 \}$. Obviously, $\sigma_g(p)=\begin{pmatrix} 1 & 2 & \ldots & {p-1} \\ g^1\pmod{p} & g^2\pmod{p} &...
Konstantinos Gaitanas's user avatar
6 votes
1 answer
391 views

Unramified non-abelian extension and Galois cohomology

Is there an example of a finite Galois extension $E/F$ of number fields, such that $G=\mathrm{Gal}(E/F)$ is non-abelian and the order of the cohomology group $H^1(G,U_E)$ is relatively prime to class ...
A. Maarefparvar's user avatar
6 votes
2 answers
559 views

Counting points on lattices in inside a box- Geometry of numbers

Let $\Lambda$ be a lattice in $\mathbb{R}^n$, and let $|\mathbf{x}|$ denote the $L^2$ norm. There is a fairly standard argument involving successive minima to obtain an estimate on $N(R)$ which is the ...
Johnny T.'s user avatar
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6 votes
1 answer
142 views

Proportion of pairs of integer polynomials with a bounded value of the resultant

Let $f(x), g(x) \in \mathbb Z[x]$ be polynomials of positive degrees $r$ and $s$ respectively, and let $\operatorname{Res}(f,g)$ denote their resultant. Further, let $H(f)$ denote the naive height of ...
Anton's user avatar
  • 1,573
6 votes
2 answers
519 views

Seeking for a meaning: a curious symmetry

Suppose $\Phi(m,n)=(2m)!^n\prod_{k=1}^n\binom{2m+2k+x}{2k+x}$. Then, algebraically, it is trivial to see that $$(2m)!^n\prod_{k=1}^n\binom{2m+2k+x}{2k+x}=(2n)!^m\prod_{k=1}^m\binom{2n+2k+x}{2k+x}.$$ ...
T. Amdeberhan's user avatar
6 votes
1 answer
323 views

Prime distribution in CM field

For any prescribed infinite set of primes $P\subseteq \mathbb{Z}$, is there a general heuristic to finding a field satisfying certain algebraic or geometric constraints such that these primes all ...
Vincent's user avatar
  • 443
6 votes
1 answer
202 views

How are the ratios of successive values of the divisor function distributed?

One motivation for this question is a paper by Erdos and Hall "Values of the divisor function on short intervals", in which the authors obtain the leading asymptotics $$x(\log x)^{2(\sqrt 2-...
Kevin Smith's user avatar
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6 votes
1 answer
364 views

How small can the smallest of the three "weak Goldbach" primes always be?

I've checked here for discussions of Helfgott's proof of the weak GC and found nothing that helps me with the following; apologies if I missed something. I'm probably being naive here (please ...
Aubrey de Grey's user avatar
6 votes
2 answers
453 views

Divisibility labeling on a boolean lattice and positive Euler totient

Let $B_n$ be the rank $n$ boolean lattice (i.e. the subset lattice of $\{1,2, \dots , n \}$). Let $\hat{0}$ and $\hat{1}$ be the minimum and the maximum of $B_n$. Let $f: B_n \to \mathbb{N}$ be a ...
Sebastien Palcoux's user avatar
6 votes
2 answers
736 views

Intuition behind the definition of the Siegel-Eichler transformation

Recently I am reading Wall's paper "On the Orthogonal Groups of Unimodular Quadratic Forms II". In this paper, I encountered with the map $E^1_\omega$, which now I am interested in. Let $X$ be an ...
Shinichiro Nakamura's user avatar
6 votes
1 answer
347 views

Applications of Level Lowering

What are some applications/consequences of level lowering of Galois representations? I understand the application of Ribet's theorem in the proof of Fermat's last theorem but I am wondering what other ...
Eins Null's user avatar
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6 votes
1 answer
356 views

Lifting of Frobenius on torsors over abelian varieties

This is related to my previous question Assume that $A$ is an abelian variety over a field $k$ of characteristic $p$, $\mathcal{L}$ is a line bundle on $A$. Assume that $A$ is ordinary and $\mathcal{L}...
SashaP's user avatar
  • 7,027
6 votes
1 answer
386 views

Levelt-Turrittin Theorem over p-adics (or the monodromy theorem)

Let $V$ be a finite dimensional vector space over $\mathbb{C}((t))$. Let $D:V\rightarrow V$ be a differential operator; i.e., an additive $\mathbb{C}$-linear map satisfying $$ D(a.v)=(t\frac{d}{dt}a)....
Dr. Evil's user avatar
  • 2,641
6 votes
1 answer
427 views

Applications of the Galois embedding problem

Given a finite Galois extension of number fields $L/K$ with Galois group $G$ and a surjection $E\twoheadrightarrow G$ of finite groups, the Galois embedding problem is the question of whether there ...
Bear's user avatar
  • 845
6 votes
1 answer
506 views

Why does $GL_n(F)\backslash GL_n(\mathbb{A}_F)/\prod_xGL_n(O_x)$ classify vector bundles over $X$?

Suppose $X$ is an algebraic curve, $F$ its function field, $\mathbb{A}_F$ its adele, $O_x$ the ring of integers at local field of $x$. Why does $GL_n(F)\backslash GL_n(\mathbb{A}_F)/\prod_xGL_n(O_x)$ ...
user avatar
6 votes
1 answer
651 views

Counting number of points in a lattice with bounded sup norm

Let $\Lambda$ be a lattice in $\mathbb{R}^n$. For $\bar{x} \in \mathbb{R}^n$, let $\| \bar{x} \| = max_{1 \leq i \leq n} \{ |x_i| \}$, i.e. the sup norm. Let $\lambda_1, ..., \lambda_n$ be a ...
SJY's user avatar
  • 579
6 votes
1 answer
411 views

When is $1+a+a^2+\dotsb+a^{{\rm ord}_n(a)-1}$ divisible by $n$?

I posted this question on math.SE 10 days ago and had no answer, comment or vote. If an answer is not available, I could really use a reference point as well. For the sake of completeness, I am ...
Dosidis's user avatar
  • 111
6 votes
1 answer
380 views

Buildings associated to generalized $BN$ pairs

I'll begin by asking a general question, and then specializing to the situation I really care about. Let $G$ be a group and let $(B, N)$ be a $BN$-pair in $G$ (see, for instance, page 39 of Tits' "...
John Binder's user avatar
  • 1,453
6 votes
1 answer
423 views

Asymptotic expansion of the Mordell integral

my question concerns the Mordell integral $$h(z;\tau):=\int_{-\infty}^\infty \frac{e^{\pi i\tau w^2-2\pi zw}}{\cosh(\pi w)}dw,\qquad \Im(\tau)>0,\quad z\in\mathbb{C},$$ which frequently occurs in ...
MHMertens's user avatar
  • 189
6 votes
2 answers
458 views

Rational points and torsion points of CM elliptic curve

Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order i.e., $\mathrm{End}_K(E)\cong \mathcal{O}=\mathcal{O}_K$. Let $\mathfrak{p}$ be a prime of $K$ such that ...
Kevin.lijh's user avatar
6 votes
3 answers
917 views

Uniformly distributed sequence in $\mathbb{R}$

We say that a sequence $(x_n)_{n=1}^\infty \subseteq \mathbb{R}$ is "uniformly distributed in $[a,b]$", with $a < b$, if $(x_n)_{n=1}^\infty \cap [a,b] \neq \varnothing$ and $$\lim_{N \to \infty} \...
Fry's user avatar
  • 61
6 votes
1 answer
1k views

weight monodromy conjecture for curves?

Hi, Is there a simple proof of the weight monodromy conjecture in the case of a curve over a mixed characteristic local field? Thanks!
Nicolás's user avatar
  • 2,802
6 votes
1 answer
415 views

Explicit Casselman theory: reference needed

Let $K$ be a nonarchimedean local field with ring of integeres $R_K$, maximal ideal $m_K$ and finite residue field $\bf k$. Let $\pi$ be an admissible irreducible complex representation of ${\rm GL}_2(...
Andrea Mori's user avatar
6 votes
1 answer
434 views

Bijective proof of Ramanujan's congruence

Is there a known bijective proof of Ramanujan's congruence for the partition function modulo 5? E.g., is there a construction that for every $n$ congruent to 4 mod 5 gives a permutation of the ...
James Propp's user avatar
  • 19.4k
6 votes
5 answers
584 views

Uniqueness of values in recurrence relations

Given an integer $k > 1$, define the sequences $X(k,n), Y(k,n)$ as follows: $a=4k-2,$ $y_0 = 1,$ $y_1 = a + 1,y_n = ay_{n-1} - y_{n-2}$ $b = 4k + 2,$ $ x_0 = 1,$ $x_1 = b - 1,$ $x_n = bx_{...
Jim White's user avatar
  • 225
6 votes
1 answer
449 views

Half integral weight Hecke operators

I would like to find a source giving the exact formula for the product of two Hecke operators $T_{\kappa}(n^2)$ and $T_\kappa(m^2)$ of half integral weight. That is, $\kappa \in \frac 12 \mathbb{Z} - \...
Eren Mehmet Kiral's user avatar
6 votes
1 answer
284 views

Projective modules over non-rational group rings

Let $G$ be a finite group. We know that the $K$-group $K_0(QG)$ of the rational group ring $QG$ is a free abelian group generated by the irreducible representations of $G$ over $Q$. Now let $R$ be a ...
yeshengkui's user avatar
  • 1,373
6 votes
1 answer
459 views

Existence of elements in an Eichler order whose norm is minus one

Let $B$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $D$, and $\mathcal{O}_N$ be an Eichler order of level $N$. Is there an element $x\in \mathcal{O}_N$ such that its reduced ...
Jiangwei Xue's user avatar
6 votes
1 answer
1k views

Reference request: Dickman, On the frequency of numbers containing prime factors

I've been trying without success to find the paper Dickman, Karl, "On the frequency of numbers containing prime factors of a certain relative magnitude." Ark Mal., Astronomi och Physik, 22A (10), ...
Tom Dickens's user avatar
  • 1,077
6 votes
2 answers
2k views

Generalising Dirichlet's theorem in arithmetic progressions-prime combinatorics

Let $M$ be a natural number $M>1$. For every prime $p_i$ not dividing $M$ take an arithmetic progression $A_i=k_i+np_i$ , $n\geq 0$ such that $k_i>p_i^2/M$. Is there any $M$ and some choice of ...
6 votes
2 answers
557 views

If the discriminant of a binary quadratic form has high valuation, is the form "almost a square".

For a binary quadratic form $ax^2+bxy+cy^2$ over a field (characteristic not 2), the discriminant $b^2-4ac$ is 0 if and only if that form is the square of a linear form. I am curious about an ...
Daniel Erman's user avatar
  • 2,895
6 votes
1 answer
1k views

Searching for an inhomogeneous diophantine approximation algorithm

Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, what is an algorithm that will provide coprime integers $a$ ...
Keenan Pepper's user avatar
6 votes
1 answer
1k views

Jacquet Langlands correspondence

I have one issue with the Jacquet Langlands correspondence. The Weyl law for $H$ modulo a congruence subgroup and the Weyl law for cocompact groups are different. So why does this not contradict this ...
Marc Palm's user avatar
  • 11.1k
6 votes
2 answers
412 views

Diophantine approximation

Let $P$ be a polynomial of fixed degree $d$ with integer coefficients of absolute values at most $n$. Assume that $P(\cos 2\pi/n)$ is no zero. Is there a lower bound for $|P(\cos 2\pi/n)|$ ? For ...
vanvu's user avatar
  • 353
6 votes
1 answer
233 views

proportion of positive integers which number of prime divisors has a special remainder.

Let $s$ be an integer greater than 1. For each natural number $n$, let $\omega(n)$ be the number of prime divisors (counted with multiplicities) of $n$ modulo $s$. For $i \in \{0,1,\dots, s-1\}$ and a ...
Jens Reinhold's user avatar
6 votes
1 answer
2k views

Do the tails of the decimal expansion of pi form a dense set in [0,1]?

Let $a_n=10^n \cdot \pi$. Is the set of numbers $\{a_n-\lfloor a_n \rfloor : n \in \mathbb{N}\}$ dense in [0,1]? What is the best known result near this question? Apparently John Nash asked this on ...
Ben Golub's user avatar
  • 1,058
6 votes
1 answer
873 views

Is it possible to recover the degree of a field extension from a list of elements and the ground field?

I'm interested to know if there is anything known about recovering the degree of a field extension, $E/k$, given $E=k(\alpha_1,\ldots, \alpha_n)$ (here I'm assuming that the extension is of finite ...
Adam Hughes's user avatar
  • 1,049
6 votes
1 answer
532 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 ...
Jonah Sinick's user avatar
  • 6,912
6 votes
1 answer
182 views

A combinatorial game with seemingly curious arithmetic properties

We consider the following combinatorial game (with two players alternatively playing optimally). Posititions are given by heaps containing $b\geq 0$ black and $w\geq 0$ white stones and are encoded by ...
Roland Bacher's user avatar
6 votes
1 answer
254 views

Sequence that sums up to the number of permutations avoiding the pattern $1-23-4$

Let $a(n)$ be A113227, i.e., the number of permutations on $[n]\equiv \{1, \ldots, n\}$ avoiding the pattern $1-23-4$. The sequence begins with $$1, 1, 2, 6, 23, 105, 549, 3207, 20577, 143239, 1071704,...
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