Questions tagged [algebraic-number-theory]

Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces

Filter by
Sorted by
Tagged with
6 votes
0 answers
392 views

Conditions under which an $\eta$-quotient becomes a **weak** modular form (reference request for theorems similar to Ligozat's theorem)

For any $z \in \mathcal{H}$, let $q = e^{2\pi iz}$; and the eta function is defined as ${\displaystyle \eta (q) =q^{\frac {1}{24}}\prod _{n=1}^{\infty }\left(1-q^{n}\right).}$ By an $\eta$-quotient ...
Davood Khajehpour's user avatar
6 votes
0 answers
197 views

$\mathbb{Z}$-points in a given $\widehat{\mathbb{Z}}$-isomorphism class

Given a finite type $\mathbb{Z}$-scheme $X$ with $X(\widehat{\mathbb{Z}})\neq\emptyset$ can we find a finite type $\mathbb{Z}$-scheme $Y$ with $X\times \widehat{\mathbb{Z}}\cong Y\times\widehat{\...
user avatar
6 votes
0 answers
150 views

Subalgebra of group algebra generated by idempotents

Let $G$ be a finite group, and let $A$ and $B$ be two abelian subgroups of $G$. Let $K$ be a number field such that all characters of $A$ and of $B$ take values in $K$. Let $\mathcal{O}_K$ be the ring ...
Ehud Meir's user avatar
  • 4,969
6 votes
0 answers
156 views

Certificate for computation of ideal class group

Is there a known way of producing a certificate that can be used to more quickly verify that an ideal class group of a number field was computed correctly? More formally, I would like to know if there'...
Daniel Hast's user avatar
  • 1,806
6 votes
0 answers
618 views

Generalized prime number theorem and Riemann Hypothesis for non-number math objects

My question is about some math objects (matrices, polynomials) and operators that satisfy a number of properties which can lead to a theory similar to PNT, RH, Dirichlet functions, abscissa of ...
Vincent Granville's user avatar
6 votes
0 answers
184 views

The power of Archimedean spirals: is there an algebraic characterization of Archimedean numbers?

I asked this question over a year ago on Math.StackExchange but I didn't get an answer. In his famous treatise On spirals, Archimedes used a spiral to square the circle and trisect an angle. There are ...
J.-E. Pin's user avatar
  • 851
6 votes
0 answers
440 views

Galois cohomology with coefficients in the unit group of a cyclotomic field

While understanding Fermat's last theorem's proof for regular primes, I bumped into a proposition (http://www2.biglobe.ne.jp/~optimist/algebra/fermat2_proof.html#proof10, written in Japanese) stating: ...
H Koba's user avatar
  • 369
6 votes
0 answers
115 views

Norm forms, slicing, and ideal classes

Let $K$ be a number field, which we may suppose satisfies $n = [K : \mathbb{Q}] \geq 3$. Let $\mathcal{O}_K$ be the ring of integers of $K$, and let $\{\omega_1, \cdots, \omega_{n}\}$ be a basis of $\...
Stanley Yao Xiao's user avatar
6 votes
0 answers
99 views

Class number of certain polynomials

Let $f_n(x)=x^n-\sum\limits_{i=0}^{n-1}{x^i}$ and $A_n$ the number field corresponding to $f_n$. Question: Is the class number of $A_n$ always equal to one, or equivalently, is the ring of integers ...
Mare's user avatar
  • 25.8k
6 votes
0 answers
148 views

$SL_2(\mathbb{Z}_p)$ extension of a local field

Let $G$ be an arbitrary open group of $SL_2(\mathbb{Z}_p)$ and $K$ be a finite extension of $\mathbb{Q}_p$. Can we construct a Galois extension field $E$ of $K$ such that $\text{Gal}(E/K)\cong G$? ...
user avatar
6 votes
0 answers
351 views

Coherent cohomology of the generic fiber of Lubin-Tate space vs. of Lubin-Tate space considered rationally?

I am trying to compare the coherent cohomology of the generic fiber of Lubin-Tate space to the coherent cohomology of Lubin-Tate space considered rationally, and I am going in circles! I would be very ...
Catherine Ray's user avatar
6 votes
0 answers
148 views

How to (easily) obtain the splitting field for dihedral extensions

Let $f(x) \in {\mathbb Q}[x]$ be a polynomial that is irreducible over ${\mathbb Q}$ with $D_{n}$ (the dihedral group of order $2n$) as its Galois group. Let $\alpha$ be a root of $f(x)$ and put $K={\...
user150983's user avatar
6 votes
0 answers
186 views

Quadratic fields with moderately large fundamental units

Let $d > 1$ be a fundamental discriminant, and let $K_d = \mathbb{Q}(\sqrt{d})$. Denote by $\varepsilon_d$ the fundamental unit of $\mathcal{O}_{K_d}$, namely the smallest algebraic integer $\...
Stanley Yao Xiao's user avatar
6 votes
0 answers
139 views

Newer versions of Mahler's Lemma

I'm trying to find a way to numerically ensure that two constructible numbers are equal (this would be done by a computer). The idea is to find a polynomial $p(x)$ that contains both numbers as roots ...
André Porto's user avatar
6 votes
0 answers
101 views

Cubic, abelian analogue of a result of Mertens-Siegel

For a number field $K/\mathbb{Q}$, put $h_K, R_K$ respectively for the class number (of the ring of integers $\mathcal{O}_K$ of $K$) and the regulator of $K$ respectively. Moreover, put $\zeta_K(s)$ ...
Stanley Yao Xiao's user avatar
6 votes
0 answers
210 views

A refinement of Faltings' lemma

In his proof of the Mordell conjecture, Faltings proved the following important result: Let $K$ be a number field and $S$ a finite set of primes in $K$. Then for any $g \geq 2$ there exists a number $...
Stanley Yao Xiao's user avatar
6 votes
0 answers
219 views

Which pair of ternary quadratic forms in Bhargava's theory parameterize the ring of integers of the quartic number field of discriminant $225$?

The binary quartic form $\mathcal{V} = (a, b, c, d, e)$ has the same discriminant $D$ as the binary cubic form \begin{equation}\label{resolvform} \mathcal{C} = \left( 1, - c, b d - 4 a e, 4 a c e - a ...
Samuel Hambleton's user avatar
6 votes
0 answers
138 views

$p >2$ is a prime, any facts about congruence relation between the class number of $Q(\sqrt p)$ and $Q(\sqrt-p)$?

Let $p$ be an odd prime. This is a question about the class number of $Q(\sqrt p)$ and $Q(\sqrt-p)$,which we denote by $h(p)$ and $h(-p)$ respectively. While doing my research on number theory I came ...
王李远's user avatar
  • 343
6 votes
0 answers
690 views

From Bhargava to Gauss -- Why does correspondence of cubes and ideal classes imply Gauss correspondence?

In his seminal 2004 paper "Higher Composition Laws I" in the Annals of Mathematics, (doi:10.4007/annals.2004.159.217), Bhargava proves that for fixed $D \neq 0$, there is a bijective correspondence ...
Ashvin Swaminathan's user avatar
6 votes
0 answers
681 views

What are the fastest ways to calculate class number of number fields?

Given a number field $K$, which approaches help us to calculate the class number $h(K)$ of $K$? I am aware that the question is broad but any argument would be helpful. Some basic approaches I know:...
Ninja's user avatar
  • 161
6 votes
0 answers
350 views

What happens to Neron-Ogg-Shfarevich when characteristic of the residue field equals the prime at which Tate module is considered?

Neron-Ogg-Shafarevich criterion states that an elliptic curve $E$ over a local field $K$ has a good reduction if and only if the Tate module $T_{\ell}(E)$ is unramified for some prime $\ell$ which ...
Johnny T.'s user avatar
  • 3,547
6 votes
0 answers
95 views

parity of index of subgroup of unit group generated by a Minkowski unit and its conjugates

Let $K$ be a number field which is Galois over $\mathbb{Q}$. Let $u$ be a Minkowski unit of $K$ in the (weak) sense as in [AV]. That is, the subgroup $S\subseteq\mathcal{O}_K^\times$ generated by $u$ ...
Christine McMeekin's user avatar
6 votes
0 answers
367 views

Two Definitions of Barsotti-Tate Representations

In different articles I have seen different definitions of Barsotti-Tate representations. I am wondering if and how these definitions are equivalent. In Section 1.1 of Conrad-Diamond-Taylor they say ...
Misja's user avatar
  • 161
6 votes
0 answers
179 views

Torsion in number field modulo rationals

Let $E$ be a quadratic number field. Is there a nice description of the torsion subgroup of $E^\times/{\mathbb Q}^\times$? Is it finite? How does it change when varying the field $E$?
user avatar
6 votes
0 answers
107 views

Class number parity for certain CM fields

I am interested in knowing the parity of the class number of a number field of the shape $L = K(\omega)$, where $\omega$ is a primitive cubic root of unity and $K$ is a totally real field (of class ...
Cindy Tsang's user avatar
6 votes
0 answers
268 views

$G$ is quasisplit at almost all places

Let $G$ be a connected, reductive group over a global field $k$. I am trying to understand why $G_v = G \times_k k_v$ is quasisplit for almost all places $v$ of $k$. There are several equivalent ...
D_S's user avatar
  • 6,100
6 votes
0 answers
357 views

Cohomology and Riemann-Roch in Number Theory (Neukirch Chapter 3)

In chapter 3 of Neukirch's Algebraic Number Theory, an analogue of the classical Riemann-Roch theorem is developed for number fields. To achieve this, notation suggestive of cohomology with sheaves of ...
Anton Hilado's user avatar
  • 3,269
6 votes
0 answers
414 views

What is the precise relationship between primitive Hida families and the connected components of the ordinary locus of the eigencurve?

In the references I've found discussing this question, I have not found any statements that I can understand and that are as precise as I would like. I'm more familiar with Hida families than with the ...
Will Dukeminier's user avatar
6 votes
0 answers
291 views

Iwasawa theory, $\mathbb{Z}_p^{2}$-extension, Greenberg module

Take $H\subset \bar{\mathbb{Q}}$ be a quartic imaginary number field such that $\operatorname{Gal}(H/\mathbb{Q})=\mathbb{Z}_2 \times \mathbb{Z}_2$. Denote by $F$ the quadratic real subfield of $H$ and ...
Adel BETINA's user avatar
  • 1,046
6 votes
0 answers
206 views

On a theorem of Dwork and totally ramified extensions

Suppose that $K \subset L$ is a totally abelian ramified extension of local fields. Let $\pi_L$ be a prime element of $L^*.$ $F \in Gal(\tilde{L}/L)$ is the Frobenius, where $\tilde{L}$ is the maximal ...
RiemannRock's user avatar
6 votes
0 answers
219 views

Furtwangler's Principal ideal theorem in number fields

Does anyone know a simple proof, using cohomological method of the fact that the verlagerung from a finite group G. to its commutator subgroup G', i.e. $$G/G'->(G')^{ab}$$ vanishes? The simplest ...
Eran's user avatar
  • 61
6 votes
0 answers
468 views

For which rational values of $c$ and $d$ are the numbers $\sin{(\pi\cdot c)}$, $\sin{(\pi\cdot d)}$ and $1$ linearly dependent over $\mathbb{Q}$?

A year ago, I posted this problem on [MSE]. After a number of edits, I have arrived at the following more general problem (suggested by Hjalmar Rosengren; see the comments below). For which ...
math110's user avatar
  • 4,230
6 votes
0 answers
113 views

Constructing a polyhedron of maximal possible volume from given bounds on areas of its faces

Consider $n$ variables $a_1,...,a_n$ ranging over $\mathbb{R}^+$. Suppose we are given $n$ pairs of positive rational numbers $(p_1,q_1),...,(p_n,q_n)$ where each pair imposes bounds on the ...
Frida Mauer's user avatar
6 votes
0 answers
224 views

How to construct Weil numbers in a given CM quartic field?

Let $L$ be a CM field of degree $4$ over the rationals, and let $p$ be a prime number. If $q$ is a power of $p$, I would like to know if it is possible to characterize (in some way) all Weil ${\bf F}...
Tommaso Centeleghe's user avatar
5 votes
5 answers
826 views

For any $n$, does there exist a number field with at least $n$ solutions to the unit equation

Let $n$ be a positive integer. Does there exist a number field $K$ such that the number of solutions of the unit equation $$a+b =1, \quad a,b\in O_{K}^\ast$$ is at least $n$? Can we write down such a ...
Taicho's user avatar
  • 225
5 votes
3 answers
530 views

Reference request: correspondence between central simple algebras and quadratic forms

Let $A$ be an algebra over $k$, $\operatorname{tr_A}(x, y):=\operatorname{tr}(m_{xy})$ be a trace form on $A$, and $V_A$ be its restriction on the orthogonal complement to $1$. I wonder why a map $A \...
evgeny's user avatar
  • 1,980
5 votes
4 answers
1k views

The integral closure $\overline{\mathbb{Z}}$ and the group $\overline{\mathbb{Z}}^{\times}$

Let $\mathbb{Q}$ be the field of rational numbers, and let $\overline{\mathbb{Q}}$ be its algebraic closure. Assume $\overline{\mathbb{Z}}$ is the integral closure of $\mathbb{Z}$ in $\overline{\...
Yuan Yang's user avatar
  • 537
5 votes
5 answers
2k views

Connection Between Knot Theory and Number Theory

Is there any connection between knot theory and number theory in any aspects? Does anybody know any book that is about knot theory and number theory?
Peg Leg Jonathan's user avatar
5 votes
2 answers
497 views

n-th root of unity in n-th division field of abelian variety?

Let $K$ be a number field and $A/K$ an abelian variety over it. Can it be that $K(A[n])$ does not contain a primitive $n$-th rooth of unity? If the answer is yes is it always possible to ...
David84's user avatar
  • 53
5 votes
2 answers
825 views

Dirichlet's approximation only using prime power as denominator

I am not sure whether this is a suitable question for MO. We know the classical version of Dirichlet's approximation theorem that if $x$ is a real number and $Q>0$ there exist $p,q\in \mathbb{Z}$ ...
Subhajit Jana's user avatar
5 votes
2 answers
440 views

When adding several $\sqrt[n]{p}$ to the rational numbers, what is the degree of field extension?

When adding several $\sqrt[n]{p}$ to the rational numbers, what is the degree of field extension? For example, does $[\mathbb{Q}(\sqrt[n]{2},\sqrt[m]{3}):\mathbb{Q}]=mn$ hold true? Are there more ...
sofia's user avatar
  • 51
5 votes
1 answer
515 views

Disjoint images of polynomials

Are there any $f,g \in \mathbb{Q}[x]$ such that for every root of unity $\zeta$, and every $a,b \in \mathbb{Q}(\zeta)$, $f(a) \neq g(b)?$
Pablo's user avatar
  • 11.2k
5 votes
1 answer
645 views

rational points of a hyperelliptic curve of genus 3

Let $K=\mathbb{Q}(\sqrt{-1}).$ I have the following hyperelliptic curve of genus 3: $$ C : y^2 = (x^2-x+1)(x^6+x^5-6x^4 -3x^3+14x^2-7x+1) $$ I want to find $C(K)$. My first attempt was to compute the ...
bijection123's user avatar
5 votes
1 answer
419 views

Tate-Shafarevich group over number fields

Let $A$ be an abelian variety over a number field $K$, $\text{Sha}(A/K)$ its Tate-Shafarevich group, $\ell$ a prime. Is it known that the $\ell$-primary torsion subgroup $\text{Sha}(A/K)\{\ell\}$ is ...
user avatar
5 votes
1 answer
545 views

Heuristics of Cohen-Lenstra-Martinet

Let $h(d)$ be the class numbers of the real quadratic field $\mathbb{Q}(\sqrt{d})$. There are some heuristics of Cohen-Lenstra-Martinet about divisibility of class numbers. Do they say anything about ...
Shamik Das's user avatar
5 votes
1 answer
2k views

Sum of square roots of natural numbers

Recently, I've encountered the following question: Assume that $n_{1}, \ldots, n_{k}$ are (not necessary distinct) natural numbers. If $$ (\sum_{i = 1}^{k}\sqrt{n_{i}}) \in \mathbb{N},$$ can we ...
Mohammad Ali Nematollahi's user avatar
5 votes
4 answers
632 views

Examples of discrete subgroups of $PSL_2(\mathbf{R})$ with finite covolume and which are not co-compact

Is there a natural example of a discrete subgroup $\Gamma\leq PSL_2(\mathbf{R})$ such that (1) $\Gamma$ has finite covolume (2) $\mathfrak{h}/\Gamma$ is not compact ($\mathfrak{h}$ being the upper ...
Hugo Chapdelaine's user avatar
5 votes
3 answers
1k views

Orders of Number Fields

Let $K$ be a number field over $\mathbb{Q}$ of degree $n$, and $\mathcal{O} \subset \mathcal{O}_K$ an order. $\textbf{Questions:}$ $\newcommand{\Spec}{\textrm{Spec }}$ $\newcommand{\cO}{\mathcal{O}}$ ...
LMN's user avatar
  • 3,525
5 votes
2 answers
413 views

Proving certain inequality related to Primes

I was reading the following paper. But I can't understand why the last line concerning $\frac{2}{\pi}$ is true. The proof is a work of Sylvester. I would be happy if someone helps me in understanding ...
math is fun's user avatar
5 votes
2 answers
492 views

Even unimodular lattices with root system $32 A_1$

I'm studying Venkov's proof of the classification of even unimodular rank 24 lattices, and it prompted the following question. For an even unimodular lattice $L$, let $R(L)= \{ x \in L : (x,x) =2\}$ ...
Ariyan Javanpeykar's user avatar

1
13 14
15
16 17
44