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
2,164
questions
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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 ...
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{\...
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 ...
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'...
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 ...
6
votes
0
answers
184
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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 ...
6
votes
0
answers
440
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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:
...
6
votes
0
answers
115
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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 $\...
6
votes
0
answers
99
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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 ...
6
votes
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148
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$SL_2(\mathbb{Z}_p)$ 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 $\text{Gal}(E/K)\cong G$? ...
6
votes
0
answers
351
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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 ...
6
votes
0
answers
148
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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={\...
6
votes
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answers
186
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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 $\...
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 ...
6
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0
answers
101
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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 $\zeta_K(s)$ ...
6
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0
answers
210
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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 $...
6
votes
0
answers
219
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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 ...
6
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138
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$p >2$ 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 ...
6
votes
0
answers
690
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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 ...
6
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681
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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:...
6
votes
0
answers
350
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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 ...
6
votes
0
answers
95
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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 $u$ ...
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 ...
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 $E$?
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 ...
6
votes
0
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268
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$G$ 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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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}...
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 ...
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 \...
5
votes
4
answers
1k
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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{\...
5
votes
5
answers
2k
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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?
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 ...
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 $p,q\in \mathbb{Z}$ ...
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 ...
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 $a,b \in \mathbb{Q}(\zeta)$, $f(a) \neq g(b)?$
5
votes
1
answer
645
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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 ...
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 ...
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 ...
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 ...
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 ...
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.
$\textbf{Questions:}$
$\newcommand{\Spec}{\textrm{Spec }}$
$\newcommand{\cO}{\mathcal{O}}$
...
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 ...
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 $R(L)= \{ x \in L : (x,x) =2\}$ ...