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

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What is the quotient group $D^*/{F^*(1+P_D)}$ for a quaternion division algebra $D$ over a local field $F$?

Let $F$ be the non-archimedean local field $\mathbb{Q}_p$ for some prime $p$ and $D$ be a quaternion division algebra over $F$. Let $\mathcal{O}_D$ and $\mathcal{P}_D$ denote the ring of integers of $...
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1 vote
0 answers
135 views

Explanation of a step in a preprinted work

I have been studying this preprinted paper and the references therein. I believe that there are some typos; However, since I am not an expert yet, I would like to make sure I am correct. I do not ...
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18 votes
1 answer
887 views

Can we state the Riemann Hypothesis part of the Weil conjectures directly in terms of the count of points?

For algebraic curves we can state the Riemann hypothesis part of the Weil conjectures directly as a formula for the number of points on the curves, sidestepping the zeta function. Namely, given a ...
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  • 19.3k
3 votes
0 answers
43 views

Minimal Norm Vectors in certain Cyclotomic Ideal Lattices

Let $q$ be an odd prime and let $\zeta$ be a primitive $q^{th}$ root of unity. Let $I$ be the ideal in $\mathbb{Z}[\zeta]$ generated by $1-\zeta$. It is known that we can give $I$ the structure of an ...
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4 votes
1 answer
127 views

On presentations of universal rings of deformations

Let $k$ be a finite field of characteristic $p$, and $R$ a complete local noetherian algebra with residue field $k$. It is well known that $R$ has a natural structure of an algebra over the ring Witt ...
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2 votes
0 answers
69 views

Differential equation of Van Gorder type for zeta of global fields, or: Does the zeta function of a global field satisfy a differential equation?

Let \begin{equation*} \zeta(s):=\prod_{p\text{ prime}}\frac{1}{1-p^{-s}} \end{equation*} be the Riemann zeta function. Van Gorder has shown that $\zeta$ satisfies a differential equation \begin{...
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1 vote
1 answer
124 views

Completion reducing to localization on Noetherian rings

It is quite easy to show that if $A$ is a Dedekind domain and $\mathfrak{p}\in \operatorname{Spec} A$, then if $A_{\mathfrak{p}}$ is the completion of $A$ at $\mathfrak{p}$ and $A_{(\mathfrak{p})}=(A\...
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  • 615
4 votes
1 answer
256 views

Rationality of field embeddings

After my earlier question question turned out to have a negative answer (Thank you to all respondents!), here is a more modest one. Both a positive answer and a counterexample would help my work. If ...
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  • 105
1 vote
0 answers
199 views

Advanced texts on analytic number theory?

So a friend of mine is very interested in analytic number theory, and is looking for resources past the basic level. He has studied analytic number theory from several books, among them are Hardy’s ...
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  • 1,149
7 votes
0 answers
128 views

Finding when a certain product in a cyclotomic field is equal to one

For the following, fix $m\geq 2$, and let $a_{0},\dots,a_{m-1}\in\mathbb{Z}$ be such that $\sum_{k=0}^{m-1}a_{k}=0$. I would like to find the exact conditions on the $a_{k}$ so that the following ...
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2 votes
0 answers
65 views

Alternative formulation of the Ferrero-Washington Theorem

The Ferrero-Washington theorem says that if $K/\mathbf{Q}$ is an abelian extension, then the Iwasawa $\mu$-invariant of the cyclotomic $\mathbf{Z}_p$ extension $K_{cyc}/K$ equals $0$. In the paper &...
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  • 695
6 votes
2 answers
253 views

Cancellation of irreducibility for Galois conjugates

Motivation: Take an algebraic number $\lambda$. In my research, I've stumbled upon the question in which cases the expression $\sum_{\sigma \in S} \sigma(\lambda)$, where $S$ is a subset of the field ...
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  • 105
7 votes
0 answers
232 views

Analogs of the Weil conjectures for non-archimedian fields

Suppose that $X$ is a smooth and proper variety defined over a perfect non-archimedian valued field $k$ of characteristic $p$.  Then one can consider the action of Frobenius on crystalline cohomology. ...
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7 votes
2 answers
261 views

Given that $n > 3$ and $z$ is a Gaussian integer, when can $z^n \pm z$ be a rational integer?

I came across the following conjecture. If you have any thoughts on how to approach it, let me know. Conjecture. For any integer $n > 3$ and any Gaussian integer $z$ that is not a unit, if $z^n - z$...
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  • 1,369
2 votes
0 answers
233 views

How soon can we represent a number as the sum of two primes?

Posting in MO since it was unanswered in MSE. Goldbach's conjecture says that every even number can be represented as the sum of two primes. But how soon can we find such a representation. Taking $20 =...
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4 votes
0 answers
100 views

On the pro-category of finite local artinian algebras

Let $\mathbb{F}$ be a finite field, and $W(\mathbb{F})$ its associated ring of Witt's vectors. On page 6 of the following lecture notes Deformations of Galois Representations, the category $\mathfrak{...
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3 votes
1 answer
208 views

What's the class group of $\mathbb{Q}^{\mathrm{ab}}$?

If $K/\mathbb{Q}$ is an infinite algebraic extension, define as usual the class group $Cl_K$ by the direct limit via the natural (conorm) map $Cl_K := \lim\limits_{\rightarrow} Cl_F$, where $F$ runs ...
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  • 741
30 votes
2 answers
959 views

Is equation $xy(x+y)=7z^2+1$ solvable in integers?

Do there exist integers $x,y,z$ such that $$ xy(x+y)=7z^2 + 1 ? $$ The motivation is simple. Together with Aubrey de Grey, we developed a computer program that incorporates all standard methods we ...
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6 votes
1 answer
337 views

Is there a conjectured dependence on $n$ in van der Waerden's conjecture?

Bhargava 2021 proves van der Waerden's conjecture about Galois groups of random integer polynomials: over all $x^n + a_{n-1} x^{n-1} + \cdots + a_0 = 0$ with $a_k \in \{-H, \ldots, H\}$, the number of ...
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5 votes
2 answers
107 views

Dihedral extension unramified at primes dividing order of group?

Consider dihedral Galois extensions $L/\mathbb{Q}$ of degree $n$ (and we know they exist thanks to Shafarevich), can we show there always exists an extension $L/\mathbb{Q}$ unramified at $p$, for all $...
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2 votes
1 answer
200 views

Lang's proof concerning ray class fields of imaginary quadratic number fields

Crosspost from Math.SE as I did not receive an answer there: In Lang's book Elliptic Functions, he shows how to generate the ray class field with conductor $N$ of an imaginary quadratic number field $...
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  • 179
5 votes
1 answer
299 views

About the structure of unit groups appearing in number theory

I think the following statement is not true in the general situations, but consider it: $R$ is a ring, $\mathfrak{p}$ is a prime ideal, then the unit group of $\dfrac{R}{\mathfrak{p}^nR}$ is ...
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1 vote
0 answers
77 views

A dimension problem related to an abelian simple extension of a field

$\DeclareMathOperator\Imm{Im}$Let $K=F(\alpha)$ be an abelian extension of $F$ and let $\sigma$ be a map (could be any map) from $K^\times$ (the multiplicative group of $K$) to itself. Define an $F$-...
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  • 213
1 vote
0 answers
57 views

The localization map for the Mordell-Weil group of elliptic curves over finite Galois extensions

Let $L/K$ be a finite Galois extensions of number fields and $E/K$ be an elliptic curve. Denote by $\mathcal{F}$ the localization map \begin{equation} \mathcal{F}: H^1(G,E(L)) \rightarrow \bigoplus_{v ...
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1 vote
0 answers
76 views

How to describe the automorphisms of $\mathbb{Q}(\theta)$ that fix $\mathbb{Q}(\sqrt{-m})$

Suppose $m > 0$ is a square free integer and $m \equiv 1 $mod $4$. Then the $K = \mathbb{Q}(\sqrt{-m})$ is a subfield of $L = \mathbb{Q}(\theta)$, where $\theta$ is a primitive $4m$-th root of ...
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2 votes
0 answers
66 views

Number fields with given discriminant

In the special case of number fields that are the splitting fields of irreducible polynomials of degree 5 or 6 with Symmetric Galois group (so degree 120 or 720 over Q), is there a good upper bound on ...
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4 votes
0 answers
315 views

Do we expect the Langlands correspondence to be a functor?

In the literature I've read, it is often said that to a Hecke eigenclass, one would like (and sometimes succeeds) to "associate" or show the existence of a Galois representation such that ...
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1 vote
1 answer
151 views

Integers in residue classes $\mathcal{O}_K/\mathfrak{p}$

Let $K$ be a number field and $\mathcal{O}_K$ be its ring of integers. For any prime ideal $\mathfrak{p}$ in $\mathcal{O}_K$ is it true that every residue class in $\mathcal{O}_K/\mathfrak{p}$ ...
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1 vote
1 answer
159 views

Counting cubic residues mod p

Given a prime $p=3m+1$, $(p-1)/3$ of the residues mod $p$ are cubic residues. So heuristically, for any given integer $k>1$ not a perfect cube, we would expect that about 1/3 of the primes $\equiv1\...
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  • 8,358
6 votes
0 answers
162 views

Reference request: projectivity of the absolute Galois group of $\mathbb{Q}^{\mathrm{ab}}$

$\newcommand{\ab}{\mathrm{ab}}$Let $\mathbb{Q}^{\ab}$ denote the maximal abelian extension of $\mathbb{Q}$. I have heard the absolute Galois group of $\mathbb{Q}^{\ab}$ is projective (e.g. see this ...
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3 votes
0 answers
116 views

Congruence of elements implies congruence of norms for central simple algebras

I was reading Eichler's "Allgemeine Kongruenzklasseneinteilungen [...]", Crelle 1938 (one of the main historical references for strong approximation theorems), and I cannot understand one of ...
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  • 697
1 vote
0 answers
95 views

Are there any connections between $a$ and $c$ where $p = a^2 + 2b^2 = c^2 + d^2$?

Let $p$ be a prime such that $p \equiv 1 \mod 8$. Then we know there exists $a,b \in \mathbb{Z}$ such that $p = a^2 + 2b^2$. But at the same time $p \equiv 1 \mod 4$, so there also exists $c,d \in \...
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0 votes
1 answer
148 views

p-adic number field $Q_p $and algebraic numbers [closed]

As we all know, the complex number field $C$ be a finite Galois extension field of the real number field that contains all algebraic numbers. I want to know the proof of the following proposition: Any ...
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  • 13
12 votes
1 answer
560 views

Geometric series in algebraic number fields

For which algebraic numbers $\alpha$ is there a valuation on the number field ${\mathbb {Q}}(\alpha)$ for which the infinite series $\sum_{n=0}^\infty \alpha^n$ converges to $1/(1-\alpha)$?
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  • 17.2k
8 votes
1 answer
357 views

Abelianization of $\mathrm{GL}_2(R)$

$\DeclareMathOperator\GL{GL}$Let $R$ be a number ring. Are there known lower bounds for $H_1(\GL_2(R);\mathbb Q)$ or $H_1(\GL_2(R),\GL_1(R);\mathbb Q)$ in terms of properties of $R$ (class number, ...
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  • 833
4 votes
0 answers
203 views

Coefficients in Hilbert's tenth problem over number rings: do they matter?

Here are two ways to define Hilbert's tenth problem over a ring $R$: Given a polynomial $p \in \mathbb Z[x_1,\ldots,x_n]$, can one decide whether it has a solution in $R^n$? Given a polynomial $p \in ...
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  • 864
4 votes
1 answer
156 views

Order of $37$-Sylow subgroup of ideal class group of $K_{37} = \Bbb Q(\mu_{37^{n}})$ is known to be $37^n$

I want to examine nontrivial examples of what we call Iwasawa class formula, $c(n)=\mu p^n + \lambda n + \nu$, where $\lambda, \mu \in \mathbf N$ and $\nu \in \mathbf Z$ are parameters depending only ...
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  • 141
1 vote
0 answers
168 views

Regarding the base for the neighbourhoods of 1 in general linear group over $p$-adic field

$\DeclareMathOperator\GL{GL}$ Let $ L/\mathbb{Q}_{p} $ be a finite extension with ring of integers $ \mathcal{O}_{L} $ and let $ \pi $ denote the uniformizer of $ \mathcal{O}_{L} $. Recall that a base ...
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5 votes
2 answers
155 views

Irreducibility of the $n$th symetric power of the reduction of the Galois representation of a non-CM newform

In "On $\ell$-adic representations attached to modular forms II", Ribet proved that the $\ell$-adic representation $\rho_{f,\ell}$ attached to a non-CM newform form $f$ satisfies $${\rm SL}...
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1 vote
1 answer
112 views

Finding a certain value of $\Gamma_p$

Let $\Gamma_p : \mathbb{Z}_p \to \mathbb{Z}_p^{\times}$ be the $p$-adic gamma function. I thought that I had successfully calculated $\Gamma_p(1 - 1/4)$, but sage is telling me I'm wrong (this is ...
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2 votes
2 answers
141 views

Value of the quadratic Gauss sum viewed in $\mathbb{C}_p$

Let $p > 2$ be a prime and $q = p^r$ for some $r \in \mathbb{Z}^+$. I will assume that all roots of unity lie in $\mathbb{C}_p^{\times}$. Let $\zeta$ a primitive $p$-th root of unity. Let $Tr : ...
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5 votes
1 answer
378 views

Galois groups of specific classes of polynomials with one coefficient fixed

Let $f = x^n + a_{n-1}x^n + \cdots + a_0$ be a monic polynomial of degree $n \geq 2$ with integer coefficients. By $\text{Gal}(f)$ we mean the Galois group over $\mathbb{Q}$ of the Galois closure of $...
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1 vote
0 answers
55 views

Image of Kudla-Millson pairing

Let $G=O(p,q)$ and $M$ the locally symmetric space obtained by taking th symmetric space of $O(p,q)$ and quotienting by an arithmetic group $\Gamma$. In INTERSECTION NUMBERS OF CYCLES ON LOCALLY ...
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7 votes
1 answer
184 views

Classical and adelic automorphic forms from SL(n) to GL(n) over number fields

It's a long post but I felt like I needed to provide some context to my problem. The explicit questions start at the bold font questions below. In the classical world, it seems that one is usually ...
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  • 697
0 votes
0 answers
58 views

Irreducibility of cyclotomic polynomial with change of variable

Consider the cyclotomic polynomial $\Phi_k(x - \alpha)$ where $\alpha$ is an algebraic integer in a number field $\mathcal{K}$. Is $\Phi_k(x - \alpha)$ irreducible in $\mathcal{K}[x]$ ? If yes, also $\...
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  • 19
2 votes
0 answers
152 views

Two basic questions on congruence subgroups

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$I have two questions related to congruence subgroups. Let $$\Gamma=\Gamma_0(N)=\Big\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \...
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  • 643
2 votes
0 answers
200 views

Solving $x^k+y^k+z^k=w^k$ non-trivially in strictly positive integers

Consider the equation $x^k+y^k+z^k=w^k$ in $x$, $y$, $z$ and $w$ with $k\in\mathbb{N}_{\geq2}$. If we look for solutions that are strictly positive and non-trivial i.e. $x\neq-y$, $x\neq w$ etc... ...
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  • 4,350
2 votes
0 answers
51 views

Filtration of norm-one elements of quaternion algebra over local field with respect to an involution

Let $K$ be a local non-archimedean field, with ring of integers $\mathcal{O}_K$, uniformizing element $\varpi_K$, and residue field $\mathcal{O}_K/\varpi_K\mathcal{O}_K \cong \mathbb{F}_q$. For ...
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  • 71
1 vote
0 answers
120 views

On closed subsets in spaces of adèlic points

Consider as in Adèlic points and algebraic closure $\mathcal{X}$ a projective and flat scheme over $\text{Spec}(\mathcal{O}_K)$, with $\mathcal{O}_K$ the ring of integers of a number field $K$. ...
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6 votes
1 answer
326 views

Adèlic points and algebraic closure

Consider $\mathcal{X}$ a projective and flat scheme over $\text{Spec}(\mathcal{O}_K)$, with $\mathcal{O}_K$ the ring of integers of a number field $K$. Let $F/K$ vary over all finite Galois number ...
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