# 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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### 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 ...
1 vote
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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### 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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### 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 &...
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 ...
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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### 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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### 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 ...
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 ...
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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### 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 ...
1 vote
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### 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 $F$-...
1 vote
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### 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 ...
1 vote
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### 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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### 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 ...
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 ...
1 vote
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 $\mathcal{O}_K/\mathfrak{p}$ ...
1 vote
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### 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 ...
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 $1/(1-\alpha)$?
### 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, ...
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 ... 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 ... 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} $. Recall that a base ... 5 votes 2 answers 155 views ### Irreducibility of the$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 $${\rm SL}... 1 vote 1 answer 112 views ### Finding a certain value of \Gamma_p Let \Gamma_p : \mathbb{Z}_p \to \mathbb{Z}_p^{\times} be the p-adic gamma function. I thought that I had successfully calculated \Gamma_p(1 - 1/4), but sage is telling me I'm wrong (this is ... 2 votes 2 answers 141 views ### Value of the quadratic Gauss sum viewed in \mathbb{C}_p Let p > 2 be a prime and q = p^r for some r \in \mathbb{Z}^+. I will assume that all roots of unity lie in \mathbb{C}_p^{\times}. Let \zeta a primitive p-th root of unity. Let Tr : ... 5 votes 1 answer 378 views ### Galois groups of specific classes of polynomials with one coefficient fixed Let f = x^n + a_{n-1}x^n + \cdots + a_0 be a monic polynomial of degree n \geq 2 with integer coefficients. By \text{Gal}(f) we mean the Galois group over \mathbb{Q} of the Galois closure of ... 1 vote 0 answers 55 views ### Image of Kudla-Millson pairing Let G=O(p,q) and M the locally symmetric space obtained by taking th symmetric space of O(p,q) and quotienting by an arithmetic group \Gamma. In INTERSECTION NUMBERS OF CYCLES ON LOCALLY ... 7 votes 1 answer 184 views ### Classical and adelic automorphic forms from SL(n) to GL(n) over number fields It's a long post but I felt like I needed to provide some context to my problem. The explicit questions start at the bold font questions below. In the classical world, it seems that one is usually ... 0 votes 0 answers 58 views ### Irreducibility of cyclotomic polynomial with change of variable Consider the cyclotomic polynomial \Phi_k(x - \alpha) where \alpha is an algebraic integer in a number field \mathcal{K}. Is \Phi_k(x - \alpha) irreducible in \mathcal{K}[x] ? If yes, also \... 2 votes 0 answers 152 views ### Two basic questions on congruence subgroups \DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}I have two questions related to congruence subgroups. Let$$\Gamma=\Gamma_0(N)=\Big\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \... 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... ... 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 ... 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$K$. ... 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 ... 