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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4
votes
1answer
159 views

Deformations of the Riemann zeta function

Consider the Dirichlet series (for fixed $0 < a \leq 1$): $$\zeta_a(s) = \sum_{n\geq 1}\frac{a^n}{n^s}$$ which reduces to the Riemann zeta function for $a=1$. What is known about this function, ...
0
votes
0answers
41 views

Image of extension ideal classes homomorphism in ideal class group under Artin map in class field theory

Let $K/P$ be a finite extension of number fields and $\epsilon_{K/P}:[\mathfrak{a}] \in Cl(P) \rightarrow [\mathfrak{a}.\mathcal{O}_K]\in Cl(K)$ be the ideal class transfer homomorphism. It's well ...
11
votes
2answers
335 views

Conceptual explanations of the class numbers for the first few $\mathbb{Q}(\sqrt{p})$ with odd conductor

It's known that the class number of $\mathbb{Q}(\sqrt{p})$ is $1$ for all primes $p<229$. Question: What would it be like for conceptual explanations of $h(\mathbb{Q}(\sqrt{p}))=1$ for the first ...
0
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0answers
68 views

Image of Frobenius element under irreducible representation is diagonalizable

Let $K/ \mathbb Q$ be a Galois extension, and $\rho$ be an irreducible representation of the Galois group $Gal(K/ \mathbb Q)$. Consider an integer prime $p$ which doesn't ramify in $K$, and let $\...
4
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0answers
98 views

Discriminant mod 8

Let $f,g\in\mathbb{Z}[X]$ be monic polynomials of degree $n$, with discriminants $\Delta_{f}$ and $\Delta_{g}$, such that $f=g$ in $\mathbb{F}_{2}[X]$ and $\Delta_{f}$ is odd. Is there an elementary ...
0
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0answers
84 views

Status of the $n$ conjecture, and as a reference request what about ideas or techniques that could be potentially interesting

The n conjecture is a generalization of the abc conjecture. What is the current status of the $n$ conjecture? See also [1] Question 1. Can you tell us what about the current status of the $n$ ...
4
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0answers
59 views

Maximal $\mathbb{Q}$-split torus in center of Weil restriction of $\text{GL}_n$ over a number ring

I'm a topologist writing a paper where I have to do a bit of work with algebraic groups, and I've managed to confuse myself about something very basic. Let $K$ be an algebraic number field. Regard $\...
1
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1answer
89 views

Theory of extensions of non-archimedian local fields

I'm searching for a recommendable reference dealing with theory of non-Archimedean local fields where I can find proofs of the following claims about finite extensions $L/K$ of non-Archimedean local ...
1
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1answer
89 views

Looking for a paper by Landau and one by Watson

For the purposes of a project, I've been looking for the following two papers referred to in Serre's "Divisibilité de certaines fonctions arithmétiques": Landau (E.), - Über die Eitenlung der ...
0
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0answers
67 views

Distribution of prime ideals by norm

I was wondering if there exist results in arithmetic statistics which count the number of prime ideals of bounded degree in a given number field, for instance a (real or imaginary) quadratic number ...
4
votes
1answer
183 views

How is class of composition of two quadratic fields is related class numbers of quadratic field?

Let $K_1=\Bbb Q(\sqrt{d_1})$ , $K_2=\Bbb Q(\sqrt{d_2})$ and $K=\Bbb Q(\sqrt{d_1},\sqrt{d_2})$.Suppose $h_1,h_2,h$ be class number of $K_1,K_2,K$ respectively. (i) Can we express $h$ in terms of $...
0
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1answer
174 views

On $L$-function of permutation representation

I came across the statement in a book: Let $k$ be a number field and $K$ be a Galois extension of $\mathbb Q$ containing $k$, with Galois group $G=\operatorname{Gal}(K/\mathbb Q)$ and let $G_k:=\...
0
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0answers
73 views

On Prime Numbers which can be Norms of an Integral Ideal of a Number Field

We know that since the ring $\mathbb Z [i]$ of Gaussian integers is a Principal Ideal Domain, the only integer primes which can norms of some ideal of $\mathbb Z [i]$ are those which can be expressed ...
1
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0answers
66 views

Small pants in arithmetic hyperbolic surfaces of high degree

Does the following statement hold: Statement: For any $\epsilon > 0$, there exist a number field $k$ of degree $d_{\epsilon}$ over $\mathbb{Q}$ and an arithmetic hyperbolic surface  $\Gamma$ ...
0
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1answer
64 views

Logarithms of $L$-functions of irreducible characters of Galois group

We know that the $L$ functions of Dirichlet characters $\chi$ of $(\mathbb Z / m\mathbb Z)^\times$ satisfy the property that $\log L(s, \chi)$ is holomorphic for $\Re(s) \geq 1$ if $\chi$ is a ...
3
votes
1answer
116 views

Level vs. conductor of a supercuspidal representation

What is the relation between level and conductor of a supercuspidal representation of $\operatorname{GL}_2(\mathbb{Q}_p)$ for some prime $p$? Proposition 3.4 in Loeffler and Weinstein - On the ...
2
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1answer
76 views

Existence of analytic continuation of Dirichlet series corresponding to the indicator sequence of a complement of a special multiplicative set

Let $K/ \mathbb Q $ be a finite Galois extension and let $X$ be a proper non-empty subset of the Galois group $G=Gal(K/ \mathbb Q)$ that is closed under conjugation. Consider a set of integer primes $...
1
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1answer
161 views

The existence of rational points [closed]

Solving some problem parametrically, I got the following answer: $$ \dfrac{5x}{4} + \sqrt{\dfrac{y^2}{4} - \dfrac{x^2}{16}} + \dfrac{1}{10} \sqrt{10x^2 + 9y^2} + \dfrac{1}{5} \sqrt{5x^2 + 16y^2} $$ ...
13
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0answers
142 views
+50

Galois groups of special polynomials

This question is motivated by long experiments with GAP. Call a monic polynomial with integer coefficients special in case it is irreducible and has only coefficients $-1$, $0$ or $1$. Let $n \geq 5$....
3
votes
1answer
112 views

English reference for the Brauer-Kuroda formula

I'm currently trying to understand the Brauer-Kuroda formula. Although there are many recent papers on the formula but they seem to be purely algebraic. They say that original analytic approach is ...
1
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2answers
397 views

Methods of sheaf theory for solving Diophantine equations

What are some examples of sheaf theory used to either provide solutions to Diophantine equations, or to state that no such solutions exist?
2
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0answers
115 views

Studying the vast world of Number Theory [closed]

I'm a high school student, interested in mathematics, especially in number theory. While preparing for the IMO test, and thinking about generalizations or the root of many olympiad problems led me to ...
2
votes
1answer
52 views

Extension of Dedekind domains and their codifferent

Let $A\subset B$ be a finite extension of Dedekind domains. Let $0\neq b\in B$ and $0\neq a\in A$ such that $(a)=(b)\cap A$. In particular, we have $a=b\cdot c$ for some $c\in B$. Now for any $A$-...
19
votes
2answers
515 views

Number of real roots of irreducible polynomials that are solvable by radicals

Let $n \geq 3$ be a natural number. Define the set $X_n$ as the set of natural numbers that appear as the number of real roots an irreducible polynomial of degree $n$ over $\mathbb{Q}$ which is ...
2
votes
0answers
161 views

Where is the flaw in this argument with $p$-adic extensions?

I cannot find what I am missing in the following computation. Let $K=\mathbb{Q}_p(p^{1/{(p-1)p^{\infty}}})$ and $L=\mathbb{Q}_p(\zeta_{p^{\infty}})$, where $\zeta_{p^n}$ is a primitive $p^n$-th root ...
6
votes
0answers
284 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 ...
2
votes
2answers
395 views

Natural number solutions for equations of the form $\frac{a^2}{a^2-1} \cdot \frac{b^2}{b^2-1} = \frac{c^2}{c^2-1}$

Consider the equation $$\frac{a^2}{a^2-1} \cdot \frac{b^2}{b^2-1} = \frac{c^2}{c^2-1}.$$ Of course, there are solutions to this like $(a,b,c) = (9,8,6)$. Is there any known approximation for the ...
4
votes
1answer
331 views

Do number rings tend to Z?

I want to know whether number rings tend to the integers as the discriminant tends to infinity. In detail, let $n$ be a natural number and let $C(n)$ be the set of all number fields $K$ of degree $n$. ...
2
votes
0answers
39 views

Compatible systems of $l$-adic Galois representations as sheaves

I have a simple translation question: Suppose we fix embeddings $\iota_l: \mathbb{Q} → \mathbb{Q}_l$ for every rational prime $l$, and take a finite-dimensional vector space $V$ over a number field $...
0
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0answers
52 views

properties of anti-cyclotomic extension

Let $K$ be an imaginary quadratic field, and $p$ be a primes number, there exists an unique $\mathbb{Z}_p$-extension of $K$, we denote it by $K_{p,\infty}$ such that the action of complex conjugate $c$...
1
vote
0answers
32 views

self dual character of local fields and global fields

There are two concepts of self dual character, one is for global and another is for local. Let $K$ be an imaginary quadratic number field, and a Hecke character $\chi : \mathbb{A}_K^{\times}/K^{\...
6
votes
1answer
88 views

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 ...
1
vote
0answers
74 views

Regulator of number fields of a special form

Is it possible to estimate the regulator of a number field of a special form (e.g. cyclotomic fields)? My motivation mainly comes from the investigation of the number fields $K_n=\mathbb{Q}[x]/(x^n+x+...
7
votes
0answers
189 views

Cohn's eight diophantine equations

Today I was reading J.H.E. Cohn's Eight diophantine equations (1966). On p. 158 he comes across the equation $y^2 = a^3 + 3a$ for odd values of $a$ and writes that this is equivalent to $x^3 + (x+1)^3 ...
0
votes
0answers
42 views

Commensurability of arithmetic, irreducible, nonuniform lattices

Let $n \in \mathbb{Z}_{\geq 2}$ be arbitrary. Let $r_1$ and $r_2$ be arbitrary elements of $\mathbb{Z}_{\geq 0}$ that satisfy $r_1 + r_2 > 0.$ Let $G := {\rm SL}_n(\mathbb{R})^{r_1} \times {\rm SL}...
1
vote
0answers
127 views

A structure on the groupoid of algebraic closures

Given a field $k$ let $\Omega(k)$ be the set of algebraic closures of $k$. $\Omega(k)$ is obviously a groupoid. At each element $\bar{k}$ of $\Omega(k)$ we have its automorphism group over $k$, ...
1
vote
0answers
115 views

$n$-variable polynomials modulo $p$

The Hasse-Weil bound implies that for any 2-variable polynomial $P(x,y)$, there exists approximately $p$ solutions in $\mathbb{F}_p$ of $P(x,y) \equiv a \pmod p$ for sufficiently large $p$, and any ...
8
votes
2answers
396 views

Irrationality measure of arctan(1/3)

I recently came across the concept of the irrationality measure. It really fascinated me and when I was looking for known values $\mu(x)$ for mathematical constants $x$, I also came across this paper: ...
3
votes
0answers
109 views

How does Langlands define Artin L-functions?

Let $\rho: \operatorname{Gal}(K/F) \rightarrow \operatorname{GL}_n(\mathbb C)$ be a representation for an unramified extension $K/F$ of $p$-adic fields. Let $\operatorname{Frob}_{K/F}$ be the (...
2
votes
0answers
79 views

Does Langlands use the geometric Frobenius or the classical Frobenius in his papers?

In several of Langlands' papers: Representations of Abelian Algebraic Groups, On Artin's L-functions, On the Functional Equation of Artin's L-functions, Langlands takes a finite Galois extension $K/F$ ...
4
votes
1answer
162 views

Which $p$-adic valuations of Weil numbers (that is, eigenvalues of Frobenius) are possible?

Let $C$ be a smooth projective curve over a finite field $\mathbb F_q$, $q$ is a power of the characteristic $p$. It is well-known that if $\alpha$ is an eigenvalue of Frobenius acting on $H^1_{et}(C,\...
1
vote
1answer
110 views

Factoring cyclotomic polynomials over quadratic subfield

The quadratic subfield of $\mathbb{Q}(\zeta_p)$ is given by $\mathbb{Q}(\sqrt{p^*})$, where $p^*$ is the choice of $\pm p$ which is $1$ mod $4$. By some elementary Galois theory, the cyclotomic ...
3
votes
1answer
274 views

Siegel's bad character

Let $K$ be an imaginary quadratic field with discriminant $d_K$. Suppose that $d_K=gt$, where either $g,t$ are discriminants or have the value $g=1,t=d$. Let $f$ be an additional discriminant of a ...
2
votes
0answers
91 views

The definition of Langlands' L-function $L(s,\pi,r)$ in the case of $\operatorname{GL}_1$

Let $G$ be a split reductive group over a $p$-adic local field $k$. For $\pi$ an unramified representation of $G(k)$, and $r$ a finite dimensional representation of the L-group $^LG$, Langlands ...
9
votes
3answers
507 views

Does the expression $x^4 +y^4$ take on all values in $\mathbb{Z}/p\mathbb{Z}$?

As the title asks: does there exist $N$ such that, for any prime $p$ larger than $N$, the expression $x^4 +y^4$ takes on all values in $\mathbb{Z}/p\mathbb{Z}$? I have been thinking about this ...
1
vote
0answers
143 views

Computing the kernel of some Artin-Map

let $K = \mathbb{Q}(\sqrt{-5})$ and $L = K(i)$. $\mathcal{O}_K$ is the ring of integers of K. I would like to show that the kernel of the Artin-Map $\phi_{L/K}: I_K \rightarrow Gal(L/K)$ is $P_K$, ...
7
votes
1answer
195 views

Finding $Q(\sqrt{-2})$-rational points on $X_0(33)$

Let $K = Q(\sqrt{-2})$. How can I compute the $K$-rational points on the modular curve $X_0(33)$? Recall that $X_0(33)$ is of genus $3$ and has the following affine model, $$y^2 +(-x^4-x^2-1)y = 2x^...
1
vote
0answers
116 views

Transcendance in function fields

Denote by $\Omega$ the completion of an algebraic closure of $\mathbb F_q\left(\!\left(\frac1T\right)\!\right)$ for the valuation $-\deg$. Let $(a_n)_n$ be a sequence of $\overline{\mathbb F_q(T)}\...
5
votes
1answer
280 views

A problem about real quadratic field

I have calculated some real quadratic field 's Hilbert class field with class number $2$,and I found they were satisfied $Gal(H_{K}/Q)\cong Z/2Z\oplus Z/2Z$,here $H_{K}$ is the Hilbert class field of ...
3
votes
0answers
152 views

Is the Cassels “$x - \theta$” map algebraic in some sense?

Setup: Let $k$ be a field of characteristic $0$, let $f(x) \in k[x]$ be a monic separable polynomial of degree $n \geq 4$, and let $\theta$ denote the image of $x$ under the map $k[x] \to K_f := k[x]/(...

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