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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Central division algebras over $ \mathbb{Q} $

Quaternions over $ \mathbb{Q} $ are an example of a Central Division algebra over $\mathbb{Q} $ for which the basis elements $\{ i,j,ij \} $ other than $1$ are represented by skew-symmetric matrices ...
4
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1answer
212 views

Torsion points on $E/\mathbb{Q}$ with large coordinates

Let $E/\mathbb{Q}$ be an elliptic curve with finitely many rational points. What are some examples where at least one rational point has large coordinates (compared to the height of $E$)?
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1answer
58 views

How to compute the intersection of an ideal with the maximal order of a subfield?

I asked this earlier on math.stackexchange but I think this is a better place for this question. Computing the intersection of ideals belonging to the same maximal order of a number field $K$ can be ...
9
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1answer
446 views

Absolute values of non-unit algebraic integers at all infinite places

Let $K$ be a number field and $O_K$ its ring of integers. Given any non-unit element $\alpha\in O_K$, does there exist a unit $u\in O^\times_K$ such that $$ |\sigma(u \alpha)| >1 \quad \text{ for ...
4
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0answers
146 views

Is every Dedekind domain the integral closure of some principal ideal domain?

I mean that $B$ is a Dedekind domain with fraction field $L$, which is a finite separable extension of a field $K$ that is the fraction field of a PID $A$ such that $B$ is the integral closure of $A$ ...
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114 views

Reducibility of a cubic over a number field

Given an extension $K = \mathbb{Q}(\alpha)$ by a cubic polynomial $g(x)\in \mathbb{Q}[x]$ (not necessarily Galois extension) is there a criterion for a cubic polynomial $f(x) \in K[x]$ to be reducible ...
2
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125 views

Construction of genus class fields

Given a finite extension $K/\mathbb{Q}$, the genus class field $L$ is defined to be the maximal abelian extension of $\mathbb{Q}$ that is a subfield of the Hilbert class field $H$ of $K$. I am trying ...
4
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4answers
456 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{\...
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0answers
67 views

Norm groups of number fields

I came across this proposition in an article about genus class fields. I have a few questions about the parts that I have underlined in red. I don't understand why the norm map $N_{H/K}: I_H \to P_K$ ...
3
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1answer
118 views

Centralizer of the absolute Galois group of a number field

By this answer, we know that if $K/\mathbb{Q}_p$ is a finite extension, the centralizer of $G_K$ in $G_{\mathbb{Q}_p}$ is trivial. The argument there uses that the abelinization of $G_K$ is the pro-...
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0answers
125 views

For any $n-1$ elements of $\mathbb Z/n\mathbb Z$, we can make $0$ using $\{-,+,\times\}$ without parentheses

MSE: Just using $+$ ,$-$, $\times$ using any given n-1 integers, can we make a number divisible by n? (no brackets allowed) Is there any hope in proving the following? (Cross-posted here after a ...
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169 views

Explicit computations of the fundamental groups of perfectoid spaces

If $X$ is a perfectoid space then it has the same étale site as its tilt $X^\flat$. This means that the fundamental groups (suitably defined) of $X$ and $X^\flat$ are isomorphic. Can you give ...
4
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1answer
371 views

is every positive real cyclotomic number the norm of a cyclotomic?

Let $a>0$ be a real cyclotomic number. Is it always possible to solve in cyclotomics the equation $X\overline{X}=a$ ? Equivalently, one might want to express $a$ as a sum of squares of two real ...
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155 views

A question on the Bombieri-Lang conjecture

Let $X$ be a variety of general type, defined over a number field $K$. Then the Bombieri-Lang conjecture asserts that the set of rational points $X(K)$ (or $X(L)$ for any finite extension $L/K$) is ...
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78 views

Topologically finitely generated absolute Galois group is topologically finitely presented

If an absolute Galois group is topologically finitely generated is it topologically finitely presented?
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88 views

Topologically finitely generated non-abelian isomorphic absolute Galois groups

Let $K$ be a field of positive characteristic and $L$ be a field of characteristic zero. Assume the absolute Galois groups of $K$ and $L$ are topologically finitely generated, non-abelian and ...
4
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1answer
151 views

Non-abelian isomorphic absolute Galois groups of fields of different characteristic

Let $K$ be a field of positive characteristic and $L$ be a field of characteristic zero. Assume the absolute Galois groups of $K$ and $L$ are non-abelian and isomorphic as profinite groups. Can $L$ ...
5
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0answers
152 views

Goldfeld resolution of the quadratic class number problem

Goldfeld proved the following result. Let $E$ be an elliptic curve (with conductor $N$) over $\mathbb{Q}$ whose Hasse-Weil L-function has a zero at $s = 1$ with multiplicity $g$ then for sufficiently ...
4
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1answer
220 views

Can every square root be represented as a linear combination on roots of unity? [duplicate]

Messing around, I noticed that $$\sqrt{2}=e^{i\pi /4}+e^{-i\pi /4}$$ $$\sqrt{3}=e^{i\pi /6}+e^{-i\pi /6}$$ and (even more surprisingly) $$\sqrt{5}=e^{2\pi i/5}-e^{4\pi i /5}-e^{6\pi i /5}+e^{8 \pi i /...
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0answers
130 views

Reference Request - New proof of Ribet's level lowering by Khare and Wintenberger

I'm currently following the note of Sug Woo Shin's course at Berkeley with notes taken by Rong Zhou. In Section 24.3 (Page 86), Ribet's level lowering theorem is stated: [Theorem 24.7] $E = E_{a^{\...
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2answers
157 views

Galois stable elements of the Picard group of a curve and the rational divisors

Let $C$ be a (smooth,proper) curve over a field $k$. Let $\operatorname{Div}_C(k)$ be the free abelian group generated by the closed points of $C/k$ and $k(C)^\times$ be the group of rational ...
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1answer
107 views

Solving the inequality between a and b [closed]

I run into this inequality $$ (a + b)^{1 - \epsilon} \;a < b $$ where $a \in \mathbb{Z}^+$ and $\epsilon \in (0, 1)$. What value (w.r.t $a$ and $\epsilon$) should I set $b$ equal to such that this ...
6
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0answers
229 views

Rational point oracle for smooth projective varieties

You have an oracle that decides if a smooth projective variety over $\mathbb{Q}$ has a $\mathbb{Q}$-point. Can you then algorithmically decide if an arbitrary variety over $\mathbb{Q}$ has a $\mathbb{...
1
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1answer
195 views

Elliptic curve over $\mathbb{Q}(T)$ receiving only constant maps from modular curves

Is there an elliptic curve $E/\mathbb{Q}(T)$ such that any map $X_1(N)\to E$ for any $N>0$ is constant?
3
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0answers
130 views

Are class numbers of number fields with prime degree often $1$?

I have taken a look at the class number statistics of the L-functions and Modular Forms Database: https://www.lmfdb.org/NumberField/stats, table "Distribution by class number". It appears ...
5
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1answer
283 views

Explicit construction of division algebras of degree 3 over $\mathbb{Q}$

In his book Introduction to arithmetic groups, Dave Witte Morris implicitly gives a construction of central division algebras of degree 3 over $\mathbb{Q}$ in Proposition 6.7.4. More precisely, let $L/...
5
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0answers
169 views

Maximum value of newform from Galois representation

One can attach $\ell$-adic Galois representations to holomorphic cuspidal newforms of weight $2$ on the upper half-plane. If a newform is $L^2$-normalized, can one extract its maximum value from the ...
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0answers
85 views

Powers of automorphic Eisenstein series

Let $G$ be a reductive group defined over $\mathbb{Q}$. Let $P$ be a standard parabolic subgroup of $G$ with Levi decomposition $$P = MN.$$ We denote by $R_{disc,M}$ the discrete spectrum of $M$. Let $...
9
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0answers
276 views

Commutative algebra details on patching when proving $R = \mathbb{T}$ theorem (Calegari-Geraghty Paper)

I have originally posted this on math.SE and been suggested to post this here. I'm merely an undergraduate student and it is the first time for me to ask questions here. I'm sincerely sorry if these ...
4
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1answer
162 views

Computing the genus of certain ternary indefinite lattices

For $k$ a positive integer, let us consider the rank 3 ternary indefinite lattice $L=L_k$ with quadratic form $$6kx^2-2(y^2+yz+z^2).$$ Its discriminant group has length $2$. Question. Is this lattice ...
3
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2answers
277 views

Dedekind Zeta functions of Biquadratic fields

Let $F/ \mathbb{Q}$ be a biquadratic field of Galois group $C_2 \times C_2$. Then I know that the Dedekind Zeta function of $F$ can be factored into $L$-functions as; $$\zeta_F(s) = \zeta(s) L(s, \...
2
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0answers
74 views

Weierstrass points in Artin-Schreier extensions of the rational function field in characteristic $p$

I encountered a problem when proving a result regarding the orders at non-Weierstrass points ($n$ is an order of the point $P$ if there is some holomorphic differential $\omega$ with order $n$ at $P$) ...
2
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0answers
100 views

Has there been much research on the Iwasawa theory of bi-quadratic fields?

The simplest non-trivial example of $\mathbb{Z}_p$-extensions happen over imaginary quadratic fields and I've seen a good amount of research done here (from my understanding still a lot to learn). I ...
16
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1answer
527 views

Principal ideal domains with finitely many units

Question: What are the (in characteristic 0 if needed) principal ideal domains that have finitely many units? Can such rings be classified? (This is a more specialised version of the question in ...
6
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0answers
298 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: ...
2
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0answers
253 views

Hard “one-dimensional” cases of Fermat's Last Theorem

Are there computable functions $p, q, r:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ and an unbounded computable function $s:\mathbb{Z}_{\geq 1}\to \mathbb{Z}_{\geq 3}$ such that the statement For all $...
3
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0answers
186 views

Why are critical points important for dynamical systems?

I have just started reading a little about (arithmetic) dynamics and it seems like critical points are very important - for instance, rational maps so that critical points have finite forward orbit (...
3
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1answer
155 views

Absolute Galois group with unique closed non-open subgroup

Is there an absolute Galois group that is not a subgroup of $\hat{\mathbb{Z}}$ and that has one and only one closed non-open subgroup?
0
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1answer
152 views

Faithful irreducible representations of the dihedral group $D_{2p}$, of dimension at most $p-1$

I am curious about the irreducible representations $\rho: D_{2p} \rightarrow GL_n(\mathbb{Q})$ of dimension at most $p-1$, not the real or complex representations. My mind is occupied with these two ...
1
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1answer
122 views

“Violent” field isomorphisms and $p$-adic “degrees” of complex numbers

It is known that if $\mathbb{F}, \mathbb{G}$ are two algebraically closed fields with characteristic zero and equal cardinality, then $\mathbb{F}$ and $\mathbb{G}$ are isomorphic as fields. If $\...
3
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1answer
116 views

Connecting different ways of constructing cubic extensions of $\mathbb{Q}$

There are at least two ways to construct cyclic cubic extensions of $\mathbb{Q}$ as explained below. (A third one is given in the answer to an earlier question). Given $A, B, C$ integers with $A\neq ...
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0answers
120 views

$\frac{1}{\pi} \int_{0}^\infty \frac{\log|\left(\frac{1}{2}+it\right)\zeta\left(\frac{1}{2}+it\right)|}{\frac{1}{4}+t^2}dt $

Consider,$$I=\frac{1}{\pi} \int_{0}^\infty \frac{\log|\left(\frac{1}{2}+it\right)\zeta\left(\frac{1}{2}+it\right)|}{\frac{1}{4}+t^2}dt $$ where $\zeta$ is the Riemann Zeta function. Since , Hardy (...
5
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2answers
323 views

Why are these sets divisible by n?

Suppose we have a polynomial $z \to f_c(z)$ defined over $\mathbb Z$ with a free parameter $c$, for instance $z \to z^2 + c$ and we consider the iterates $z \to f_c^{(n)}(z)$ and define the ...
31
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6answers
3k views

Using algebraic geometry to understand class field theory

In Algebraic Number Theory, S. Lang says "[a geometrical approach] allows one to have a much clearer insight into the whole class field theory, since the existence theorem and the reciprocity law ...
22
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1answer
684 views

Why is the regulator of the degree 11 extension of $\mathbb{Q}$ so large?

Let $K_\infty/\mathbb{Q}$ denote the $\hat{\mathbb{Z}}$-extension of $\mathbb{Q}$. Then for each $n\geq1$, $K_\infty$ has a unique subfield $K_n$ of degree $n$ over $\mathbb{Q}$. The fields $K_n$ are ...
2
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0answers
145 views

Generalizations of Artin–Verdier duality?

Constructible étale abelian sheafs on $Spec\ O_\mathbb K$, for number fields $\mathbb K$, satisfy Artin-Verdier duality. Are there known any algebraic schemes or algebraic stacks, other than $Spec\ O_\...
10
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1answer
443 views

How Dirichlet proved Dirichlet's unit theorem for general number fields?

For a general number field $K$, Dirichlet's unit theorem states that the unit group of the ring of integers of $K$ is a finitely generated group of rank $r_1+r_2-1$. It seems that standard algebraic ...
5
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0answers
134 views

Truncation and weighted orbital integrals in hyperbolic term of trace formula for $\mathrm{GL}(2)$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PGL{PGL}$I am looking at Gelbart--Jacquet's article in the first Corvallis volume (the article entitled Forms of $\GL(2)$ from an analytic point of ...
1
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0answers
94 views

What conditions on primes makes it non principal?

Let $K$ be a number field and $\mathfrak{p}$ is prime of $K$ what condition I can have on $\mathfrak{p}$ so it becomes non-principal ideal?
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0answers
79 views

How to determine if a unramifed prime split or not?

Let $K$ be the Number field and $L$ be finite extension where $\mathfrak{p}$ prime of K is unramified. Are there any conditions on $\mathfrak{p}$ so that I can say $\mathfrak{p}$ splits completely in ...

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