# 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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### Recent results about Galois groups of number fields

I recently stumbled upon The m-step solvable anabelian geometry of number fields, and I got the following question: Have there been any other significant results about Galois groups of number fields ...
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### The group PSL(2,q) [closed]

Consider the group PSL$(2, q)$, where $q\equiv 3$ or $5($mod$\ 8)$. Can we let $q=p^n$ with $p$ is a prime?
1 vote
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### Minimum possible value of $\sum_{i=1}^n \binom{x_i}{r}$

Suppose an unknown sequence $x$ with $n$ non-negative integers such that the sum of elements of that is fixed. In other words $\sum_{i=1}^n x_i = c$ for some constant $c$. What is the minimum possible ...
156 views

### Minimum possible value of $\sum_{i=1}^n x_i(x_i-1)(x_i-2)$ for fixed sum

Suppose $x$ is a unknown sequence of non-negative integers with $n$ elements and $\sum_{i=1}^n x_i = c$ for a constant $c$. What is the minimum possible value of $\sum_{i=1}^n x_i(x_i-1)(x_i-2)$? I ...
1 vote
23 views

### Natural Density of Norms of ideals in a given ideal class

Some time ago, Landau proved the following formula for general number fields: $I_K(x)=U_Kx+O(x^{\delta})$, where $\delta=1-2/(1+[K:\mathbb{Q}])$, where $I_K(x)$ is the number of ideals with norm below ...
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### Irreducibility of the residual p-adic representation attached to an elliptic curve

I am trying to understand the proof of Serre of the irreducibility of the residual representation of an elliptic curve (Frey curve) $E$ when $p \geq 5$ as follows; Suppose it is reducible. Then $E$ ...
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### English references for SGA 7 chapter I

I am struggling to read SGA 7 chapter I mostly because it is in French. Actually my aim is to understand the theorem 6.1 and the proof of it. I noticed that the author sometimes refers to the book &...
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### “Sheaf cohomology” of Galois groups

Apparently the Galois group $G$ of a Galois extension $E/F$ can be viewed as an “étale sheaf” on the set $X$ of intermediate Galois extensions equipped with an appropriate Grothendieck topology (see ...
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### How do I extend the $2$-adic absolute value to prove Monsky's Theorem?

In proving Monsky's Theorem, it is required that we define the $2$-adic absolute value on an arbitrary finitely generated extension of $\mathbb{Q}$ say $\mathbb{K}=\mathbb{Q}(\alpha_1,\ldots,\alpha_n)$... 555 views

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### One question on linear combinations of roots of unity

For $n \geq 1$, I want to find all solutions $x_i$ of the equation \begin{equation} \begin{array}l x_i \in \mathbb{Z}, i=0,1,2\dotsc,n-1 \\ x_i^2 = 1, i=0,1,2\dotsc,n-1 \\ \...
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### Class numbers of cyclotomic fields and their maximal totally real subfields

Let $\zeta_p$ be a $p$-th root of unity for a prime $p$, let $L:=\mathbb{Q}(\zeta_p)$ and $K$ the maximal totally real subfield of $L$, i.e. $K:=\mathbb{Q}(\zeta_p+\zeta_p^{-1})$. I am trying to prove ...
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1 vote
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### Embedding number fields in fields with class number prime to $p$

Let $p$ be a fixed prime. Question: For any number field $K$, is there always a finite extension $L$ of $K$ of $p$-power order such that the class number of $L$ is prime to $p$? Moreover, for any ...
• 307
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### Basic question on the Langlands conjectures for $GL_n$ over global field of positive characteristic

My field is far from the Langlands conjectures. I am just trying to understand some basic ideas. At the moment I am interested in a global field $K$ of positive characteristic and the group $G=GL_n$. ...
• 20k
51 views

### Continuous morphism in function fields with extra conditions

Let $\Omega$ be the completion of $\overline{\mathbb F_q\left(\left(\frac1T\right)\right)}$ for the topology induced by the valuation $-\deg$ on $\mathbb F_q\left(\left(\frac1T\right)\right)$. ...
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### Ramification of primes and order of $\smash{\hat{H}}^0$ in ray class fields with one finite prime divisor

Let $K$ be a number field, $\mathfrak{p}$ be a prime of it, and $L=K(\mathfrak{p}^n)$ be the ray class field of $K$ with finite conductor $\mathfrak{p}^n$ (we do not care about the infinite part of ...