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
1,999
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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 ...
- 143
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90
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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
2
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181
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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 ...
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2
votes
1
answer
156
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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 ...
- 23
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0
answers
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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 ...
- 111
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88
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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 $E$ ...
2
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0
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199
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Galois cohomology of $\breve{\mathbb Q}_p \otimes_{\mathbb Q_p} \breve{\mathbb Q}_p$
Let $\breve{\mathbb Q}_p$ denote the completion of the maximal unramified extension of $\mathbb Q_p$. I‘d like to compute Galois cohomology groups and sets related to $\breve{\mathbb Q}_p \otimes_{\...
- 191
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111
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Map between Mordell-Weil group and Ext of (Mixed) Motives
We know that the motivic cohomology of an abelian variety $A$ over a number field $k$ computes the Mordell-Weil group up to torsion, and so if we were to grant the existence and nice behaviour of ...
- 3,843
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54
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An identity for lattices in vector spaces over non-Archimedean local field
Let $V$ be a finite dimensional vector space of a non-Archimedean local field $\mathbb{F}$. Let $\Lambda\subset V$ be a lattice, i.e. an open compact $\mathcal{O}$-submodule. Let $W_1,W_2\subset V$ be ...
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What is the residue field of the integer ring of $\mathbb{C}_p$?
Fix a prime $p$. Let $\mathbb{C}_p$ be the completion of the algebraic closure of $\mathbb{Q}_p$, and $\mathcal{O}_{\mathbb{C}_p}$ the integer ring of $\mathbb{C}_p$. I know $\mathcal{O}_{\mathbb{C}_p}...
- 389
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140
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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 &...
- 163
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votes
1
answer
266
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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 ...
- 224
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1
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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 $\mathbb{K}=\mathbb{Q}(\alpha_1,\ldots,\alpha_n)$...
user500557
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1
answer
555
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Quotients of number fields by certain prime powers
I apologise in advance for what must be a naive question. Let $\mathcal O_K$ be the ring of integers of the algebraic number field $K.$ Let $p$ be a rational prime, and factorize $$(p)=\mathfrak p_1^{...
- 322
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1
answer
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Constituents of $C_0^\infty(F^\times)$ for the regular action
Let $F$ be a $p$-adic field, and $C_0^\infty(F^\times)$ the space of smooth compactly supported functions on $F^\times$. Under the regular action of $F^\times$ on $C_0^\infty(F^\times)$, I believe we ...
- 689
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Algebraic numbers with a polynomial property
In my research I faced with an intricate construction of an algebraic number with some properties.
Problem. For which classes of polynomials $P(X,Y)\in \mathbb{Z}[X,Y]$, we have the following property....
- 467
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votes
3
answers
378
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Irreducibility of polynomials over some number fields
Recently, a problem in my research has appeared and now I need to construct some algebraic numbers with special properties (related to its degree and some other fields extensions).
Now, in order to ...
- 467
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0
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An important proposition that relies on the group of fractional ideals of a Dedekind domain and localization [migrated]
Does the map from the group of fractional ideals of a Dedekind domain $A$ to the group of fractional ideals of $S^{-1}A$ where $S$ is a multiplicative subset is surjective? And what about its kernel? ...
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2
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Does $x_0=1/3$ lead to periodicity in the logistic map $x_{k+1}=4x_k(1-x_k)$?
Does $x_0=1/3$ lead to periodicity in the logistic map $x_{k+1}=4x_k(1-x_k)$?
I believe it does not, but this is equivalent to proving that $(2\pi)^{-1}\arcsin(\sqrt{1/3})$ is irrational. I am ...
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1
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Surjectivity of the norm of units in Galois extensions ramified exactly at one finite prime
Let $L/K$ be a finite Galois extension of number fields that is ramified exactly at one finite prime and is unramified at all infinite primes. Let $U_K$ and $U_L$ denote the units of the ring of ...
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Units in Abelian extensions which are not in the subgroup of cyclotomic units
This question is motivated by a Quora post and the top answer to it.
The post wishes to make explicit the difference between units in cyclotomic fields and cyclotomic units.
One problem with answering ...
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0
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97
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When is a prime considered to be ramified, split or inert in a non-maximal order of an imaginary quadratic number field?
I am reading this paper on "Averages of Elliptic curve constants" here and in section 2.2 page no. 693 the formula for the conjectural constant in the asymptotics of the Lang-Trotter ...
- 239
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votes
0
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97
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Local global principle over infinite extension of $\Bbb{Q}$ which is not algebraically closed
Let $A$ be an algebraic variety over a field $K$, which is finite extension of $ \Bbb{Q}$.
We say local global principle holds if $A(K_v) \neq \emptyset$ implies $A(K) \neq \emptyset$, where $K_v$ is ...
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158
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Maximal abelian subgroups of absolute Galois group
For both local and global fields, we have a good handle on the abelianization of the absolute Galois group of $K$. Essentially this allows us to "understand" all maps from $G_K$ to abelian ...
- 3,843
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114
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Invariants of Iwasawa modules
Let $\Lambda$ denote Iwasawa algebra $\mathbb{Z}_p[[\Gamma]]$, where $\Gamma$ is a group isomorphic to $\mathbb{Z}_p$(ring of $p$-adic integers). The structure theorem of the Iwasawa module says: If M ...
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votes
1
answer
118
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Explicit tensor product of isocrystals
Let $L$ be the completion of the maximal unramified extension of $\mathbb Q_p$. Let $\sigma$ be a topological generator for the Galois group of $L/\mathbb Q_p$. Say that $(D, \phi)$ is an isocrystal ...
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answer
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Conductor at 2 of abelian surfaces with real multiplication
Let $A/\mathbb{Q}$ be an abelian surface such that $\text{End}_{\mathbb{Q}}(A)\otimes \mathbb{Q}$ is a real quadratic field $E$. I am interested in bounding the conductor of $A$ at $2$.
Let $\mathfrak{...
- 1,309
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What is the conductor of $K(\sqrt{2})$ over $K$?
Let $ K=\Bbb Q\left(\sqrt{(-1)^\frac{N+1}{2}N}\right)$. I want to find the ray class field of $K$ containing $\sqrt{2}$. I considered $L=\Bbb Q(\sqrt{2})$. By Artin reciprocity, there exist modulus $\...
- 693
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A detail in I. Piatetski-Shapiro and S. Rallis's "Doubling paper": computing the integral on negligible orbits
I'm currently reading the paper "L-functions for the classical groups" by I. Piatetski-Shapiro and S. Rallis, where they introduced the doubling method over classical groups.
I'm confused at ...
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On the Erratum to P. Scholze's "$p$-adic Hodge theory for rigid-analytic varieties"
I am trying to understand section (3) of the Erratum to P. Scholze's "$p$-adic Hodge theory for rigid-analytic varieties" in detail. In particular, there is the following sentence on page ...
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What's the use of group cohomology for class field theory?
I'm a graduate student studying now for the first time class field theory.
It seems that how to teach class field theory is a problem over which many have already written on MathOverflow.
For example ...
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Is every prime $q$ of the form $x^2 + py^2$ for some prime $p<q$?
For every odd prime $q \geq 3$, does there exist a prime $p < q$ and integers $x,y$ such that
$$\displaystyle x^2 + py^2 = q?$$
One can easily show that all primes $q \not \equiv -1 \pmod{3}$ can ...
- 22.7k
1
vote
0
answers
104
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Integral points in smooth cubic curves
Let $X$ be a smooth affine cubic curve in $\mathbb A^2$ defined by $f(T_1,T_2)\in\mathbb Z[T_1,T_2]$ (of course $\deg(f)=3$ by definition), and
$$n(f, B)=\{(x_1,x_2)\in\mathbb Z^2| |x_1|\leq B, |x_2|\...
- 393
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1
answer
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Ramifications in Galois closures of number fields
Let $K/\mathbb{Q}$ be a finite Galois extension, and $L/K$ an infinite Galois extension such that the Galois group $ \operatorname{Gal}(L/K) $ is isomorphic to a closed subgroup of $ \operatorname{GL}...
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votes
1
answer
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Why is Fontaine's infinitesimal period ring $A_{\text{inf}}$ complete?
Fix a perfectoid field $K$ in mixed characteristic with ring of integers $\mathcal{O}$ and pseudo-uniformizer $\varpi$. Its tilt is the fraction field of $\mathcal{O}^{\flat}=\varprojlim_{x\mapsto x^{...
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votes
2
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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 \\
\...
- 200
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votes
1
answer
151
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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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vote
1
answer
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Quadratic extension of local field
Let $F$ be a nonarchimedean local field of characteristic zero, and $E$ an extension of $F$ with $[E:F]=2^n$ for some $n$. Is it always possible to find a quadratic extension $M$ of $F$ such that $F\...
- 689
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0
answers
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Minimal algebraic norm in the codifferent of a number field
Let $k$ be a number field of absolute degree $d$.
I consider the absolute codifferent $\mathfrak{d}^{-1}$, which is
the dual of the maximal order of $k$ for the form induced by the trace map.
I am ...
- 313
6
votes
0
answers
108
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Finiteness of wildly ramified cohomology
$\newcommand\p[1]{\left(#1\right)}\newcommand\Char{\operatorname{char}}\newcommand\Gal{\operatorname{Gal}}\newcommand\b[1]{\left\{#1\right\}}$
Let $K$ be a global field. All cohomology below is fppf-...
- 161
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0
answers
102
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Measure on the places of $\bar{\mathbb Q}$
Consider the set $S$ of all places of $\mathbb Q$ (i.e. the set of all absolute values up to equivalence). Then we can consider $S$ as a measure space with the counting measure $\mu$. Therefore $\mu(\{...
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The behaviour of a prime over quadratic fields
Let $d$ be a positive integer such that $-d$ is a fundamental discriminant, and let $p$ be a rational prime. It is well-known how to detect whether $p$ splits in the quadratic field $K = \mathbb{Q}(\...
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2
votes
0
answers
118
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relating class number and narrow class number of a real field
I am interested in finding out when the narrow class number of $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ is the same as the class number of $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ where $\zeta_p$ is a primitive $...
- 429
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votes
0
answers
102
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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
2
votes
0
answers
173
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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
0
votes
0
answers
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 $\mathbb F_q\left(\left(\frac1T\right)\right)$.
...
- 3,208
2
votes
1
answer
68
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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 ...
- 303
4
votes
0
answers
126
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"Reference Request" for a lecture note by C. Skinner: Galois Representations, Iwasawa Theory, and Special Values of $L$-functions
This was originally posted on math.stackexchange as https://math.stackexchange.com/questions/4589793, where I was suggested to move it here.
I'm searching a lecture note by C. Skinner named "...
- 41
4
votes
1
answer
461
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Proving $\zeta_K\left(\frac12\right)\neq 0 \implies \zeta_K'\left(\frac12\right)\neq0?$
For an algebraic number field $K$, let $\zeta_K(s)$ be the Dedekind zeta function associated to $K$, and let $\zeta_K'(s)$ be its derivative.
I believe that the following statement is true:
$$\zeta_K\...
- 67
3
votes
1
answer
252
views
Positive system of algebraic integers
Let $\mathbb{A}$ be the ring of algebraic integers. Consider a sequence $(d_i)_{i \in I}$, with $I$ a finite set and $d_i \in \mathbb{A} \cap \mathbb{R}_{\ge 1}$, such that $$d_i d_j = \sum_{k \in I} ...
- 24.4k