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
2,171
questions
3
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A question concerning the paper "Safarevic's theorem on solvable groups as Galois groups"
Theorem 15 in the paper Safarevic's theorem on solvable groups as Galois groups by Schmidt and Wingberg.
The proof has four steps.
In the fourth step, $x$ is a class in $H^1(k_S|K,\mathcal E(n, ν))$, ...
- 39
2
votes
1
answer
190
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Irreducible components of a cyclic extension over $ \mathbb{Q} $
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gal{Gal}$Let $ L $ be a cyclic Galois extension of $ \mathbb{Q} $ of degree $ 6 $. So $ G = \Gal(L/\mathbb{Q}) $ is a cyclic group of order $ 6 $. Then ...
- 913
-1
votes
1
answer
327
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On pi being transcendental [closed]
There are probably 101 reasons why this argument is plain wrong.
At the same time however there probably is a subtle truth to it I imagine:
We note that $\sin(x) \in \mathbb{Q}[[x]]$
Suppose $\pi$ is ...
- 113
2
votes
0
answers
246
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Is the absolute Galois group $\text{Gal}(\bar K/K)$ isomorphic to $\text{Gal}(K(S)/K)$?
Let $K$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}$, maximal ideal $\mathfrak{m}$ and uniformizer $\pi$. Let $\bar K$ be the algebraic closure of $K$ and $\bar{\...
- 870
0
votes
0
answers
124
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Denesting of sum of multiple square roots
It was well known that a nested radical with form $\sqrt{a+b\sqrt{q}}$ can be "denested" if and only if either $a^2-b^2q$ or $q(b^2q-a^2)$ is a square.
However, I can't make any ...
- 153
7
votes
1
answer
451
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When must a set of sections which is Zariski dense in the generic fiber also be dense in some special fiber?
Let $f : X\rightarrow S$ be a flat finite type morphism of schemes with $S$ integral and Noetherian. Let $\eta\in S$ be the generic point.
Let $\{\sigma_i\}$ be a collection of sections of $f$ (...
- 7,161
0
votes
1
answer
144
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Analogues of an identity involving quadratic characters
Let $d$ be a positive integer, and suppose $c$ is an integer such that $\gcd(c,d) = 1$. Then the following identity holds:
$$\displaystyle \left \lvert \{b \pmod{d} : b^2 \equiv c \pmod{d} \}\right \...
- 25.5k
0
votes
0
answers
270
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Does $abc$ preclude very smooth solutions?
Recall the $abc$-conjecture, which asserts that for any $\epsilon > 0$ there exists a positive number $C(\epsilon)$ such that for any coprime integers $a,b,c$ with $a + b = c$ and $\max\{|a|, |b|, |...
- 25.5k
6
votes
0
answers
400
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Conditions under which an $\eta$-quotient becomes a **weak** modular form (reference request for theorems similar to Ligozat's theorem)
For any $z \in \mathcal{H}$, let $q = e^{2\pi iz}$; and the eta function is defined as
${\displaystyle \eta (q)
=q^{\frac {1}{24}}\prod _{n=1}^{\infty }\left(1-q^{n}\right).}$
By an $\eta$-quotient ...
5
votes
0
answers
76
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Conjugacy classes in normalized unit group of a group ring
Let $V(FA_4)$ be the normalized unit group of the group ring $FA_4$, where $F$ is the field containing 4 elements and $A_4$ is the alternating group on 4 symbols. How can I find conjugacy classes of ...
- 381
5
votes
0
answers
270
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Exactness of a term after taking Pontryagin dual: a step in the proof of Poitou-Tate duality
I'm reading the proof of Poitou-Tate duality in the book Galois Cohomology and Class Field Theory by David Harari.
After some arguments, we get a exact sequence
$$
\mathbf{P}^1_S(k,M^{'})^* \...
- 151
2
votes
1
answer
157
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On some claims on cyclic modules over Hecke algebra used in Serre's "Quelques applications du théorème de densité de Chebotarev"
I have been reading section 7 of Serre's "Quelques applications du théorème de densité de Chebotarev" (http://www.numdam.org/item/PMIHES_1981__54__123_0/), and in particular have been trying ...
- 719
2
votes
0
answers
235
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Correspondence between class group of binary quadratic forms and the narrow class group via Dirichlet composition: an elementary approach?
I have been trying to explore and learn about connections between the form class group and the ideal class group. To be on the same page, we define the form class group of a negative discriminant $D \...
- 719
2
votes
0
answers
242
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Ambiguity about the exact definition of coefficients of modular forms
You can see the parts after my questions in the boxes. I received the answer to my first question in the comments.
I am confused about the definition of $a_n$ and $b_n$ in Part II below. I know the ...
2
votes
0
answers
119
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Finding elements in a real extension of $\mathbb{Q}$ that are close to some number in $\mathbb{R}$
Let's consider a set of numbers that one knows to high precision, and one knows or has a strong suspicion that `exact versions of these numbers' (see below) belong to a real extension of $\mathbb{Q}$. ...
- 361
3
votes
0
answers
181
views
Decomposition of primes in cyclotomic extensions and their ramifications
Let $p$ be a prime. Suppose $L$ is a degree $p$ Galois extension over a number field $K$. Suppose $p$ splits both in $K$ and $L$.
So there will be $[K:\mathbb{Q}]$ primes of $K$ over $p$. Call them $...
- 313
3
votes
0
answers
226
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Difficulty about Jordan decomposition, (and also an ambiguity about the quadratic forms in indecomposable Jordan components of quadratic modules)
I am trying to understand a concept through solving some exercises, but I can't solve one of them, and I need a hint and guide.
I asked my questions in the boxes (See the end of this question). (I ...
4
votes
0
answers
279
views
Explicit Chebotarev in function fields
Let $K/\mathbb F_q(T)$ be a finite Galoisian extension of degree $d$ and $n\in\mathbb N$. Does one have a completely explicit bound on the number of irreducible $P\in\mathbb F_q[T]$ of degree $\le n$ ...
- 3,739
1
vote
0
answers
231
views
Globalization of a local field
I am reading the paper ''Endoscopic classification of representations of quasi-split unitary groups'' by Chung Pang Mok, and cannot come up with the proof of theorem 7.2.1.
Here is the statement.
...
- 347
3
votes
1
answer
460
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Algebraic numbers in all $\mathbb Q_p$ [duplicate]
Do there exist non-rational algebraic numbers that belong to $\mathbb Q_p$ for all prime $p$? If yes, can one characterize them?
I spent several days for the first question, and I found nothing. The ...
- 3,739
3
votes
0
answers
169
views
Characterizing polynomials which behave like a logarithm modulo $1$
This is a version of a question I asked in Math StackExchange about two weeks ago, which is still unanswered. (UPDATE: the original question for $R=\mathbb{Z}$ has been finally answered, but its ...
- 976
1
vote
0
answers
120
views
Galois extension of $ C_{k} $ field and irreducible polynomial over $ C_{k} $ field
A field is called a $ C_{k}$- field if every form i.e. every homogeneous polynomial of degree $ d $ in $ n > d^{k} $ indeterminates has a nontrivial zero. We know that any finite field is a $ C_{...
- 913
15
votes
0
answers
520
views
Does the $\mathbb{F}_1$ point of view lead to any testable predictions?
In number theory we can informally consider number rings as curves over something like a field with one element. For example it is mentioned here by Kedlaya.
The question is does this perspective lead ...
- 441
4
votes
1
answer
275
views
A condition such that $p\mid\sum_{f(\theta)=0}\theta^n$ for all $n$?
If $f$ is any monic polynomial/$\mathbb{Z}$ with non-zero constant coefficient. I wish to study the quantities
$$t_n=\sum_{i}\theta_i^n\in\mathbb{Z}$$
where $(\theta_i)_{i=1}^{d}$ are the roots of $f$ ...
- 2,817
3
votes
1
answer
225
views
base change of adele rings
I read Neukirch’s book “Algebraic Number Theory”, and its remark following to proposition VI.2.3, there is an assertion that natural map $\mathbb{A}_K \otimes_K L \to \mathbb{A}_L$ is isomorphism. How ...
- 65
3
votes
1
answer
434
views
Topology of multiplication groups of local fields
In Neukirch’s book “Algebraic Number Theory”, Proposition II.5.7, the following is insisted: for a mixed characteristic local field $K$ with a residue field $\mathbb{F}_q$, $q = p^f$, then one has an ...
- 65
2
votes
0
answers
264
views
Generalized Siegel Weil formula
I am studying the following Poincare-like series,
\begin{equation}
F_k(\tau,\bar{\tau})=\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma}\sqrt{\text{Im}\gamma\tau}(q_{\gamma}\bar{q}_{\gamma})^k,
\end{...
- 103
4
votes
1
answer
478
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The real part of the period of an elliptic curve
Let $E$ be an elliptic curve over $\mathbf{Q}$. Then we can base-change $E$ to $\mathbf{C}$ and apply the uniformization theorem to obtain:
$$E(\mathbf{C}) \cong \mathbf{C}/(\mathbf{Z} + \mathbf{Z} \...
- 1,869
1
vote
0
answers
187
views
bound norm of algebraic integers in cyclotomic field
Let $\zeta$ be the $p$th root of unity, with $p$ an odd prime number.
Let $\mathbb{Q}(\zeta)$ be the $p$th cyclotomic field and let $\mathcal{O}=\mathbb{Z}(\zeta)$ the ring of integers of $\mathbb{Q}(\...
- 19
1
vote
0
answers
283
views
Number of roots over the rationals of a multivariate polynomial
Let $P(x_1,\dots,x_m)$ be a polynomial with $N$ roots over the rationals. If $N$ is finite, is there a known upper bound on $N$ in terms of $m$ and the degree $d$ of the polynomial? For $m=1$, an ...
- 1,159
7
votes
1
answer
316
views
Explicit cocycles for the first Galois cohomology of a $p$-adic torus
Let $K$ be a $p$-adic field (a finite extension of the field of $p$-adic numbers ${\mathbb Q}_p$).
Let $T$ be a $K$-torus with character group $X={\sf X}^*(T)$ and cocharacter group $Y={\sf X}_*(T)=X^\...
- 13.6k
6
votes
1
answer
273
views
Existence of genus 0 solution for linear ordinary differential equation
This question is about the linear differential equations with polynomial coefficients. I am interested in the necessary and sufficient conditions for the existence of genus 0 for linear differential ...
- 685
8
votes
0
answers
232
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Hilbert 10th problem for genus 2 equations
Hilbert 10th problem, while undecidable in general, remains open for 2-variable equations: we do not know if there is an algorithm that, for polynomial $P(x,y)$ with integer coefficients, decides ...
- 5,957
1
vote
0
answers
125
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What is the preimage of the maximal ideal under certain exponential functions?
I'm taking a shot in the dark with this question, so I apologize if it makes no sense.
Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $K_n$ be the field obtained by adjoining the $n$-th ...
6
votes
0
answers
197
views
$\mathbb{Z}$-points in a given $\widehat{\mathbb{Z}}$-isomorphism class
Given a finite type $\mathbb{Z}$-scheme $X$ with $X(\widehat{\mathbb{Z}})\neq\emptyset$ can we find a finite type $\mathbb{Z}$-scheme $Y$ with $X\times \widehat{\mathbb{Z}}\cong Y\times\widehat{\...
user409739
2
votes
1
answer
399
views
Why geometric generic point (in abstract algebraic geometry) replace general points in the unit disk?
In section 4.1, chapter 4 of Pierre Deligne's paper La conjecture de Weil : I (french version, translation to English) he states:
On $\mathbb{C}$ Lefshietz local results are as follows. Let $X$ be a ...
- 519
2
votes
0
answers
311
views
Existence of "nth root function" which is analytic
Let $K$ be a finite extension of $Q_p$. Let $q$ be the size of the residue field of $K$, and let $\pi$ be a uniformizer of $K$. Then $q/\pi$ is some power of $\pi$ up to a unit $u$ in $K$, say $q/\pi =...
3
votes
0
answers
98
views
LCM for Pochhammer symbols
For $x\in\mathbb C$ and $n\in\mathbb N_0$, one defines the Pochhammer symbol $(x,n)$ by $(x,n)=\frac{\Gamma(x+n)}{\Gamma(n)}=x(x+1)\cdots(x+n-1)$. For $\alpha\in\overline{\mathbb Q}\setminus\mathbb Z^-...
- 3,739
2
votes
0
answers
137
views
Dirichlet unit theorem for finite rings
Let us fix a square free positive integer $n\in\mathbb{N}$ and consider the number field $\mathbb{Q}(\sqrt n)$ with ring of integers $K=\mathbb{Z}[\sqrt n]$. Let us denote the Galois norm of elements ...
1
vote
0
answers
120
views
Topology of direct product of $ F_{p}$
Let's assume that $F$ is a number field, $R^{*}$ =$\prod_{P\in S}$$F_{P}$, $R$ is the Adele ring of $F$ and
$$
R' = \prod_{P\in S_{0}}\mathfrak{o}_{\mathfrak{p}}\times\prod_{P\in S_{\infty}}F_{P}.$$
...
- 89
3
votes
1
answer
114
views
Can we construct composite Fermat pseudoprimes to integral algebraic bases?
Let $0\neq \beta\in\overline{\mathbb{Z}}$ and let $n$ be a positive integer coprime to $N_{\mathbb{Q}(\beta)/\mathbb{Q}}(\beta)$. Say that $n$ is a Fermat pseudoprime to base $\beta$ if
$$\beta^{n^{[\...
- 458
1
vote
0
answers
129
views
Solution of an equation in cyclotomic extension over $ \mathbb{Q} $ of degree $6$
Let us consider a primitive $7^{\text{th}}$ root of unity $\eta$. Then the minimal polynomial of $ \eta $ over $ \mathbb{Q} $ is $1 + \eta +.....+ \eta^{6}$. So the dimension of the $\mathbb{Q}$-...
- 913
1
vote
0
answers
170
views
Does Lemma 5.4 in Deligne's Ramanujan paper generalize to Shimura varieties of PEL type?
It is generally not known if a smooth variety over a perfect field embeds into a smooth proper variety.
Lemma 5.4 in Formes modulaires et représentations $\ell$-adiques provides such an embedding for ...
- 31
5
votes
1
answer
184
views
Can Frobenius traces jump like crazy in non-geometric Galois representations?
If I have a continuous representation $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \mathrm{GL}_n(\mathbb{Q}_l)$ ramified at finitely many places how can the Frobenius traces behave?
Assuming ...
- 61
6
votes
2
answers
262
views
Curve with a rational point but no new points in number fields of low degree
Given an integer $d\geq 2$ is there an algebraic curve $C/\mathbb{Q}$ with $C(\mathbb{Q})\neq\emptyset$ and the natural map $C(\mathbb{Q})\to C(F)$ bijective for all number fields of degree at most $d$...
- 61
2
votes
0
answers
121
views
Conditions for being an entry in a trace compatible sequence
$\DeclareMathOperator\Tr{Tr}$Let $K$ be a local field and let $q$ be the size of the residue field of $K$. $\pi$ will be a uniformizer of $K$. Let $f(X) = \pi X + X^q$. Then there is a unique formal ...
3
votes
1
answer
155
views
Fiberwise isomorphism of number rings $\mathcal{O}_E\otimes \mathbb{F}_p\cong \mathcal{O}_F\otimes \mathbb{F}_p$
Are there two different number fields $E$ and $F$ such that $\mathcal{O}_E\otimes \mathbb{F}_p\cong \mathcal{O}_F\otimes \mathbb{F}_p$ for all primes $p$?
- 325
1
vote
0
answers
90
views
Number of points in number fields on curve of genus at least 2
A smooth projective curve over $\mathbb{Q}$ of genus at least 2 has finitely many $K$-points for any number field $K$.
What functions from number fields to $\mathbb{N}$ come up as the number of points ...
- 11
2
votes
0
answers
84
views
Tate module whose maximal semisimple subrepresentation is a line
An abelian variety over $\mathbb{Q}_p$ is cool if the maximal semisimple subrepresentation of its Tate module is a line.
Are there cool abelian varieties of arbitrarily high dimension? What about the ...
user371522
1
vote
0
answers
192
views
Fields such that every finite Galois extension is solvable
What are the fields such that every finite Galois extension is solvable?
We have algebraically closed fields, real closed fields, p-adic fields. Anything else?
A more pointed question after comments:
...
- 21