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, Homogeneous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces
2,421 questions
18
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2
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915
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Irreducibility of a family of integer polynomials
"Let $f(x)$ be a polynomial of degree at least 2 with $f(\mathbb{N})\subset \mathbb{N}$. Then the set of natural numbers $n$ such that $f(x)-n$ is reducible over $\mathbb{Q}$ has density 0."
...
- 345
12
votes
2
answers
1k
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Using calculus to prove the fundamental theorem of arithmetic?
How can one use calculus to prove the fundamental theorem of arithmetic?
The late John Conway was known for many proofs, among these is one of the standard lemma on algebraic integers that asserts ...
7
votes
3
answers
1k
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Relationship between algebraic number theory and analytic number theory
I’m curious about the relationship between algebraic number theory and analytic number theory from the following point of view:
Is it common for the two fields to have joint conferences?
Is it ...
7
votes
3
answers
552
views
Integer solutions of $1+x+x^2+x^3=y^2$?
Long time ago, I saw this problem in the section “for junior kids” of the
Kvant magazine, where it was asked in which base system $1111$ is a perfect square (binary, decimal, etc.). Naturally this ...
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16
votes
1
answer
427
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Is the topology on the adeles (of the rationals, say) given by a metric analogous to the Fréchet $C^\infty$ metric?
In a discussion with Kevin Buzzard it arose that the adeles of $\mathbb{Q}$ are a Polish space, and before it was realised how easy this was to see (they are a locally compact Hausdorff second ...
- 37k
6
votes
1
answer
230
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Global to local solutions of $x^n-y=0$
Let $n,y$ be positive integers that are greater than 1. Suppose that $x^n=y$ does not have a solution in positive integers. Then is it true that for infinitely many primes $p$ there are no solutions $...
- 345
7
votes
1
answer
330
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On an interesting product involving Jacobi sums over a finite field
$\newcommand\Legendre{\genfrac(){}{}}$Let $p$ be an odd prime, $\mathbb{F}_p$ be the finite field with $p$ elements and $\mathbb{F}_p^*=\mathbb{F}_p\setminus\{0\}$ be the multiplicative group of all ...
- 169
2
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0
answers
144
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Can a holomorphic cusp form become a CM form？
Given a holomorphic cusp form $f$ with weight $k \geq 2$, level $N$ and trivial nebentypus. I am wondering if $f$ can be a CM (dihedral) cusp form, i.e. $f$ is isomorphic to its quadratic twist. ...
- 479
17
votes
0
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670
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Dirichlet series that gives power of $\pi$ at positive even integer
Let $f(s)$ be a Dirichlet series with algebraic coefficients, and suppose that:
it admits a meromorphic continuation to $\mathbb{C}$;
there exists $d \in \mathbb{N}$ such that $f(2n) \in
\...
- 1,073
4
votes
1
answer
237
views
Number field with rational real points
Consider a number field $K$. How to classify or characterize those $K$ intersecting the real line only in the rationals: $K\cap \mathbb{R}=\mathbb{Q}$ ?
An example are quadratic imaginary fields.
In ...
- 43
0
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0
answers
50
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Linear-Disjointness of the field obtained upon iterated pre-images
Suppose $K$ is a number field and we have finitely many elements $\alpha_{1}, \ldots, \alpha_{k} \in K$ and a polynomial $f(x)\in K[x]$ of degree $\geq 2$. Given $m \geq 1$, we define $K_{m}(\alpha_{i}...
- 33
6
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0
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344
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For a prime, is there always a prime number for which it is a primitive root?
Artin’s primitive root conjecture states that for any integer $a\neq \pm1$ which is not a square,there are infinitely many primes $p$ such that $a$ is a primitive root mod $p$. By Heath-Brown's result,...
- 538
1
vote
0
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53
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Effective bounds for degree and height in algebraic number enumeration
I am considering a specific enumeration of all complex algebraic numbers within the unit ball, defined as follows:
Enumerate all primitive, irreducible polynomials in $\mathbb{Z}[x]$ with positive ...
- 545
2
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0
answers
183
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Irreducibility of a degree-$27$ polynomial through Newton polygon or residual reduction
Let $K=\mathbb{Q}_3(t)$ be the finite extension of the $3$-adic number field $\mathbb{Q}_3$, where $t=3^{1/13}$. I want to know if the polynomial $$f(x)=(x^9-t^2)^3-3^4x+3^3t^5 \in K[x]$$ is ...
- 450
4
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0
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100
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A finiteness question concerning adelic Galois cohomology
$\newcommand{\Fbar}{{\overline F}}
\newcommand {\A}{{\mathbb A}}
\newcommand{\Abar}{{\bar {\mathbb A}}}
\newcommand{\Gal}{{\rm Gal}}
$Let $F$ be a number field (for example, $F=\mathbb Q$), and let $B$...
- 15.4k
2
votes
1
answer
370
views
Reduction of an abelian variety
Let $T_1, T_2, ..., T_g$ be the local co-ordinates of Tangent space of an Abelian variety of dimension $g$ around the identity $O$. Let $P \in A(K)$ and $v$ be a non-archimedean place of number field $...
- 317
1
vote
0
answers
72
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What replaces index calculus / n-descent for higher-dimensional abelian varieties?
Kummer theory gives a unifying perspective on several 'discriminant factorization' phenomena. In the setting of a number field $K$, there is the short exact sequence of Galois modules
$$
1 \...
- 273
4
votes
0
answers
236
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On a sufficient condition for all extensions of a discrete valuation to be unramified
Let $\mathcal O$ be a discrete valuation ring with maximal ideal $\mathfrak m$, fraction field $\mathbb K:=\mathrm{Frac}\,\mathcal O$, and residue field $\mathbb F:=\mathcal O/\mathfrak m$. Suppose ...
0
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0
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142
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Sign function in finite field
We build a finite field $\mathbb Z/p\mathbb Z$, $p>2$. Then we introduce a sign function, which is $0$ at $0$, $+1$ at $1\dots(p-1)/2$, and $-1$ at $(p+1)/2\dots p-1$. Now we want to generalize the ...
- 109
0
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0
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178
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On quadratic equations over the Gaussian ring $\mathbb Z[i]$
In 1972 C. L. Siegel proved that there is an algorithm to decide for any polynomial $P(x_1,\ldots,x_n)\in\mathbb Z[x_1,\ldots,x_n]$ with $\deg P=2$ whether
$$P(x_1,\ldots,x_n)=0$$ for some $x_1,\ldots,...
- 18.1k
3
votes
2
answers
317
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Problem understanding the proof of $L(s,\operatorname{Ind}_N^H(\eta))=L(s,\eta)$ (Edit)
(I majorly editted the question to improve clarity)
I'm following Jurgen Neukirch's proof of $L(s,\operatorname{Ind}_H^G(\eta))=L(s,\eta)$ and I am having trouble understanding one key step.
The set ...
2
votes
0
answers
127
views
Ramification in Kummer extensions associated to CM Elliptic curves
Let $p$ be a prime that splits as $p\mathcal{O}_F=\mathfrak{p}\bar{\mathfrak{p}}$ where $F$ is an imaginary quadratic field. Let $H_F$ denote the Hilbert class field of $F$.
Finally let $E/H_F$ be an ...
- 21
3
votes
1
answer
365
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On pairs of natural numbers that satisfy certain equations with real numbers
This question has come up in my physics research. Assume the following function
$$
f(N,M,k) = \frac{1}{MN}\frac{N\tanh(Mk)-M\tanh(Nk)}{N\tanh(Nk)-M\tanh(Mk)},
$$
where $N$ and $M$ are natural numbers ...
- 660
0
votes
0
answers
68
views
Checking if a supersingular elliptic curve shares an edge with the Spine
Let $E$ be supersingular elliptic curve defined over $\mathbb{F}_{p^2}$ and let $\ell$ be a prime. I'd like to know how one can check if E shares an edge, in the $\ell$-isogeny graph, with the set of ...
- 387
2
votes
1
answer
195
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Finiteness of cycles for $T_k(n):=\operatorname{rad}\bigl(\sigma^{\circ k}(n)\bigr)$ when $k$ is fixed
$\DeclareMathOperator\rad{rad}$Let $\sigma(n)=\sum_{d\mid n} d$ be the sum-of-divisors function, and let $\rad(m)=\prod_{p\mid m}p$ be the radical (with $\rad(1)=1$). For a fixed integer $k\ge 1$, ...
11
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2
answers
1k
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Is every number field generated by a trinomial?
Can every algebraic number field be constructed by adjoining a root of a trinomial (with rational coefficients) to $\mathbb{Q}$?
- 3,213
2
votes
0
answers
115
views
Inverse limit of homology groups of the unit group of number fields
Let $K$ be a number field and $p$ be a prime number. Let $K_{\emptyset}(p)$ be the maximal unramified pro-$p$ extension of $K$. If $L$ is a finite Galois extension of $K$ with Galois group $G(L/K)$, ...
- 51
0
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0
answers
73
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Principal Ideal Theorem for the genus fields of imaginary abelian number fields
According to Lemmermeyer's answer to my previous question, which provides some counterexamples for "real" quadratic fields, I re-ask it here for a more specific case, say for "imaginary&...
- 597
5
votes
1
answer
265
views
Principal Ideal Theorem for the genus field
Let $K$ be an abelian number field and denote its genus field and Hilbert class field by $\Gamma_K$ and $H_K$, respectively. By Principal Ideal Theorem (PIT), every fractional ideal $\mathfrak{a}$ ...
- 597
1
vote
0
answers
280
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Is Alexander Stolin's proof of the Kummer–Vandiver conjecture valid? [closed]
In https://arxiv.org/abs/2001.09702 Alexander Stolin announced a proof of the Kummer–Vandiver conjecture.
My questions are: Is his proof valid?
And what is the status of the Kummer–Vandiver conjecture?...
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13
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0
answers
530
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On mechanically checking Fermat's last theorem for large exponents
I first note that I will solely focus on the second case of FLT, as the concept of Wieferich prime pretty much satisfies my curiosity in the first case and suggests what kind of answers I am looking ...
- 480
3
votes
1
answer
336
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Definition of Adelic points of a scheme over $\mathbb{Q}$
In Milne's notes on Shimura Varieties (https://www.jmilne.org/math/xnotes/svi.pdf) at the start of section 4, it defines the ring of finite adeles. Then for an affine variety $V=\mathrm{Spm}(A)$ over $...
- 129
4
votes
1
answer
200
views
Congruent subgroups of $\mathrm{PGL}_2$ vs $\mathrm{SL}_2$
Exercise 4.3 of Mine's notes on Shimura Varieties (https://www.jmilne.org/math/xnotes/svi.pdf) says:
Show that the image in $\mathrm{PGL}_2(\mathbb{Q})$ of a congruence subgroup in $\mathrm{SL}_2(\...
- 129
1
vote
1
answer
188
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Supersingular reduction of CM elliptic curves: what are the possible values of $\operatorname{tr}(\widetilde{[\sqrt{d}]}\pi_{\tilde{E}})$?
Let $E$ be an elliptic curve of good reduction over a local field with residue field characteristic $p\ge 5$ and suppose $E$ has complex multiplication by $\mathcal{O}_d$ for some discriminant $d$ ...
- 470
16
votes
1
answer
611
views
When is the ring of integers of a character field the ‘character ring’?
Let $G$ be a finite group with an irreducible complex character $\chi$.
Let $\mathbb Q(\chi)$ denote the field extension of the rationals generated by the values of $\chi(g)$ for $g \in G$.
A theorem ...
- 333
6
votes
1
answer
489
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A question on the paper "Maximal unramified extensions of imaginary quadratic number fields of small conductors"
Let $K$ be a number field and let $K_{ur}$ its maximal unramified extension. Let $K=\mathbb{Q}(\sqrt{-105})$. In Yamamura, K. (1997). Maximal unramified extensions of imaginary quadratic number fields ...
- 925
2
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0
answers
242
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On the Cartier dual étale sheaf
Let $K$ be a number field and $S$ be a finite set of places of $K$ containing all archimedean places. Let $G_K$ be the absolute Galois group of $K$. Let $\mathcal{O}_{K,S}:=\{a\in K~|~\text{ord}_{v}(a)...
- 51
0
votes
0
answers
124
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Describing imaginary abelian fields in terms of Dirichlet characters
I am reading the paper, titled "The imaginary abelian number fields with class numbers equal to their genus class numbers" in which the authors describe an imaginary abelian field $N$ by the ...
- 597
2
votes
0
answers
134
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Reference request: compact representations of p-adic groups
I'm looking for a proof of the following claim: Let $G$ be a $p$-adic group and $W$ a smooth representation with compactly supported matrix coefficients (Bernstein calls these 'compact'). I want to ...
1
vote
0
answers
230
views
Two conjectures concerning the distribution of small numbers in coprime pairs (a, b)
The research findings of mathematician Ivan Niven imply that the distribution of the smallest exponents in the prime factorization of natural numbers is $1$, while the average largest exponent in the ...
- 2,148
3
votes
0
answers
234
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If resultant $T(y) = \mathrm{Res}_{x}(f(x), y - g(x))$ have a nontrivial factors then can $f(x)$ also have a nontrivial factors?
This is a follow-up question to this question. In that question, we learned that if, $T(y) = \mathrm{Res}_{x}(f(x), y - g(x))$ , then $T(g(z))$ is divisible by $f(z)$.
Now, my question is:
If $T(y) = ...
- 311
8
votes
2
answers
490
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How do I write down an explicit primitive 9th root of unity in the invariant 1/3 central division algebra over $Q_3(\zeta_3)$?
Here is a very naive question about division algebras. Let $K$ denote the $p$-adic number field given by adjoining a primitive third root of unity to $\mathbb{Q}_3$. Let $D$ denote the central ...
- 3,448
1
vote
1
answer
162
views
Principalization and local conditions
Let $K$ be a number field and let $v$ be a finite place of $K$ (i.e., a prime). Does there exist a finite Galois extension $L/K$ such that every ideal in $K$ becomes principal in $L$ and such that $L\...
- 281
1
vote
1
answer
135
views
Surjectivity of the map $h_1:K_2 F / l K_2 F\to H^2(F,\mu_l\otimes\mu_l)$, where l is prime, and F is a global field
This is a crosspost from a MSE question I asked a time ago, but have not gotten an answer that satisfies me. I appreciate any help.
I am new to algebraic number theory, and I am reading a paper by ...
- 13
1
vote
0
answers
265
views
Vanishing of Weil–Châtelet group over solvably closed field
Let $A$ be a connected algebraic group defined over a solvably closed field $K$. Are there any results concerning the vanishing of the WC-group $H^1(G_K, A)$?
- 161
0
votes
0
answers
165
views
How does the volume factor in Kedlaya's Poisson summation formula disappear when applying it to the theta function they derived?
I've been trying to craft a small proof of the analytic continuation of the Dedekind zeta function on my own, mainly studying the theta function and the Mellin transform method.
When going through my ...
1
vote
0
answers
246
views
Does $\text{Gal}(K(F[\pi^{\infty}])/K \cong \text{Gal}(K(G[\pi^{\infty}])/K$ imply $F[\pi^n] \cong G[\pi^n]$?
Let $K$ be a $p$-adic number field with ring of integers $\mathcal{O}_K$ and uniformizer $\pi$. Let $F$ and $G$ be two $d$-dimensional formal groups of height $h$ over $\mathcal{O}_K$. Denote:
\begin{...
- 450
2
votes
2
answers
401
views
Quadratic Hecke characters with certain given local components
Let $F$ be a fixed number field, and $S$ be a non-empty fixed finite set of places of $F$. We further assume that $S$ contains all the infinity places and at least one finite place. Suppose that for ...
- 479
2
votes
1
answer
184
views
The genus number of real biquadratic fields
Let $K=\mathbb{Q}(\sqrt{m},\sqrt{n})$ be a real biquadratic field, where $m,n>0$ are two distinct squarefree integers. Denote by $g_K$ and $g_K^+$ the genus number and the narrow genus number of $K$...
- 597
2
votes
0
answers
307
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Course notes for Galois Representations of Frazer Jarvis from 2006
I have been looking for the course notes of Prof. Frazer Jarvis (who has a good algebraic number theory book by Springer in the series Springer Undergraduate Mathematics Series) on Galois ...
