All Questions
118 questions
3
votes
1
answer
121
views
Reference request: ray class group as quotient of finite ideles
Let $K$ be a number field, and write $\mathbb{A}_{K,f}^\times$ for the group of finite ideles of $K$. That is
$$
\mathbb{A}_{K,f}^\times = \{(u_v)_v \in \prod_{v \nmid \infty} K_v^\times : v(u_v) = 0 \...
- 411
1
vote
1
answer
241
views
The equation $ax^2 +by^2 =1 \mod P$ in cyclotomic field
Let $L$ be a cyclotomic field, and $P$ a prime ideal of $\mathcal{O}_L$.
is there any symbol for the equation $ax^2 + by^2 =1 \mod P$ and if so, is it computable in polynomial time?
if $a$ is ...
- 133
3
votes
3
answers
383
views
On subfields of the cyclotomic field $\mathbb{Q}(\zeta_p)$
Let $p$ be an odd prime. Let $\zeta_p=e^{2\pi{\bf i}/p}$ and let $1\le k\le p-1$ be a divisor of $p-1$. Recently, when I learnt algebraic number theory, I met the following problem.
If we let
$$U_k=\{...
- 87
2
votes
1
answer
101
views
Bound on number of extensions of Q unramified outside a fixed prime
Are there any known asymptotic bounds on the number of degree $d$ extensions of $\mathbb{Q}$ unramified outside a fixed prime $p$?
- 2,907
0
votes
2
answers
215
views
Papers related to a diophantine equations about Magic square of squares for $n=3$
The open problem of magic squares of squares explained here. Consider the following magic square of squares:
$$
\begin{aligned}
&a^2&b^2&&c^2\\\\
&d^2&e^2&&f^2\\\\
&...
1
vote
0
answers
59
views
A question on generalized bases
I just came to know that it is possible to define a generalized base as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\...
3
votes
1
answer
98
views
Reference Request: Possible generalizations of the stability of $\gamma$-factors
$\DeclareMathOperator\GL{GL}$
Let $F$ be a nonarchimedean local field. Suppose $\pi, \sigma$ are irreducible admissible representations of $\GL_{n}(F)$ and $\GL_{m}(F)$ respectively, with $n \geq m$. ...
- 639
2
votes
0
answers
129
views
Imaginary quadratic fields with prime class number
Let $K$ be an imaginary quadratic field, with class number equal to an odd prime, say $h_K = p$.
In the proof Proposition 2.4 of this paper, Fukuda and Komatsu write,
"Since $h_K = p$, there ...
- 707
2
votes
2
answers
293
views
Cohomology of $S$-arithmetic groups with trivial coefficients such as $H^n(\rm{PGL}_2(\mathbb{Z}[1/N]);\mathbb{Z})$
As $\rm{PSL}(2,\mathbb{Z})=(\mathbb{Z}/2\mathbb{Z})*(\mathbb{Z}/3\mathbb{Z})$, its cohomology groups $H^n(\rm{PSL}(2,\mathbb{Z});\mathbb{Z})$ are easy to get.
Let $N$ be a product of distinct primes.
...
- 391
3
votes
0
answers
117
views
Reference Request: Local decomposition of GGP period integrals of cuspidal forms on unitary groups
Setup: Let $E/F$ be a CM-extension of global number fields. Let $(V,\phi)$ be an Hermitian space of dimension $n$ over $E$. Let $(V^{\flat}, \phi^{\flat})$ be a subspace of $V$ of dimension $n-1$ on ...
- 639
2
votes
2
answers
432
views
Expositions of the classical approach to local class field theory (Brauer group and Hasse invariant)
I've posted this question already on MSE and didn't get much out of it, so I hope it's OK to repost here.
I'm an undergraduate trying to learn local class field theory from the corresponding chapter ...
- 121
3
votes
1
answer
245
views
Integration against Eisenstein series can be regarded as a cup product
This summer, I was very fortunate and honored to attend the conference "Iwasawa 2023" at the University of Cambridge as a young Ph.D. student on Iwasawa theory. There, one of the speakers, ...
- 639
3
votes
0
answers
122
views
Description of $\operatorname{Gal}(K(E[n])/K)$ as a subgroup of $\operatorname{GL}_2(\mathbb{Z}/n\mathbb{Z})$ for a CM elliptic curve $E$
I am looking for a specific description of the Galois groups $\operatorname{Gal}(K(E[n])/K)$ as a subgroup of $\operatorname{GL}_2(\mathbb{Z}/n\mathbb{Z})$ for an elliptic curve $E$ with complex ...
- 309
10
votes
1
answer
480
views
Questions about ray class groups
Let $K$ be an imaginary quadratic number field (so there are no real embeddings) with ring of integers $\mathcal{O}_K$ . Let $w$ be the number of units in $K$ and $h$ be the class number of $K$. Let $\...
- 1,990
4
votes
0
answers
149
views
Infinite family of monogenic cyclic quartic fields
I am currently reading a paper (masters' thesis, in fact) from 2008 where it is mentioned that it is not currently known whether there exist infinitely many monogenic cyclic quartic fields (i.e., ...
- 26.9k
2
votes
0
answers
103
views
On equidistribution of primes in positive characteristic
In S. Lang's book "Algebraic Number Theory" (1986), page 317, Theorem 6 states essentially that given $P$ a set of primes, let $\tau:P\longrightarrow J$ be the typical idèle map taking ...
- 675
0
votes
1
answer
112
views
Statistics of action of Galois group of number field on primes over unramified rational primes
Let $p \in \mathbb{Z}$ be prime and $K / \mathbb{Q}$ be a finite Galois extension. The Galois group $G$ of $K$ acts on the primes of $\mathcal{O}_K$ over $p$. Do we know any statistical information ...
- 658
4
votes
0
answers
181
views
The order of the global Galois group
For any profinite group $G$, we can define the order of $G$ using the notion of supernatural number. Now let $K$ be a number field, $S$ a finite set of primes of $K$ and $ G_{K,S} $ the Galois group ...
- 863
1
vote
0
answers
98
views
Existence of countable dense normal subgroups of global Galois group
Let $K$ be a number field, $S$ a finite set of primes of $K$ and $ G_{K,S} $ the Galois group of the maximal extension of $ K $ (inside a fixed algebraic closure of $K$) unramified outside $ S $. In ...
- 863
5
votes
2
answers
349
views
Geometric interpretation of Iwasawa algebras: $\mathbb{Z}_p[[T]]$ as a disk?
I am a student learning Iwasawa theory. I am so sorry if this post is too trivial for this site. I posted it on math.stackexchange yesterday but obtained no responce.
A quite basic object is the ...
- 639
4
votes
0
answers
135
views
Analog of a theorem on equidistribution in adeles
Is there a reference anywhere for the analog of Theorem 6 in chapter XV of Langs Algebraic Number Theory for global function fields?
In my research I have been using this theorem to prove density ...
- 337
7
votes
3
answers
611
views
Question on a crucial lemma in Euler's approach to Fermat's Last Theorem for $n=3$
As many of you may know, the illustrious L. Euler put forward a proof of the case $n=3$ of Fermat's Last Theorem via infinite descent. The thing is that, at a certain point, he resorted to the ...
- 8,842
7
votes
0
answers
265
views
"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 "...
- 71
1
vote
1
answer
364
views
Good references to study Baker's theory
I am studying diophantine equations and I need the theory of Bakers, Can you advise me about good books, or lectures on Baker's theory?
- 17
2
votes
0
answers
182
views
On the relative class number of a cyclotomic extension
Let $\Bbb Z[\zeta_p]$ denote the cyclotomic integers where $p$ is a prime and let $h_1 = h_1(p)$ denote its relative class number.
Question: Is it known whether there are infinitely many primes $p$ ...
- 18.8k
3
votes
1
answer
280
views
Computing mth power residue symbols
Let's say I have a two odd primes, $p, q$ and $K$ is the field $\mathbb{Q}(\zeta_{pq})$. Let's say $\alpha \in \mathcal{O}$ is an arbitrary element in the ring of integers of $K$, $\frak{b} \subset \...
- 565
2
votes
0
answers
1k
views
Advanced texts on analytic number theory?
So a friend of mine is very interested in analytic number theory, and is looking for resources past the basic level.
He has studied analytic number theory from several books, among them are Hardy’s ...
3
votes
1
answer
418
views
Counting cubic residues mod p
Given a prime $p=3m+1$, $(p-1)/3$ of the residues mod $p$ are cubic residues. So heuristically, for any given integer $k>1$ not a perfect cube, we would expect that about 1/3 of the primes $\equiv1\...
- 9,114
6
votes
0
answers
456
views
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 ...
2
votes
0
answers
245
views
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 ...
1
vote
0
answers
255
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.
...
user163485
18
votes
1
answer
1k
views
Distinct integer roots for a degree 7+ polynomial and its derivative
Question: Is there a polynomial $f \in \mathbb{Z}[x]$ with $\deg(f) \geq 7$ such that
all roots of $f$ are distinct integers; and
all roots of $f'$ are distinct integers?
Background:
I asked a ...
- 7,831
6
votes
1
answer
2k
views
Sum of square roots of natural numbers
Recently, I've encountered the following question:
Assume that $n_{1}, \ldots, n_{k}$ are (not necessary distinct) natural numbers. If
$$ (\sum_{i = 1}^{k}\sqrt{n_{i}}) \in \mathbb{N},$$ can we ...
3
votes
0
answers
97
views
Study of relative class number of 'non-abelian' CM field by using L-functions
I'm currently interested in finding good upper bounds for the relative class numbers of non-abelian CM-fields.
So I'm looking for some references to learn the techniques that can be useful.
So far, I ...
- 1,013
2
votes
0
answers
161
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 ...
- 707
4
votes
1
answer
205
views
Multiplicative set of positive algebraic integers
Let $S$ be a set of algebraic integers such that:
$\mathbb{N}_{\ge 1} \subseteq S \subset \mathbb{R}_{\ge 1}$,
$\alpha, \beta \in S \Rightarrow \alpha \beta \in S$,
$\alpha, \beta \in S \Rightarrow ...
2
votes
0
answers
110
views
Spectral decomposition of idele class group $L^2(\mathbb{I}_F/ F^{\times})$
Let $F$ be a number field. Let $\mathbb{A}_F$ be the ring of adeles. The group of units of $\mathbb{A}_F$ is called the group of ideles $\mathbb{I}_F=\mathbb{A}_F^{\times}= GL_1(\mathbb{A}_F)$. The ...
- 421
3
votes
1
answer
758
views
Looking for a paper of Lagarias and Odlyzko
I have been studying about the Chebotarev Density Theorem and have been hunting for the following paper of Lagarias and Odlyzko for quite a while:
Effective versions of the Chebotarev density theorem, ...
- 739
2
votes
0
answers
491
views
Examples of almost Dedekind domains that are not Dedekind
All I know about almost Dedekind domains (which I have come to learn about only recently) is that they are integral domains whose localization at every prime is a discrete valuation ring. In other ...
- 739
15
votes
1
answer
484
views
Looking for a paper on transfinite diameter by David Cantor
I have been reading about transfinite diameter and its applications to number theory and have been hunting for the following paper for quite a while:
Cantor D.: On an extension of the definition of ...
- 739
1
vote
0
answers
188
views
I'm looking for a proof of Polya-Bertrandias Theorem
I'm looking for a proof of Polya-Bertrandias rationality criterion in english (not the one from Amice).
- 11
14
votes
3
answers
2k
views
Norms in quadratic fields
This should be well-known, but I can't find a reference (or a proof, or a counter-example...). Let $d$ be a positive square-free integer. Suppose that there is no element in the ring of integers of $\...
- 38k
2
votes
0
answers
97
views
Is there any good reference on the Bayesian view that can be helpful for reading papers on the number theory using heuristic arguments?
Nowadays there are many papers on the number theory using heuristics.
I have read some of them.
But I have no clear understanding of the Bayesian Probability(subjective probability).
The concept of ...
- 1,013
2
votes
1
answer
162
views
On a transcendental number defined as a variation involving the Lambert $W$ function in the nested square root representation of the golden ratio
Define the real number $\xi$ satisfying
$$\xi=\sqrt{1+W\left(1+\sqrt{1+W(1+\ldots)}\right)}\tag{1}$$
where $W(x)$ denotes the main branch of the Lambert $W$ function, as reference I add that Wikipedia ...
0
votes
0
answers
91
views
Image of Frobenius element under irreducible representation is diagonalizable
Let $K/ \mathbb Q$ be a Galois extension, and $\rho$ be an irreducible representation of the Galois group $Gal(K/ \mathbb Q)$. Consider an integer prime $p$ which doesn't ramify in $K$, and let $\...
- 739
0
votes
0
answers
177
views
Status of the $n$ conjecture and, as secondary question or reference request, what about a transfer method for this conjecture $n>3$
The n conjecture is a generalization of the abc conjecture. What is the current status of the $n$ conjecture? See also [1]
Question 1. Can you tell us what about the current status of the $n$ ...
2
votes
1
answer
214
views
Looking for a paper by Landau and one by Watson
For the purposes of a project, I've been looking for the following two papers referred to in Serre's "Divisibilité de certaines fonctions arithmétiques":
Landau (E.), - Über die Eitenlung der ...
- 739
0
votes
0
answers
226
views
On Prime Numbers which can be Norms of an Integral Ideal of a Number Field
We know that since the ring $\mathbb Z [i]$ of Gaussian integers is a Principal Ideal Domain, the only integer primes which can norms of some ideal of $\mathbb Z [i]$ are those which can be expressed ...
- 739
3
votes
1
answer
195
views
English reference for the Brauer-Kuroda formula
I'm currently trying to understand the Brauer-Kuroda formula.
Although there are many recent papers on the formula but they seem to be purely algebraic.
They say that original analytic approach is ...
- 1,013
5
votes
0
answers
354
views
Modern reference for Andre Weil's 'Sur les courbes algébriques et les variétés qui s'en déduisent'
I'm currently interested in the cardinality of the set of values of a polynomial over a finite field.
I found a paper
Saburo Uchiyama, Sur le Nombre des Valeurs Distinctes d'un Polynome a ...
- 1,013