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
20
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How does assuming GRH help us calculate class group?
It seems that, almost all computer programs assume GRH to calculate $\mathbb{Q}(\zeta_p)$ for $p > 23$. I'm very curious how assuming the GRH, helps us to calculate class groups in practice. Can ...
8
votes
3
answers
2k
views
Counter example of a radical extension that is not Galois/normal over $\mathbb{Q}(\omega)$?
Most proofs of Galois theorem stating that "an equation is solvable in radicals if and only if its Galois group is solvable," show the left to right direction by induction on the height of ...
- 221
8
votes
1
answer
404
views
p-adic versions of log concavity for graphs (or matroids)
It was recently shown using techniques inspired by algebraic geometry (by Huh and Adiprasito-Huh-Katz) that the chromatic polynomial of a graph (or matroid) has coefficients that satisfy log-concavity....
- 7,646
2
votes
0
answers
126
views
How do elliptic units generate the module of Euler systems over abelian extensions of imaginary quadratic fields?
I am trying to undesrtand the analogy between the Euler systems over abelian extensions of the rationals and the Euler systems over abelian extensions of imaginary quadratic fields.
As Soogil Seo ...
- 99
8
votes
0
answers
176
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Simple abelian varieties of $\mathrm{GL}_2$ type with positive rank and large dimension
$\DeclareMathOperator\GL{GL}$I would like know if there are known constructions of simple abelian varieties of $\GL_2$ type of arbitrarily large dimension and positive Mordell-Weil rank, whose rank is ...
- 2,185
4
votes
1
answer
346
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Clarification regarding a claim in Heilbronn’s 1934 paper
I was reading Heilbronn’s 1934 paper where he proves that $H(d) \to \infty$ as $d \to -\infty$, where $H(d)$ is the ideal class number of the imaginary quadratic field with discriminant $d$. I couldn'...
- 577
9
votes
1
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227
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Are the class numbers of $\mathbb{Q}(\cos(2\pi / m))$ $O(m^n)$ for some fixed $n$?
Question: Are the class numbers of $\mathbb{Q}(\cos(\frac{2\pi}m))$ $O(m^n)$ for some fixed $n$?
Evidences (e.g. a recent paper) showing that the question above is open are also OK.
Remark: If such $n$...
- 9,421
3
votes
0
answers
99
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algebraic number with explicit base two digits
I am looking for an irrational algebraic number $\alpha \in [0,1[$ whose base two expansion
$$
\alpha = \sum_{i=1}^\infty {1 \over 2^{\varphi(i)}},
$$
is easily computable. By this I mean $\varphi : {...
- 18.5k
2
votes
1
answer
150
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Are the maximal cyclotomic field contained in a number field and its Hilbert class group the same?
Let $K$ be a number field. If $d$ be the smallest even integer such that $\Bbb Q (\zeta_d) \subset K,$ then I wanted to prove that if $d'>d$ then $\Bbb Q (\zeta_{d'}) \not\subset H(K),$ where $H(...
- 723
2
votes
0
answers
98
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The Guinand-Weil explicit formula for Hecke characters
The Guinand-Weil formula for the Riemann zeta function is
\begin{aligned}&\Phi (1)+\Phi (0)-\sum _{\rho }\Phi (\rho )\\={}&\sum _{p,m}{\frac {\log(p)}{p^{m/2}}}{\Big (}F(\log(p^{m}))+F(-\log(p^...
- 9,421
5
votes
1
answer
471
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Action of the symmetric group $S_3$ on an elliptic curve $E$ defined over $\mathbb{Z}$
I came up with the following question on a facebook group: find the positive integer solutions of the equation $$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=4$$
Now clearly this is very difficult, ...
- 1,333
4
votes
1
answer
388
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What are the known number-theoretic functions, that are related to "the number of ideals of norm $n$, that belong to the class $[c]$"?
Let $L$ be a number field, $\mathcal{O}_L$ its ring of integers, and $\mathcal{Cl(O}_L)$ its ideal class group. Let's fix an arbitrary class $[c] \in \mathcal{Cl(O}_L)$. By $r(n)=r([c], n)$, I mean ...
14
votes
2
answers
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Euclid-style proof of Dirichlet’s theorem on primes in certain arithmetic progression
The well-known theorem of Dirichlet on primes in arithmetic progression states that given coprime natural numbers $a\le q$, there are infinitely many prime numbers congruent to $a\pmod q$. The ...
- 1,423
2
votes
0
answers
54
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Naive height of the truncation of taylor series of the exponential
Let $\alpha\in\overline{\mathbb Q}$. One calls naive height of $\alpha$ and one notes $H(\alpha)$ the maximum of the $|a_i|$ ($0\le i\le d$) where $P(X)=a_dX^d+a_{d-1}X^{d-1}+\cdots+a_0$ is the unique ...
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7
votes
1
answer
468
views
How to compute Hilbert class field of $\Bbb Q(\zeta_{31})$?
I tried constructing the Hilbert class field of $\Bbb Q(\zeta_{31})$ by imitating one of the problems from MO. I failed miserably as a quadratic field inside the cyclotomic field $\Bbb Q(\zeta_{31})$ ...
- 723
4
votes
1
answer
189
views
Generalization of $\lim_{n \rightarrow \infty} \prod_{i=1}^{n}\frac{2i-1}{2i}$ for a character $\chi:\mathbb{Z}/s \mathbb{Z} \rightarrow \mathbb{C}^*$
Playing with some infinite products I came up with this problem, that I'm not able to figure it out by myself. Moreover in the internet it doesn't seem to appear anywhere.
Maybe it is just an easy ...
- 1,333
9
votes
1
answer
729
views
how to compute Hilbert class field of $\Bbb Q(\zeta_{23})$?
I want to construct the Hilbert class field of $K=\Bbb Q(\zeta_{23}).$ I have no clue how to construct it except that I know that $[H(K):K]=3$ from Sage. Any references or comments are appreciated.
- 343
9
votes
1
answer
491
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How can I transform $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^k\sin(\pi rn)}$ into a modular form?
Let
$$f_k(z)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^k\sin(\pi zn)}$$
be a family of holomorphic functions on the upper-half plane $\mathbb{H}=\{a+bi|b>0\}$ for each odd natural number $k$. These ...
- 2,817
2
votes
1
answer
207
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If $a_{g}(1)=g(x)$ and $a_{g}(r)=g(a_{g}(r-1))$ for $r>1$ then is it true that $\limsup\limits_{r\to\infty}\gamma(a_{r})=\infty?$
Let $g(x)$ be a polynomial with integral coefficients.We define $\gamma(g(x))$ to be the degree of the non constant polynomial $r(x)$ which divides $g(x)$ for all $x$ and also has minimal degree.
...
- 1,032
2
votes
0
answers
667
views
On Serre's "Local fields"
While I was reading J.-P. Serre's book "Local Fields" I found something strange in Chapter V. When Serre discusses properties of norm for unramified extensions, he says it is possible to ...
- 445
2
votes
0
answers
99
views
Question on the algebraic structure of the set $\mathcal{P} = \{ \sum_{i=1}^n a_i x^i = 0, a_i \in \pm \mathbb{P}~or~0\}$
It is well-known that the set
$$
\mathcal{A} = \bigg\{ x\in \Bbb C: \sum_{i=1}^n a_i x^i = 0, a_i \in \mathbb{Z}\text{ and } n \text{ is a positive integer}\bigg\},
$$
is the set of all algebraic ...
- 395
1
vote
1
answer
225
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Krull dimension and elimination theory over the integers
Let $K:=\mathbb{C}$, and let $R:=K[x_1,\dots , x_n]$.
Then, a system of polynomial equations $p_1=0, p_2=0, \dots , p_r = 0$, where the $p_i$ are polynomials in the $x_j$, has finitely many solutions $...
- 311
10
votes
0
answers
473
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Is class group of cyclotomic fields cyclic?
What are the cyclotomics fields with a cyclic class group. I read that there are only 29 cyclotomic extensions of $\Bbb Q$ with class number one. But I wanted to know what condition on $n$ would make $...
- 723
3
votes
1
answer
675
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, ...
- 719
5
votes
1
answer
651
views
rational points of a hyperelliptic curve of genus 3
Let $K=\mathbb{Q}(\sqrt{-1}).$ I have the following hyperelliptic curve of genus 3:
$$ C : y^2 = (x^2-x+1)(x^6+x^5-6x^4 -3x^3+14x^2-7x+1) $$
I want to find $C(K)$. My first attempt was to compute the ...
- 51
1
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0
answers
94
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Irreducibility of $\frac{x^{n+1}-(n+1) x+n}{(x-1)^2}$ [duplicate]
The question is motivated by this question.
Consider the polynomials
$$\dfrac{x^{n+1}-(n+1) x+n}{(x-1)^2} = \displaystyle \sum _{k=0}^n (n-k) x^k, n=1,2,3,\dots,$$
Are they all irreducible (over $\...
- 282
2
votes
2
answers
409
views
Relation between the Selmer group and the ideal class group
Let $E/K$ be an elliptic curve defined over the number field $K$. Does exist any relation between the $p$-Selmer groups of $E/K$ and the ideal class group $Cl(K)$ of $K$?
- 385
1
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1
answer
367
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Proving that $m\leq 4$ when $\sqrt{a+\sqrt{b}+\sqrt{c}}=\sum_{i=1}^m \sqrt{d_i}$ with each $d_i/d_j$ non-square
The question has been posted on math.SE but had no response.
There are positive integers $a,b,c,d_i$, s.t. $\sqrt{a+\sqrt{b}+\sqrt{c}}=\sum_{i=1}^m \sqrt{d_i}$, and for any $i\ne j$, $\sqrt{d_i/d_j}$ ...
- 282
1
vote
0
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87
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Invariant subfield of $\overline{\mathbb F_q(T)}$
Let $G$ be the group of automorphisms of $\overline{\mathbb F_q(T)}$ defind by $G=\{T\mapsto T+\xi\mid\xi\in\mathbb F_q\}$. Can one describe ${\overline{\mathbb F_q(T)}}^G$?
- 3,739
0
votes
0
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137
views
Elliptic units as Euler systems
I’m trying to understand elliptic units in order to work with the Euler systems of the abelian extensions of quadratic imaginary number fields. I’ve looked at few references about the topic, but they ...
- 99
1
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0
answers
123
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Siegel's formula for generalized theta series with characteristics?
Siegel's formula(Siegel-Weil) directly relates the weighted sum of theta functions to Eisenstein series. (Or equivalently, the weighted sum of the cusp form is zero). I wonder if there is a ...
- 11
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votes
0
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216
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Have the following summations been studied before?
Suppose $q^2-4pr<0$, and consider the set of integral points
$$\mathcal Z=\{(X,Y)\in\mathbb Z^2:$$
$$px^2+qxy+ry^2+sx+ty+u=0\}$$
which lie on an ellipse. Then define $$M=\sum_{(X,Y)\in\mathcal Z}\...
- 13.7k
9
votes
1
answer
705
views
Forms of ${\rm SL}(2)$
I know all real forms of ${\rm SL}(2,{\Bbb C}$). They are ${\rm SL}(2,{\Bbb R})$ and ${\rm SU}(2)$.
Moreover, ${\rm SL}(2,{\Bbb R})$ is isomorphic to ${\rm SU}(1,1)$. Thus I can say that all real ...
- 13.6k
1
vote
1
answer
580
views
If $\gcd(x,y)=1$ find necessary and sufficient condition(s) such that $\gcd (x-1,y-1)>1$
Let, $xy=n^{\underline k} = n(n-1)(n-2)\cdot\dotsm\cdot (n-k+1)$ and it is given that $ \gcd(x,y)=1$ with one of $x$, $y$ is odd, another is even. When is $\gcd (x-1,y-1)=z>1$?
In other words, what ...
1
vote
0
answers
198
views
Realization of a p-adic field as a completion of a number field
Let $F$ be a $p$-adic field of characteristic 0. Is it always possible to find a number field $K$ such that $K$ has only one place lying above $p$ and such that its completion at this place is $F$?
If ...
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7
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0
answers
88
views
Is the set of conjugates of Pisot numbers dense?
Let $S$ be the set of Pisot numbers. It is known that $S$ is closed and has infinitely many limit points. However, I want to know if there are are results about the set of conjugates of Pisot numbers. ...
- 171
4
votes
0
answers
210
views
Traces vs. Determinants in Artin's $L$-functions
Loosely put, my question is:
What happens if we swap determinant by the trace in an Artin $L$-function?
This question is not very precise and can be a little misleading, so I explain the specific ...
- 79
1
vote
0
answers
114
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Simultaneous $S$-unit equations
In this question I am primarily interested in rational integers and rational primes, but the same question can be easily extended to number fields.
Let $S = \{p_1, \cdots, p_k\}$ be a finite set of ...
- 25.5k
1
vote
0
answers
419
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 ...
- 719
3
votes
0
answers
79
views
The type number of an algebra
I've been reading On the existence of maximal orders, by C.F. Yu, in which he discusses maximal $R$-orders in semisimple algebras over a field $K$, where $R$ is a Noetherian integral domain and $K = \...
- 323
18
votes
1
answer
661
views
Is it true that $\{x^4+y^2+z^2:\ x,y,z\in\mathbb Z[i]\}=\{a+2bi:\ a,b\in\mathbb Z\}$?
Recall that the ring of Gaussian integers is
$$\mathbb Z[i]=\{a+bi:\ a,b\in\mathbb Z\}.$$
Clearly
$$(a+bi)^2=a^2-b^2+2abi\ \ \mbox{and}\ \ (a+bi)^4=(a^2-b^2)^2-4a^2b^2+4ab(a^2-b^2)i.$$
Question. Is it ...
- 14.5k
2
votes
1
answer
154
views
Quaternions algebras whose class groupoid is actually a class group
Let $A$ be a finite-dimensional $\mathbb{Q}$-algebra, and let $\mathcal{O}$ be a maximal order in $A$. Like the case of number fields, we can define an equivalence relation to define something similar ...
5
votes
1
answer
227
views
Can a product of Cohn matrices over the Eisenstein integers with non-zero, non-unit coefficients be a Cohn matrix?
For $k > 1$, is it possible that $\begin{pmatrix} a_1 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} a_2 & 1 \\ -1 & 0 \end{pmatrix}\ldots \begin{pmatrix} a_k & 1 \\ -1 & 0 \end{...
3
votes
0
answers
99
views
Binary quartic forms with vanishing invariants: ring theoretic interpretation
Let $F(x,y) = a_4 x^4 + a_3 x^3 y + a_2 x^2 y^2 + a_1 xy^3 + a_0 y^4 \in \mathbb{R}[x,y]$ be a binary quartic form, and let $V(\mathbb{R})$ be the 5-dimensional $\mathbb{R}$-vector space of such forms....
- 25.5k
4
votes
0
answers
119
views
Linear relation between polynomial roots
Consider an irreducible polynomial $P\in\mathbb{Q}[x]$ of degree $n$ whose second leading coefficient is $0$ and $\alpha_1,\dots,\alpha_n$ its $n$ distincts roots. I am interested on the problem of ...
- 231
26
votes
5
answers
1k
views
Condition for a matrix to be a perfect power of an integer matrix
I have a question that seems to be rather simple but for I got no clue so far.
Let's say I have a matrix $A$ of size $2\times 2$ and integer entries. I want to know if there is a kind of test or ...
- 1,879
7
votes
1
answer
255
views
Why is the regulator of a number field normalized the way it is?
The regulator of a number field is essentially the covolume of the unit group embedded into the vector space $\{(x_1, \ldots, x_{r+s}): \sum_i x_i=0\}$ under the log embedding: $$x \mapsto (\log |\...
- 27.8k
0
votes
0
answers
248
views
How to prove the map of rings $\mathcal{R} \to \mathcal{R'}$ is flat?
We fix a finite extension $K$ of $p$-adic field $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_K$ and residue field $\kappa$. Consider the ring of witt vectors $W(\kappa)$ over the residue field $\...
- 870
2
votes
0
answers
111
views
Iwasawa's results about relation between Galois cohomology and principal factorization
Let $K$ be a Galois number field with Galois group and units group $G$ and $U$, respectively. How we can relate the first cohomology group $H^1(G,U)$ to principal factorization in K?
I'd try to find ...
- 385
0
votes
1
answer
172
views
When are the Artin symbols of two rational primes equal?
Let $K$ be a number field Which is Galios over $\Bbb Q$.The group $Gal(K/\Bbb Q)$ is not neccesarly abelian. Let $p_1$, $p_2$ be rational primes.
In this link they show if $p_1\equiv p_2$ modulo ...
- 723