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
2,169
questions
5
votes
0
answers
147
views
Generalization of Deuring's theorem
Deuring has the following important theorem on elliptic curves over a finite field: Let $p$ be a prime at least $5$, and $N=p+1-a$ an integer with $|a|\leq 2\sqrt{p}$, then the number of elliptic ...
- 151
0
votes
1
answer
176
views
Leech lattice shortest vector vs other 23 cases and E8 cases
In this paper by Viazovska, she said that:
"The E8-lattice sphere packing 𝒫E8 is the union of open Euclidean balls with centers at
the lattice points and radius $1/\sqrt{2}$." So I think ...
- 337
4
votes
1
answer
395
views
4-manifold $M$ with intersection form of Leech lattice
Do we know any 4-manifold $M$ with intersection form of rank-24 Leech lattice?
Is $M$ smooth? Differentiable to which $n$-order?
Can $M$ be obtained by any surgery on 3 copies of E8 4-manifolds with ...
- 337
2
votes
1
answer
324
views
Lang's remark on Lindemann-Weierstrass theorem
On his book "Introduction to transcendental numbers", page 99-100, Lang wrote
"Finally, we note that Lindemann actually proves something slightly stronger than the algebraic ...
- 3,737
5
votes
2
answers
308
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 ...
- 579
4
votes
0
answers
144
views
Epstein zeta function for non-fundamental discriminant to L-series
Let $Q(x,y) = ax^2+b xy + cy^2$ be a primitive integral positive-definite quadratic form, with associated number field $K$. If $D=b^2-4ac$ is a fundamental discriminant, then it's well-known that $$\...
- 341
0
votes
0
answers
108
views
What is the quotient group $\mathfrak{q}^2/\mathfrak{p}^2 \mathbb{Z}_p$?
Let $p \geq 2$ be prime and $K=\mathbb{Q} (\zeta_p),~ \zeta^{p-1}=1$ with ring of integers $\mathcal{O}_K$. We denote $\mathfrak{p} \mid p$ the prime ideal of $K$ dividing $p$. Let $K_{\mathfrak{p}}$ ...
- 870
2
votes
0
answers
166
views
Ramanujan graphs in Polynomial time
I am a research scholar with a computer science background, currently working on graph theory. I am working on a reduction to prove that a problem is NP-complete. I need to include the Ramanujan graph ...
5
votes
0
answers
183
views
Image of $\gamma-1$ on etale $(\varphi,\Gamma)$-modules
Let $p\geq 3$ be a prime, $D$ be an etale $(\varphi,\Gamma)$-modules over the classical period ring $A_{\mathbb{Q}_p}=\mathbb{Z}_p[\![T]\!][1/T]^{\widehat{\phantom{xx}}}_p$ and $\gamma$ be a ...
- 569
2
votes
1
answer
200
views
Computing explicit isogenies between elliptic curves over different kinds of fields
I have some questions about isogenies of elliptic curves in two settings:
1. Elliptic curves defined over the rationals.
1.1. Given two elliptic curves $E/\mathbb{Q}$ and $E'/\mathbb{Q}$ we can decide ...
- 595
4
votes
0
answers
130
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
1
vote
0
answers
128
views
Characterization of Selmer group in terms of two descent
This question is about p 337 of Silverman's book ''The arithmetic of elliptic curves'', p 337, http://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf.
Let $E:y^2=x^3+ax^2+bx$ and $E':Y^2=...
- 1,405
-2
votes
2
answers
149
views
Calculate the great common factor between $2^{2n+1}-1$ and $2^{4m+2}+1$ [closed]
How to calculate the great common factor between $2^{2n+1}-1$ and $2^{4m+2}+1$, where $n$ and $m$ are positive numbers.
We guess that: the great common factor is $1$.
- 577
6
votes
1
answer
503
views
Need for Drinfeld modules compared to elliptic curves over function field
In a sense ever since they were invented that Drinfeld modules and later shtukas are the "right" objects to study and play the role of elliptic curves over function fields by virtue that ...
- 4,246
7
votes
3
answers
565
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 ...
- 8,722
2
votes
2
answers
358
views
A ring map from algebraic integers to algebraic closure of $\mathbb F_p$
Let $p$ be a prime and ${\mathbb F}_p$ the finite field with $p$ elements. There is a canonical ring map ${\mathbb Z} \to {\mathbb F}_p \cong {\mathbb Z}/ p {\mathbb Z}$. Denote the image of $n$ by $[...
- 971
2
votes
1
answer
173
views
Can we encode a torsor as a binary function on the isomorphism classes of objects?
Let $G$ be a group object in a topos $\mathcal{T}$. Then we have the notion of a $G$-torsor in $\mathcal{T}$, and the set of isomorphism classes of such objects is denoted $H^1(\mathcal{T};G)$. For ...
- 15.1k
2
votes
1
answer
266
views
On Elkies' $\text{9T32}$ nonic and a shared property with j-function formulas
I. First Set
Before going to Elkies' nonic, we start with something a bit simpler. There is a list of j-function formulas in this MSE post. For example, for prime levels $p = 5,7,13,$ we have,
$$j=\...
- 12.1k
3
votes
0
answers
174
views
Extending a theorem of Washington
In Class numbers of the simplest cubic fields, Larry Washington states the following theorem (I have added some hypotheses to make the statement more self-contained), which is Theorem 2 in said paper:
...
- 25.5k
0
votes
0
answers
114
views
Rank growth of elliptic curve $E:y^2=x^3-17$ in quadratic number field
Let $E:y^2=x^3-17$ be an elliptic curve.
It is known that rank$(E/\Bbb{Q})=0$.
(For example, prop $6.5$, $362$p in Silverman's book 'The arithmetic of elliptic curves')
Over $K=\Bbb{Q}(i)$, what is ...
- 1,405
2
votes
1
answer
198
views
Eigenvalues of Frobenius in $l$-adic cohomology
Let $X_0$ be a smooth projective variety over a finite field $\mathbb{F}_q$. Let $X$ be the corresponding variety over the algebraic closure $\bar{\mathbb{F}}_q$. Let $Fr_q\colon X\to X$ be the ...
- 21.1k
0
votes
0
answers
69
views
Rank of infinite family of elliptic curves over the rationals without assuming finiteness of Sha
Are there any known infinite families of elliptic curves over the rationals, that are proved to have rank $\geq 2$, without assuming finiteness of their Tate-Shafarevich group?
- 71
6
votes
0
answers
371
views
How to explain the relationship between Tate–Shafarevich and Ideal Class Group, when all else fails?
In the short paper On the Tate–Shafarevich group of a number field of Sameer Kailasa, he reviews a curious phenomenon by which the class group of a number field $K$ appears as the exact kernel of the ...
- 1,242
4
votes
2
answers
262
views
Biquadratic extension of global function fields with cyclic decomposition groups
Let $F$ be a global function field, for example $F={\mathbb F}_q(t)$, the field of rational functions in one variable over a finite field ${\mathbb F}_q\,$.
Question. What would be an example of a ...
- 13.6k
2
votes
1
answer
94
views
Rationality of quasi-elementary group actions
I am learning the paper https://arxiv.org/pdf/1604.01005.pdf "Reductive group actions" by Knop and Krotz.
They defined that a linear k-algebraic group $H$ with unipotent radical $H_{u}$ is ...
- 101
3
votes
0
answers
229
views
When does the set of Frobenius conjugacy classes happen to be the whole infinite Galois group?
Let $ K$ be a number field and let $S$ be a finite set of places that contains the archimedean places. Let $G_{K,S}=\operatorname{Gal}(K_{S}/K)$ be the Galois group for a maximal extension $K_S/K$ ...
- 491
10
votes
1
answer
333
views
Vinogradov-Korobov prime number theorem for number fields
Without assuming the Riemann hypothesis, the traditional error bound of the prime-counting function $\pi(x)$ is $O(x\exp(-c(\log(x))^{1/2}))$. As shown by the Wikipedia page for the Landau prime ...
- 211
2
votes
0
answers
174
views
Interpretation of completed tensor product of algebras over lower base
Let $\mathbb F$ be a finite field of order $q = p^n$. It is known that
$$\mathbb F[[x_1]] \mathbin{\widehat{\otimes}_{\mathbb F}} \mathbb F[[x_2]] = \mathbb F[[x_1, x_2]].$$ Geometrically, this is the ...
- 251
1
vote
0
answers
151
views
The map from the decomposition group to the Galois group of the residue fields
$\DeclareMathOperator\Gal{Gal}$Let $L/K$ be a Galois field extension. Let $L$ be a field and $K$ be a number field. Let $B$ be a valuation subring of $L$ and let $A$ be the preimage of $B$ in $K$ (i.e ...
- 45
1
vote
1
answer
138
views
Does $P(\exp_p(a),\exp_p(b))=0$ imply $P=0$, where $\exp_p(\cdot)$ is $p$-adic exponential?
Classical case:
Let $\{a,b\}$ be linearly independent set over $\mathbb Q$ and $\{e^{at},e^{bt}\}$ be linearly independent set over $\mathbb Q[[t]]$. Suppose $P(x,y)$ is a polynomial over $\mathbb Q$. ...
- 870
1
vote
1
answer
77
views
Norm fixed under complex automorphisms implies algebraic
Is it true that if I have $\alpha \in \mathbb{C}$, $q,w \in \mathbb{Z}$ such that for every automorphism $\sigma$ of $\mathbb{C}$, $$|\sigma(\alpha)|=q^{w/2}$$ then $\alpha$ must in fact be algebraic?
...
- 302
1
vote
1
answer
137
views
The map from the ring of integers to the residue field of a valuation subring is surjective
Let $L$ be a number field and let ${\mathcal{O}}_L$ be its ring of integers. Let $B$ be a valuation subring of $L$ and let $k_B$ be the residue field of $B$. Then the map from ${\mathcal{O}}_L$ to $...
- 45
1
vote
1
answer
138
views
Decomposition groups for the Galois module $\mu_8$
$\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Gal}{Gal}
\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\Fbar}{{\overline F}}
\newcommand{\G}{\...
- 13.6k
16
votes
1
answer
751
views
Who first proved that algebraic numbers form an algebraically closed field?
I am interested in the history related to algebraic numbers and have two questions:
Who first proved that algebraic numbers form a field?
Who first proved that algebraic numbers form an algebraically ...
- 40.9k
1
vote
2
answers
208
views
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 ...
- 11
1
vote
0
answers
38
views
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 ...
- 211
0
votes
0
answers
102
views
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
votes
0
answers
303
views
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_{\...
- 291
2
votes
0
answers
151
views
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 ...
- 4,246
2
votes
0
answers
195
views
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}...
- 489
5
votes
1
answer
417
views
“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 ...
- 279
0
votes
1
answer
369
views
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
7
votes
1
answer
673
views
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
1
vote
1
answer
51
views
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 ...
- 833
1
vote
0
answers
170
views
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....
- 515
3
votes
3
answers
565
views
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 ...
- 515
8
votes
2
answers
339
views
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 ...
- 3,088
6
votes
1
answer
297
views
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 ...
- 313
4
votes
0
answers
55
views
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 ...
- 1,546
2
votes
0
answers
203
views
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 ...
- 311