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
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Chebotarev density theorem with certain "bounds"
Let $f(x)\in \mathbb{Z}[x]$ be a nonconstant polynomial with no rational roots (in particular, $\deg(f)\geq 2$).
By the Chebotarev density theorem, there exist infinitely many primes $p,$ such that $f(...
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0
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100
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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 ...
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0
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98
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Equidistribution of Hecke points and Steinitz classes
Let $K$ be a number field and $\mathcal{P}$ be a prime ideal of $K$. Let $k_{\mathcal{P}}$ be the residue field $\mathcal{O}_K/\mathcal{P}$.
Consider the following construction used very often in ...
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1
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116
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Integer quadratic representation subject to discriminant minimization algorithm
Let $f(x)=ax^2+bx+c$ and $f(x)=n$. Is there an algorithm to choose $a,b,c$ such that the discriminant is minimized? Where $a,b,c,n,x$ are all integers.
More concretely, is there an algorithm to find $...
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0
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59
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What is the lattice point distribution over binary quadratic forms?
Let $f(x,y)=x^2+ny^2$ be the binary quadratic form of interest and consider the lattice points $S=\{ (x,y,f(x,y)) \in \mathbb{N}^3 \}$.
For simplicity, we keep things only on quadrant I of the ...
1
vote
0
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65
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Cardinality or covolume of $S$-units in quaternion algebras
Let $D=\left(\frac{a,b}{\mathbb{Q}}\right)$ be a quaternion algebra over $\mathbb{Q}$.
Suppose $D$ is ramified at a finite set $S$ of places and $\infty\in S$.
It is known that the $S$-units (the unit ...
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78
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Hasse invariant of subalgebra of division algebra over local field
This question didn't receive an answer on MathSE.
Let $K$ be a $p$-adic field, or more generally a local field. Let $D$ be a $d^2$-dimensional division algebra over $K$. Then $D$ is necessarily of the ...
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70
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$n$-th root of character on local field
Let $F$ be a non-Archidean local field of characteristic 0, and $\zeta_n$ the set of $n$-th roots of unity in the algebraic closure of $F$. Assume $\zeta_n\subseteq F$. Let $\chi:F^\times\to\mathbb{C}^...
- 833
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209
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Abelian extensions of Q and cyclotomic fields
I have changed some notation based on the comments of Chris Wuthrich and Wojowu.
For an abelian extension $F$ of $\mathbb{Q}$, let $c(F)$ be its conductor. That is, $c(F)$ is the smallest positive ...
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1
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235
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'$\times$' or '$\otimes$' when writing $L$-functions?
Recently, I came across the Langlands correspondence theorem, there is the following line:
$$L(s,\pi(\sigma) \times \pi(\tau)) = L(s,\sigma \otimes \tau), $$
where $\sigma$ and $\tau$ are ...
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500
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Genus of a number field
I'm reading Algebraic Number Theory by Neukirch. In chapter 3, he defines the genus of a number field as
$$ g = \log \frac{ |\mu (K)| \sqrt{|d_K|}}{2^{r} (2\pi)^{s}} $$ where $|\mu(K)|$ is its ...
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What is the interpretation of the reduction modulo $p$ of the modular curve $X(N)$ for $p$ dividing $N$?
Let $N>3$ be an integer.
The modular curve $X(N)$ is the compactification of the scheme parametrising triples $(E,t,t)$ where $E$ is an elliptic curve defined over a field of characteristic 0, and $...
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Reference request: Discriminant of a $V_4$-extension of local fields is the product of discriminants of intermediate fields
Disclaimer - cross-posting: I already posted this question on MSE, here. In line with the accepted answer of this meta question, I am also asking it here, since it is a research-level question and it ...
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399
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Meaning of a result of Gauss on "Mensura" of cyclotomic numbers
(This question was asked before on mathstackexchange. I received a few useful comments there, which helped me answer it for a special case, but I did not succeed in proving the general case.)
In an ...
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50
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The decomposition forms of primes in $A_5$-fields
Let $K$ be a number field of degree $5$ whose Galois closure (over $\mathbb{Q}$) has the Galois group $A_5$, the alternating group of degree five. Is there any result concerning the decomposition ...
- 385
5
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2
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446
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When adding several $\sqrt[n]{p}$ to the rational numbers, what is the degree of field extension?
When adding several $\sqrt[n]{p}$ to the rational numbers, what is the degree of field extension?
For example, does $[\mathbb{Q}(\sqrt[n]{2},\sqrt[m]{3}):\mathbb{Q}]=mn$ hold true? Are there more ...
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1
vote
1
answer
178
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Unramified composition for every extension
Let $K$ be a number field and $S$ be a finite set of primes. Is it possible to construct a finite extension $M$ of $K$ such that $LM/M$ is unramified at (the primes above) $S$ for all degree $n$ ...
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2
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625
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Proof of remark in algebraic number theory
$\def\aroof{\overline{\hspace{-1pt}\smash[t]{|}\hspace{1pt}\alpha\hspace{1pt}\smash[t]{|}\hspace{-1pt}}}$In Hua Loo Keng's "Introduction to Number Theory" page 489, there is the following ...
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0
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43
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Asymptotic counts for imaginary quadratic discriminants with fixed splitting conditions
Let $p$ be prime and $r$ be a positive integer. I am interested in asymptotics for the number of imaginary quadratic discriminants $d$ such that
$p$ does not divide the conductor of $d$,
$p$ splits ...
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1
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1
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138
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Quadratic unramified extension of a p-adic field
Let $F$ be a non-archimedean local field of residual characteristic $p\neq 2$, and let $E=F[\sqrt{\epsilon}]$ be the quadractic unramified extension, here $\epsilon$ is a non-square element of $\...
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0
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70
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Embeddings of unitary groups over $\mathbb{Q}$
$\DeclareMathOperator\GU{GU}$$\DeclareMathOperator\GL{GL}$I'm a bit confused by the following situation:
suppose we have an Hermitian vector space $V=K^3$ of matrix $$
J=\begin{pmatrix}& & \...
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1
vote
1
answer
319
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A Mordell equation $y^3=x^2+20$ [closed]
Recently I met a problem when I'm studying algebraic number theory.
Problem. Find all positive integer solutions of $y^3=x^2+20$.
I solved the situation when $x$ is an odd because the two ideals $(x+...
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votes
1
answer
354
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How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to higher number sets
How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to higher sets (Everything circled in red is what I'm interested in (+ the Cauchy integral to make it Dedekind ...
- 19
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124
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A normal extension of a number field of given degree that does not split over a given set of finite places
Let $K$ be a number field and $S$ be a finite set of non-archimedean places of $K$. Let $n>1$ be a natural number.
Question. Does there exist a normal extension $L/K$ of degree $n$ such that $L\...
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408
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Automorphic representation of GL(1)
These might be very silly questions, but somehow I am not able to understand it or I might have misunderstood something.
I am reading automorphic forms from this book.
What I have understood till now:
...
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147
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Non-abelian ray class fields for local fields
Let $K$ be a non-Archimedean local field. Then, thanks to work of Koch (when $K$ has positive characteristic) and Jannsen-Wingberg (when $K$ has characteristic zero, and odd residual characteristic) ...
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141
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A computation of nearby cycles
I'm currently reading P.Scholze's paper "THE LANGLANDS-KOTTWITZ APPROACH FOR THE MODULAR
CURVE". In Lemma 7.7, he showed a (maybe simple) nearby cycle computation, which I can't follow.
Now ...
5
votes
1
answer
336
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Diophantine equation $\cos(2\pi x)\cos(2\pi y) = \cos(2\pi z)$
While working on finite order elements of $\operatorname{SO}_n$, I meet this question:
Find all identities of the form $\cos(2\pi x)\cos(2\pi y) = \cos(2\pi z)$ with $x, y, z$ rational numbers.
As ...
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0
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220
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Reference book on the relation between modular forms and elliptic curves
What is a modern reference book to understand the relation between modular forms and elliptic curves after the proof of the Taniyama–Shimura theorem?
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1
answer
225
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p-adic L functions from Selmer groups - how canonical are they?
For this question, I am going to be very concrete but very much appreciate broader viewpoints. Let $F$ be a number field and define $F_n = F(\mu_{p^n})$ and let us suppose for simplicity that $\mu_p \...
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votes
2
answers
401
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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 ...
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votes
1
answer
346
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Families of Galois representations over disks
Edit on Nov. 20, 2023. This question is answered below in the case that $0<r_i<1$. And indeed it is shown in the answers to not be an interesting question in that case. So please take all $r_i=1$...
anon
3
votes
0
answers
87
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Commutant of irrep of $S_n$ (over local field)
Let $k$ be a field of characteristic zero and let $(V, \rho)$ be a finite-dimensional representation over $k$ of the symmetric group $S_n$. I would like to understand the commutant $\operatorname{End}...
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0
answers
101
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For a quadratic extension $E/K$, condition on a character $\chi:E^\times/E^{\times 2} \to C_2$ to give a $C_4$-extension $L/K$
Let $K$ be a finite extension of $\mathbb{Q}_2$, and let $E/K$ be a quadratic extension. By local class field theory, quadratic extensions $L/E$ correspond to quadratic characters $\chi:E^\times \to ...
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1
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303
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Conductor of determinant of a 2-dimensional Galois representation divides conductor of representation
I am studying Serre's paper "Sur les représentations modulaires de degré 2 de $\mathrm{Gal(\overline{\mathbb Q}/\mathbb Q)}$" and I'm stuck trying to prove that $N(\det\rho)$ divides $N(\rho)...
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96
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Selmer groups associated with Drinfeld modules
Given an elliptic curve $E_{/\mathbb{Q}}$ (or more generally, a number field) and a prime $p$, there is a standard short exact sequence
$$0\rightarrow E(\mathbb{Q})\otimes \mathbb{Q}_p/\mathbb{Z}_p\...
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0
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166
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Maximal p-extension and pro-p extension
I’m studying Iwasawa theory and I meet some questions Thanks a lot for your help.
Q_1: About terminology $p$-extension.
I find many reference use maximal $p$-extension or maximal abelian p-extension ...
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6
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0
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108
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Is the minus class group isomorphic to the relative class group?
I think this is something I should have known, but if I ever did I forgot about it. Consider the field $L$ of $p$-th roots of unity ($p$ prime) and its maximal real subfield $L^+$. The transfer of ...
- 32.2k
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votes
2
answers
339
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Upper bound on number of integral solutions of elliptic curves
I was studying M. Bhargava Et al's seminal paper titled "Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves"
And came across a very fascinating ...
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1
vote
1
answer
222
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Conceptual explanation for extra/missing $p$ solutions to $x^2+y^2=a \pmod p$ at $a=0$
Throughout, $p$ will denote a prime integer, and $k$ an arbitrary integer.
I have worked through V. Lebesgue's proof of quadratic reciprocity outlined by Keith Conrad in this MO thread, and I feel ...
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1
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188
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Representations of the symmetric group with image in a given subgroup of $\operatorname{GL}_m$
Let $S_n$ be the symmetric group on $n$ elements. The irreducible representations of $S_n$ are parametrised by partitions $\lambda$ of $n$ and are defined already over the integers $\mathbb Z$.
Let $\...
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0
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207
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Global class field theory and closure of unit groups
I'm looking for a reference for the following facts from global class field theory that I found without proofs. I will state them as questions, just in case I get the statement wrong. We fix $K$ a ...
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votes
2
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258
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Expressions for binomial residue sum $\sum_{k=0}^n {n \choose k} x^k \left( \frac{k}{q} \right)$
I'm interested in the sum:
$$\sum_{k=0}^n {n \choose k} x^k \left( \frac{k}{q} \right)$$
where $q$ is a prime number. This is just the binomial expansion with an extra weight on quadratic residues ...
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1
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583
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$\mathbb{Q}$-forms of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_8(\mathbb{R})$
Let $\mathbf{G}$ be the image of the natural embedding of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_4(\mathbb{C})\subset \operatorname{SL}_8(\mathbb{R})$. Then $\mathbf{G}$ is an ...
user515519
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0
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171
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Clarifications about the Iwasawa Main Conjecture
I would like to clarify a couple of things regarding the Iwasawa main conjecture.
In the paper where Mazur and Wiles prove the main conjecture, on page 182, it is written that $h_p(\omega^i, T)$ is ...
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votes
1
answer
240
views
Irreducible integral polynomials having roots module primes in arithmetic progressions
Let $f(x)$ be an irreducible polynomial with integer coefficients. One can show (see Exercise 7.2 in this paper of Lenstra) that if $f(x)=0$ has a solution mod $p$ for all but finitely many primes $p$...
- 6,104
4
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0
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253
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Special case of Eichler–Shimura
I'm reading ‘Rational Points on Elliptic Curves’ by Silverman and Tate, and the exercise 4.6 is about the following special case of the Eichler–Shimura theorem. Let $C$ be the elliptic curve given by ...
- 41
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vote
0
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109
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Degrees of trigonometric numbers
For a rational number $r\in(0,1)$ the number $z=\sin(r\pi)$ is an algebraic number — such numbers appear to be called trigonometric numbers.
What is its degree?
That is, what is the minimal degree of ...
- 8,046
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1
answer
924
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Can a field have an irreducible polynomial of any degree?
We all know that all the irreducible polynomials in $\mathbb{C}[x]$ are linear and in $\mathbb{R}[x]$ they aren't more than 2 degree. However,in $\mathbb{Q}[x]$ we can have an irreducible polynomial ...
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votes
1
answer
193
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Number fields with prescriped prime decomposition
Pick your favorite prime $p$, as well as three positive integers $e,f,g$.
For each such choice, does there exist at least one Galois number field $K/\mathbf{Q}$ of degree $n=efg$ in which $p$ has ...
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