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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&...
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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 $\mathfrak{a}$ ...
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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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In On $l$-independence of Algebraic Monodromy Groups in Compatible Systems of Representations, they stated such an example (Example 6.3): Let $X$ be a connected normal scheme of finite type over $\...
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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 ...
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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 $L\...
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$\newcommand{\Gm}{\mathbb{G}_\mathrm{m}}$I've been told that the formulation of class field theory in étale cohomology is through Artin-Verdier duality, but I am struggling to make this connection ...
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Local class field theory concerns with norm subgroups, in particular they correspond to abelian extension of the base field. Giving that in a galois extension L/K can never be that: $N^L_K(L^\times) = ...
rico rico's user avatar
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Let $k$ be a global function field of positive characteristic $p$ (e.g. $k = \mathbb{F}_p[t]$). Let $x \in k$ be non-zero and assume that $x$ is not a $p$th power. For each place $v$ of $k$, we can ...
Daniel Loughran's user avatar
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Let $j$ be the j-invariant defined by $$j(\tau)=\frac{E_4(\tau)^3}{\Delta(\tau)},\quad \Im(\tau)>0$$ where $$E_4(\tau)=1+240\sum_{n=1}^\infty \frac{n^3 q^n}{1-q^n},\quad q=e^{2\pi i\tau},$$ $$\...
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Let $K$ be an imaginary quadratic number field. If the class number of $K$ is greater than 1 there exist non-principal ideals. Some of the ideals that are principal have prime norm, i.e. they are ...
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Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication (CM) by an order $\mathcal{O}$ in an imaginary quadratic field $K$. Let $p$ be an odd prime of good reduction for $E$, and let $\...
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Let $K$ be a number field and let $Cl(K)$ denote the ideal class group of $K$. If $L/K$ is finite extension, then the norm map induced a homomorphism $Cl(L) \to Cl(K)$ . Now let $M$ be the maximal ...
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Let $F$ be a non-Archimedean local field. For a multiplicative character $\chi$ of $F^\times$, let $a(\chi)$ be the conductor of $\chi$. For a nontrivial additive character $\phi$ of $F$, we denote ...
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Thanks for your help. I know the definition of Weil group and how to get it for $p$-adic local field cases. By reading the answer https://math.stackexchange.com/a/2898449/1007843, I write down my ...
Rellw's user avatar
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What are some examples of approaches to non-abelian class field theory that existed before the Langlands program?
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It is well known that $37$ is the first irregular prime, i.e. a prime number $p$ which divides the class number $h_K$ of $K = \mathbb Q(\zeta_{p})$. For $p=37$, the Hilbert class field $L$ of $K$ is ...
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Let $K/\mathbb{Q}$ be a Galois number field and $H/K$ its Hilbert class field. Then we have $\mathrm{Gal}(H/K) \cong \mathrm{CL}_K$ from class field theory, and moreover $H/\mathbb{Q}$ is Galois. We ...
Daniel Loughran's user avatar
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Let $k$ be the rational numbers and assume $A$ is a central simple algebra over $k$ with ind$(A) > 2$. We denote by $X$ the variety of all right ideals in $A$ of reduced dimension $2$. This variety ...
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I am trying to understand the proof of Proposition 4.25 in David Harari's book "Galois Cohomology and Class Field Theory". Proposition 4.25: Let $G$ be a profinite group. Let $H$ be a closed ...
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$ \newcommand{\Q}{{\mathbb Q}} \newcommand{\Z}{{\mathbb Z}} \newcommand{\Gal}{{\rm Gal}} \newcommand{\Br}{{\rm Br}} \newcommand{\isoto}{\overset\sim\longrightarrow} $1. Let $K$ be a $p$-adic field (a ...
Mikhail Borovoi's user avatar
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Let $F$ be a non-archimedean local field, and let $T$ be an $F$-torus with cocharacter group $M$. Let $E/F$ be the splitting field of $T$ in an algebraic closure $\overline F$. Write $\Gamma={\rm Gal}...
Mikhail Borovoi's user avatar
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3 votes
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Given positive square-free integers $m$ and $d$. Lets denote the prime factorization of $m$ by $m=\prod p_i$. I know that if each $p_i$ can be written as $x^2+dy^2$ then $m$ can be written in that ...
Alexander's user avatar
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I. Condition If there are integers $(r_1, r_2, r_3, r_4)$ such that, $$ar_1^2+br_1r_2+cr_2^2=ac\\ r_3=(ar_1+br_2)/c\\ r_4=(br_1+cr_2)/a$$ then $(a u^2+buv+cv^2)(a x^2+bxy+cz^2)= (a z_1^2+bz_1z_2+cz_2^...
Tito Piezas III's user avatar
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Bogdan Grechuk recently asked for elementary consequences of the Langlands program. His question reminded me of something that has nagged at me for a while. Before I state my question, let me give a ...
Timothy Chow's user avatar
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I'm recently reading the very beginning of David Harari's book Galois Cohomology and Class Field Theory, and I met a problem with the definition of cup-products for the modified cohomology. Before ...
Jianing Song's user avatar
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Suppose $\mathcal H$ is the upper half plane, and $\Gamma\subset \operatorname{PSL}_2(\mathbb Z)$ is a finite index subgroup. Then by Belyi's theorem we know that $\Gamma\backslash\mathcal H$ is an ...
Richard's user avatar
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Let $K$ be a number field, and write $\mathbb{A}_{K,f}^\times$ for the group of finite ideles of $K$. That is $$ \mathbb{A}_{K,f}^\times = \{(u_v)_v \in \prod_{v \nmid \infty} K_v^\times : v(u_v) = 0 \...
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Let $K/\mathbb{Q}$ be an imaginary quadratic extension of discriminant $-D_K < -3$ and fix a prime $p > 3$ that is split in $K$ as $p\mathcal{O}_K = \mathfrak{p}\overline{\mathfrak{p}}$. ...
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Let $\mathfrak{m}$ be an ideal of the ring of integers of a number field $K$ and $S$ is a subset of the real places, then the ray class group of $\mathfrak{m}$ and $S$ is the quotient group $$I^{\...
HGF's user avatar
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For $p\equiv3\pmod4$, let $h_{\mathbb{Q}(\sqrt{-p})}$ denote the class number of $\mathbb{Q}(\sqrt{-p})$, the class number formula shows $h_{\mathbb{Q}(\sqrt{-p})} < p$. I wonder if there is a pure ...
Wenhao Huang's user avatar
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3 answers
803 views

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=\{...
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It is well known that for $K/F$ a finite "cyclic" extension of number fields, we have $g_{K/F}=a_{K/F}$, where $g_{K/F}$ denotes the relative genus number, and $a_{K/F}$ denotes the ...
A. Maarefparvar's user avatar
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3 votes
1 answer
426 views

Class field theory defines an isomorphism between the Brauer group of a finite extension of p-adic fields and a cyclic group with a canonical generator. This in turn defines an isomorphism of the ...
NZK's user avatar
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1 answer
241 views

The power residue symbol is the multiplicative character $\bigl(\frac a \cdot\bigr): \mathcal I_\mathfrak m\to \mu_n$ that satistfies $\bigl(\frac a {\mathfrak p}\bigr)\equiv a^{\frac{N\mathfrak p-1}{...
roasted_cashews's user avatar
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Let $F$ be a local field, in particular a finite extension of $\mathbb{Q}_p$ for some prime $p$ and let $Art_{L}: L^\times \to Gal(F^{ab}/F)$ be its local Artin map. We know that if $L/F$ is a finite ...
Mario's user avatar
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0 answers
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I am searching a reference for local class field theory that use the Weil group instead of the absolute Galois group. In particular that the Artin map is an isomorphism between the multiplicative ...
Mario's user avatar
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Let $K$ be a finite extension of $\mathbb{Q}$ and $L/K$ be a $\mathbb{Z}_p$-extension with finite layers $L_i$, hence $L_j/L_i$ is cyclic of order $p^{j-i}$ (put $K=L_0$). Let $U_E$ be the unit group ...
Ehsan Shahoseini's user avatar
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I just read Iwasawa's review of Meyer's "Die Berechnung der Klassenzahl abelscher Körper über quadratischen Zahlkörpern" and wonder how the problems Iwasawa mentions at the end of it ...
Thomas Riepe's user avatar
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3 votes
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It is well-known that, in a real-closed field $K$, every polynomial of degree $>2$ is reducible in $K$. But in this case the characteristic of $K$ is zero. My question is: there exists a non-...
Medo's user avatar
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1 vote
1 answer
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I am curious to know if we can somehow relate to the study of local class field theory through Lubin-Tate formal group with the study of class number of a field (global field in general) in class ...
MAS's user avatar
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2 votes
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Let $K$ be a number field, and let $S=S_f\cup S_{\mathbb R}\cup S_{\mathbb C}$ be a finite set of places of $K$, where $S_f$ denotes the set of finite places in $S$, $S_{\mathbb R}$ denotes the set of ...
Mikhail Borovoi's user avatar
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4 votes
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Let $F$ be a $2$-adic field, i.e. a finite extension of the $2$-adic numbers $\mathbb{Q}_2$. Suppose that I have a quadratic extension $E = F(\sqrt{d})$ of $F$. Given a unit $\alpha \in \mathcal{O}_F^\...
Sebastian Monnet's user avatar
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Let $F$ be a global field without real places (that is, a function field or a totally imaginary number field). Let $K/F$ be a cyclic extension of degree $n$. Let $S$ be a ${\rm Gal}(K/F)$-invariant ...
Mikhail Borovoi's user avatar
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-1 votes
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The context: We consider ideal theoretic formulation of global class field theory of a number field $K$ and in following all used terminology I'm going to use is adapted from these notes: https://math....
user267839's user avatar
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Let $L_0$ be a global field without real places, that is, a global function field or a totally imaginary number field, and let $V_f(L_0)$ denote the set of finite (that is, non-archimedean) places of $...
Mikhail Borovoi's user avatar
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1 vote
1 answer
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I have following question about a remark in J. Neukirch's Algebraic Number Theory around page 397. The context: We consider ideal theoretic formulation of global class field theory of a number field $...
user267839's user avatar
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4 votes
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Let $K/k$ be an unramified biquadratic extension of number fields (i.e., $\operatorname{Gal}(K/k)\simeq V_4$), and $k_1$, $k_2$ and $k_3$ its three intermediate fields. I know, in general, we can ...
ayoub-chess's user avatar
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1 answer
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Theorem 4 of Eric Bach's "Explicit bounds for primality testing and related problems" states the following: Let $K$ be a number field of degree greater than 1. Let $d$ be the absolute value ...
Rashad Ek's user avatar
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4 votes
1 answer
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Given an abelian extension $K$ of $\mathbb{Q}$, the global Kronecker–Weber theorem tells us that there exist a positive integer $N$ and a primitive $N$-th root of unity $\zeta_N$ such that $K\subseteq ...
Yijun Yuan's user avatar
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