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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 ...

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16 views

Finite generation for a restricted ramification idele module

Let $k$ be a number field, let $\bar k \subseteq \mathbb{C}$ be a fixed algebraic closure of $k$, and let $S$ be the set of infinite primes of $k$. Denote by $k_S$ the maximal extension of $k$ inside $...
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63 views

Local factors determine Weil representations - proof of the Artin representation case

This post can be seen as a continuation of this post I created on MathOverflow. I want to understand the proof of the following Theorem from "Euler Factors determine Weil Representations" by Tim and ...
6
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1answer
176 views

A class number estimate

Let $\mathcal{D} = \{D \in \mathbb{Z} : D \equiv 0, 1 \pmod{4}\}$ be the set of discriminants. It is well-known that each element in $\mathcal{D}$ is the discriminant of a primitive binary quadratic ...
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1answer
74 views

How to show the set $\operatorname{Hom}_K(L,\bar{K})$ of all $K$-embeddings of $L$ is partitioned into $m$ equivalence classes of $d$ elements each?

Let $L|K$ be a finite separable extension. Denote the algebraic closure of $K$ by $\bar K$. $\forall x\in L$, denote $d=[L:K(x)]$ and $m=[K(x):K]$. How to show the set $\operatorname{Hom}_K(L,\bar{...
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1answer
222 views

The outer automorphism of the dihedral group $D_4$ and quartic polynomials

Let $L$ be a dihedral quartic field. That is, we have $[L : \mathbb{Q}] = 4$ and the Galois closure $M$ of $L$ is a degree 8 extension of $\mathbb{Q}$ with Galois group isomorphic to the dihedral ...
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0answers
62 views

Generalized Shimura correspondence

(Sorry for my poor english) Let $f(z)\in S_{2k}(\Gamma_0(N))$ be a newform. Let $\chi$ be a Dirichlet character modulo $N$ and $\chi'$ be an unique even Dirichlet character modulo $4N$ associated to ...
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1answer
61 views

Steinberg components of local deformation rings

Let $F=\mathbb Q_l$ and $E$ be a finite extension of $\mathbb Q_p$ with residue field $k \cong O_E/m_E$ ($p \neq l$). Let $\bar r: \Gamma_F=Gal(\bar F/F) \to GL_2(k)$ be a continuous representation. ...
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1answer
173 views

About real abelian number fields

How can I prove this: Let $K$ be a real abelian number field, $K_1$ be the Hilbert Class Field of $K$, and $J=K_1\cap K(\zeta_b)$. If a prime $p$ divided $[J:K]$ but did not divide $[K:\mathbb{Q}]$, ...
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0answers
41 views

Is there a result that connects the discriminant of an order to the discriminants of the generators of an integral basis for the order?

Let $\left\{ 1 , \omega_1, \dots , \omega_{n-1} \right\}$ be an integral basis for an order $\mathcal{O}$ of a number field of degree $n$ over $\mathbb{Q}$, let $\Delta $ be the discriminant of $\...
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95 views

Explicit algebraic constructions of Parshin covers

Let $X$ be an algebraic curve defined over a number field $K$ and has genus $g \geq 2$. Let $P$ be a $K$-point of $X$. We say that $X_P$ is a Parshin cover of $X$ if $X_P$ is defined over a finite ...
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4answers
450 views

In which cyclic cubic number fields does there exist this type of unit?

Let $K$ be a cyclic cubic number field with conductor $f$ and ring of integers $\mathcal{O}_K$. Define $K$ to be blue if and only if $$\operatorname{Norm}_{K/\mathbb{Q}}(w) = \operatorname{Norm}_{K/...
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1answer
412 views

The valuation of j-functions vs number of isomorphisms for an elliptic curve

Gross and Zagier prove the following fantastic result in their paper "Singular Moduli": Let $R$ be a discrete valuation ring over $\mathbb Z_p$ with uniformizer $\pi$ such that $k = R/\pi$ is ...
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133 views

Which pair of ternary quadratic forms in Bhargava's theory parameterize the ring of integers of the quartic number field of discriminant $225$?

The binary quartic form $\mathcal{V} = (a, b, c, d, e)$ has the same discriminant $D$ as the binary cubic form \begin{equation}\label{resolvform} \mathcal{C} = \left( 1, - c, b d - 4 a e, 4 a c e - a ...
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88 views

Does there exist an algebraic integer with discriminant dividing two numbers involving the discriminant, norm, and trace

Let $K$ be a number field of degree $n$ over $\mathbb{Q}$ and discriminant $\Delta $. Does there always exists a non-rational algebraic integer $\alpha \in \mathcal{O}_{K}$ of degree $n$ such that the ...
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1answer
133 views

Does there always exist a non-rational algebraic integer in a number field whose discriminant divides its norm?

Let $K$ be a number field of degree $n$ over the rationals. Under what conditions does there exist a non-rational algebraic integer $\alpha $ in $K$ such that the discriminant of $\alpha $ divides the ...
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182 views

Hardy-Littlewood vs heuristics on the zeta zeros

The first Hardy-Littlewood Conjecture asserts: Conjecture 1: Fix integers $0< a_1 < \ldots < a_k$. The density of those primes $p\le x$ such that $p + 2a_1,\ldots, p+2a_k$ are also primes, ...
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1answer
168 views

Local factors determine Weil representations - proof of the cyclic case

I already created this post on Math Stack Exchange but I was not so sure if this question fits better here. If it is not, I want to apologize in advance and feel free to delete my post. I want to ...
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0answers
186 views

Zeros of $p$-adic power series and rationality

Let $K$ be a non-archimedean field with valuation ring $(V,\mathfrak{m})$, and $K\langle t_1,\ldots, t_n\rangle$ a Tate algebra of convergent power series. Fix $f \in V\langle t_1,\ldots, t_n\rangle$....
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2answers
584 views

How to visualize finiteness of class number?

As the question title asks for, how do others "visualize" the finiteness of class number with algebro-geometric insight? I just think of it as a result in algebraic number theory and not one in ...
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1answer
270 views

Complex Multiplication and algebraic integers

Let $q=e^{2\pi i\tau}$ and $$E_2(\tau) = 1 - 24 \sum_{n=1}^\infty\frac{nq^n}{1-q^n}$$ be the Eisenstein Series of weight $2$ and let $E_2^*(\tau) = E_2(\tau) - \frac{3}{\pi\cdot Im(\tau)}$ be the ...
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104 views

Regular semisimple elements in $G(\mathbb Z_p)$

If $G$ is a reductive group defined over $\mathbb Z_p$ and let $K(\mathbb Z_p)=Ker(G(\mathbb Z_p) \xrightarrow{mod \: p} G(\mathbb F_p))$. Fix a Zariski-closed proper subset $Z$ of $G(\overline{\...
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1answer
159 views

Basis for modular forms with Nebentypus character

(Sorry for my poor english..) Let $N$ be a positive integer and $\chi$ be a Dirichlet character modulo 4N. I already know that the $\mathbb{C}$-vector space $S_{k}(\Gamma_1(N))$ has a basis $\{F_1,\...
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1answer
224 views

Interpolation of families of local fields

Let $\Sigma$ be the set of all places of $\mathbf{Q}$. We’ll call, for any $v\in\Sigma$, a local field complete with respect to an absolute value lying over $v$ and of finite degree over $\mathbf{Q}_v$...
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1answer
295 views

“Algebraization" of $p$-adic fields

Part 1: a single finite place. Let $K$ be a finite extension of $\mathbf{Q}_p$. Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion ...
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1answer
161 views

Explicit family of number rings $\mathcal{O}_{K_n}$ requiring $n$ generators?

Could someone provide or point me to a family of number rings $\mathcal{O}_{K_n}$ that require $n$ generators (as $\mathbb{Z}$-algebra)? Second best would be a family requiring $f(n)$ generators for a ...
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34 views

Cassels Frohlich, Module index

In the book “Algebraic Number Theory” written by Cassels and Frohlich, module index is defined. Let R be Dedekind domain, K be its quotient field, U be a n-dimensional vector space over K, and L,M be ...
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72 views

About newforms of half-integral weight

(Sorry for my poor english..) I have some questions about newforms of half-integral weight. In Mao's paper ("A generalized Shimura correspondence for newforms"), he said: "Ueda defined the set of ...
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1answer
212 views

Under What assumptions on $p$, $\mathcal{O}_K^* \simeq \mathbb{Z}_p^{*} \oplus \mathbb{Z}_p^{*}$

Let $p$ be a fixed prime number and $\mathbb{Q}_p$ be the field of $p$-adic numbers and $K$ be an extension of degree $2$ of $\mathbb{Q}_p$. Let $\mathcal{O}_K$ be the ring of integers of $K$ and $\...
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141 views

Are there partially algebraic Hecke characters?

$\newcommand{\C}{\mathbb{C}} \newcommand{\U}{\mathbb{U}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}}$ Let $F$ be a number field. Let $\chi\colon \mathbb{A}_F^\...
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125 views

Evaluate a curious determinant with Legendre symbol entries

Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. R. Chapman's conjecture on the exact value of the determinant of $$C_p:=\left[\left(\frac{i-j}p\right)\right]_{0\le i,j\le (p-...
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2answers
227 views

Inclusion of multiplicative group of one local field into the idele class group is a closed embedding, but inclusion of more than one isn't?

Let $k$ be a global field, $J$ its idele group, and $C = J/k^\times$ the idele class group. For any place $v$ of $k$, we have the familiar closed embedding $k_v^\times \hookrightarrow J$. More ...
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57 views

non $p$ part of the class group and analogous results

Washington had proven in 1978 that for $q$ a prime ($q \neq p$), if $q^{e_n}$ exactly divides the class number of $\mathbb{Q}_n$, ie the $n$-th layer in the cyclotomic $\mathbb{Z}_p$ extension, then $...
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1answer
202 views

Are all real-closed subfields of $\overline{\mathbb{Q}}$ conjugate?

Let $\overline{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. The absolute galois group $G_\mathbb{Q}$ of $\mathbb{Q}$ acts on the set of real-closed subfields of $\overline{\mathbb{Q}}$. ...
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0answers
188 views

Effective version of a lemma of Faltings

This question is about Lemma 2.3 of this paper: https://arxiv.org/abs/1807.02721, the lemma is attributed to G. Faltings. The statement is: Let $K$ be a number field and let $S$ be a finite set of ...
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412 views

What would be the simplest analog of Langlands in algebraic topology?

It is oversimplified, I know, but just as a superficial analogy, one may think of the fact that abelianization of the fundamental group is the first homology group, as some remote relative of class ...
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31 views

Minimal steps of construction for constructible number

It is known that a real number $\alpha$ is constructible if and only if it lies in a number field $K=K_{n}$ s.t. there exists a tower of field extension $\mathbb{Q}\subset K_{1}\subset K_{2}\subset \...
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135 views

Modular root of $-1$

Take two distinct coprime integers $a,b$ and take $q=a^2+b^2$. Consider the equation $(xy^{-1})^2\equiv-1\bmod q$. The solutions are two roots which correspond to $(x,y)=(a,b)$ and $(x,y)=(b,a)$. Look ...
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1answer
141 views

Parity and number of squares taken by polynomials in a range?

I have a polynomial $f(x)=a^2x^2+bx+c\in\mathbb Z[x]$ with $f(x)$ not a constant times a square and $abc\neq0$ and I want to know how many $x$ between $-a$ and $a$ the polynomial is a perfect square. ...
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1answer
138 views

references on group representation over local fields / a question on an argument of a Ralph Greenberg's paper

I'm currently studying Iwasawa theory. 1) There are many $\mathbb{Z}_p$-modules on which some Galois groups act. So I often face some facts on the group representation over local fields or p-adic ...
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190 views

The Hilbert Symbol and real algebraic geometry

Let $(a,b)_K$ be the quadratic Hilbert symbol in a local field $K$. Let $a$ be a rational number. By a consequence of the quadratic reciprocity law we have: $$\prod_{p} (a,-1)_{\mathbb{Q}_p}=\mathrm{...
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1answer
125 views

Sums of squares in global fields (Reference Request)

There is a result due to Siegel that, for a number field $K$, any totally positive element of $K$ is the sum of four squares of $K$. This is discussed in another question (sum of squares in ring of ...
12
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1answer
260 views

Determine $2^{\frac{p-1}{4}}\equiv 1\pmod p$ or $2^{\frac{p-1}{4}}\equiv -1\pmod p$ when $p\equiv 1 \pmod 8$

Let $p=8k+1\equiv 1\pmod 8$ be a prime, thus $2$ is a quadratic residue module $p$. Euler's criterion show that $$2^{\frac{p-1}{2}}\equiv 1 \pmod p.$$ So we must have $$2^{\frac{p-1}{4}}\equiv \...
10
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1answer
683 views

Books with exercises to learn Langlands program, Galois representations, modular forms

I want to develop a basic background in number theory with a goal towards contributing to some part of the Langlands program (which I know is vast, and has many different aspects; I want to do ...
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2answers
301 views

Berkovich space including both archimedean and non-archimedean worlds

From this Temkin's paper (at the end of section 1.1.3), I know that one may define Berkovich spaces that include both archimedean and non-archimedean worlds. This looks very interesting. Temkin ...
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0answers
118 views

$p >2$ is a prime, any facts about congruence relation between the class number of $Q(\sqrt p)$ and $Q(\sqrt-p)$?

Let $p$ be an odd prime. This is a question about the class number of $Q(\sqrt p)$ and $Q(\sqrt-p)$,which we denote by $h(p)$ and $h(-p)$ respectively. While doing my research on number theory I came ...
28
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2answers
2k views

How to visualize Dirichlet’s unit theorem?

As the question title asks for, how do others "visualize" Dirichlet’s unit theorem? I just think of it as a result in algebraic number theory and not one in algebraic geometry. Bonus points for ...
5
votes
1answer
177 views

Splitting of primes in cyclotomic $\mathbb{Z}_p$-extension

Let $p$ be a prime, $K$ be a finite extension of $\mathbb{Q}$ and $K_{\infty}$ be a cyclotomic $\mathbb{Z}_p$-extension of $K$ i.e. Gal$(K_{\infty}/K) \cong \mathbb{Z}_p$, the group of $p$-adic ...
10
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1answer
168 views

Unique factorisation of prime geodesics?

In T. Sunada's 1985 paper ``Riemannian coverings and isospectral manifolds'', it is noted that for a compact Riemannian manifold $M$, prime (closed and non self-intersecting) geodesics behave like ...
11
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1answer
324 views

If $R$ is an etale extension of $\mathbb Z$, then $R = \mathbb Z^n$?

Related question. Let $A$ be a ring, and let $B$ be an $A$-algebra which is projective and finite as an $A$-module. Then the trace $B \rightarrow A$ can be defined. Let's say that $B$ is separable ...
3
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0answers
92 views

When does a number field have $p$-rank greater than $n$?

Consider $F/\mathbb{Q}$, a number field. Let $S$ be a finite set of primes of $F$ containing the Archimedean primes. Let $n$ be any natural number and $L_n$ be a finite extension of $F$ such that $\...