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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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0answers
27 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 \...
3
votes
0answers
95 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 ...
0
votes
1answer
123 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. ...
1
vote
1answer
124 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 ...
2
votes
0answers
183 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{...
3
votes
1answer
114 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 ...
11
votes
1answer
250 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 \...
-1
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0answers
80 views

A question about Ideals in Number Fields [closed]

for a finite number field K over Q suppose you have and Ideal I. Furthermore suppose I is generated by the algebraic numbers $x_1, ... , x_n$ If $N_{K/Q}(x_i)=C$ for all $i$ and some $C \in Z$ is it ...
10
votes
1answer
594 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 ...
9
votes
2answers
264 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 ...
6
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0answers
112 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 ...
26
votes
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
167 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 ...
8
votes
1answer
127 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
votes
1answer
296 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
91 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 $\...
0
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0answers
34 views

The localization of the integral closure of a valuation ring is a valuation ring

Let $L|K$ be a finite field extension, $v$ a non-archimedean valuation and $w$ an extension to $L.$ If $\mathcal{O}_L$ is the integral closure of the valuation ring $\mathcal{O}_v$ of $v$ in $L,$ show ...
-1
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0answers
65 views

Constructive Kronecker-Weber

Following up on a previous question, where should I look for an algorithm to represent elements of abelian extensions of $\mathbb{Q}$ in a cyclotomic extension of it? In particular, I am looking to ...
3
votes
0answers
98 views

Reference Request: Specialization map in Huber's Context

The specialization map $sp:\mathfrak{X}_\eta\to \mathfrak{X}_{red}$ has an important role in rigid analytic geometry. I tried looking in Huber's papers ("Continuous Valuations", "A generalization of ...
3
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0answers
102 views

The maximal unramified extension is the intersection of the maximal unramified extension of a localization?

I am reading Neukirch's book on Algebraic Number Theory, and there is a statement I can not figure out why it is true, and I hope someone here can help me. Apologies if question is at an inappropriate ...
16
votes
1answer
329 views

Centraliser of an absolute Galois group

Let $K$ be a finite extension of $\mathbb{Q}_p$. Is the centraliser of $\operatorname{Gal}(\overline{K}/K)$ in $\operatorname{Gal}(\overline{\mathbb{Q}_p} / \mathbb{Q}_p)$ trivial ? If yes, how can I ...
6
votes
1answer
203 views

Are all totally ramified $\mathbb{Z}_p$-extensions of local fields come from (relative) Lubin-Tate formal groups?

The setup is as follows: $k/\mathbb{Q}_p$ is a finite extension, $\mathfrak{p}$ is the maximal ideal of $\mathcal{O}_k$, $q=\#(\mathcal{O}_k/\mathfrak{p})$ $k'/k$ is a finite unramified extension of ...
1
vote
1answer
221 views

A question on a proof in the Ralph Greenberg's paper “On a Certain l-Adic Representation”

I'm currently reading the paper "On a Certain l-Adic Reprersentation" written by Ralph Greenberg.(Inventiones 1973) And I'm stuck with a proof of the Proposition 2. Here $k$ is a totally imaginary ...
8
votes
0answers
109 views

Local Langlands Correspondence for unramified principal series representations

Let $G$ be a connected, reductive group over a $p$-adic field $k$. Assume $G$ is quasi-split with Borel subgroup $B = TU$. Consider those irreducible admissible representations $\pi$ of $G(k)$ which ...
2
votes
0answers
110 views

Specialization map on geometric points

Let $\mathcal{X}$ be a proper and smooth scheme over $\text{Spec}(\mathbf{Z}_p)$, and let’s call $X$ the geometric generic fiber of $\mathcal{X}$, and $X_0$ the geometric special fiber of $\mathcal{X}$...
1
vote
0answers
94 views

Lefschetz trace formula and Frobenius elements

Let $\mathcal{X}$ be a smooth and proper scheme over $\text{Spec}(\mathbf{Z}_p)$ (with $\mathbf{Z}_p$ the $p$-adic integers). Let’s call $X$ its geometric generic fiber $X := \mathcal{X}_{\overline{\...
6
votes
1answer
237 views

Binomial coefficients in discrete valuation rings

Let $V$ be a complete discrete valuation ring whose residue field is a finite field $k=\mathbf{F}_q$. Let $\pi\in V$ be a uniformizer. For any integer $d,n\ge 0$, define: $${\pi^d \choose n} := \...
5
votes
1answer
193 views

Is there infinitely many prime $p$ such that the normalized trace of Frobenius $\frac{a_p(E)}{2\sqrt{p}}$ is arbitrarily small (but not zero)?

I just encountered the following problem and failed to find proper reference, could anyone kindly help to give me some illustration? For an elliptic curve $E$ without complex multiplication (just ...
0
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0answers
52 views

Definition of the local Artin conductor for a representation of the Weil group

Let $\rho$ be a continuous, finite dimensional complex representation of the Galois group $\operatorname{Gal}(\overline{F}/F)$, for $F$ a $p$-adic field. Is there a general notion of an Artin ...
3
votes
0answers
190 views

The closed subgroup of the idele corresponding to the maximal elementary $p$-extension of a global field

I want to know whether for any global number field $k$, the closed subgroup of the idele corresponding to the maximal elementary $p$-extension ($p$ is a prime number) is $k\ J^p$. The critical point ...
17
votes
2answers
470 views

Langlands correspondence for higher local fields?

Let $F$ be a one-dimensional local field. Then Langlands conjectures for $GL_n(F)$ say (among other things) that there is a unique bijection between the set of equivalence classes of irreducible ...
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0answers
103 views

A mixed of the Dedekind zeta function and the L-function

I have recently come across the following function, which seems like a "mix" between the Dedekind zeta function and the L-function: $\sum_I\frac{\chi_k(N(I))}{N(I)^s}$ where $\chi_k(n)$ is the ...
14
votes
1answer
257 views

Largest rank assumed by infinitely many elliptic curves

One of the most interesting questions in Mathematics concerns the Mordell-Weil rank of the group of rational points on elliptic curves $E/\mathbb{Q}$, namely whether this quantity is bounded as one ...
7
votes
3answers
270 views

Dirichlet/Chebotarev's Theorem for natural/analytic density

A little bird told me that Dirichlet/Chebotarev's Theorem is true not only for Dirichlet density, but also for analytic/natural density. I'm having trouble finding a citation for this. Does anyone ...
8
votes
2answers
294 views

Does $\det\left[\left(\frac{i^2-\frac{p-1}2!\,j}p\right)\right]_{1\le i,j\le(p-1)/2}$ vanish for every prime $p\equiv3\pmod 4$?

For any odd prime $p$, let $D_p$ denote the determinant $$\det\left[\left(\frac{i^2-\frac{p-1}2!\times j}p\right)\right]_{1\le i,j\le (p-1)/2},$$ where $(\frac{\cdot}p)$ is the Legendre symbol. Then \...
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0answers
51 views

Compositum and intersection of number fields

Suppose that we have three number fields $K,L$ and $M$. Assume that all of these number fields are Galois and that $M$ is abelian (for the sake of brevity we may assume that it is Cyclotomic). Edit: ...
4
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0answers
58 views

What is known about the homomorphisms from local to global Weil groups?

I have been reading Tate's article Number Theoretic Background in the Corvallis proceedings about the Weil and Weil-Deligne groups. I understand that the global Weil group $W_K$ of a number field $K$ ...
0
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0answers
39 views

Modern exposition of Igusa's Kroneckerian Model of Elliptic Modular Functions

I am trying to understand the Q model, and reduction modulo p, of modular curves, and in particular the Eichler-Shimura relation. It appears that Kroneckerian Model of Elliptic Modular Functions is ...
4
votes
1answer
161 views

Do all finite-dimensional division algebras appear as Wedderburn factors of rational group rings?

Suppose that $D$ is a division algebra that is finite-dimensional over $\Bbb Q$, does there exist a finite group $G$ such that one of the factors in the Wedderburn decomposition of $\Bbb Q[G]$ is a ...
6
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0answers
332 views

From Bhargava to Gauss — Why does correspondence of cubes and ideal classes imply Gauss correspondence?

In his seminal 2004 paper "Higher Composition Laws I" in the Annals of Mathematics, (doi:10.4007/annals.2004.159.217), Bhargava proves that for fixed $D \neq 0$, there is a bijective correspondence ...
13
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1answer
2k views

Some clarifications on Connes' approach to RH

How serious/promising is Connes' work on the Riemann Hypothesis? Connes is mostly known these days for his work in non-commutative geometry, having previously earned a Fields medal* for his work on ...
13
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0answers
178 views

Integral element over p-adic power series

Let $p$ be a prime number. and $R[[X]]$ be the ring of formal series with coefficients in a $p$-adic field $R$. Let $\Lambda=\mathbb{Z}_p[[X]]$. Question 1) Does there exist an explicit description ...
9
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1answer
537 views

Why are values of Eisenstein $E_2^*$ algebraic integers?

I'm looking for a proof that the following term is an algebraic integer whenever $\tau_N=\frac{N+\sqrt{-N}}{2}$ is a quadratic irrationality with class number $1$: $$A_N:=\sqrt{-N}\cdot\frac{E_2(\...
4
votes
0answers
102 views

Supercuspidals and representations of $\operatorname{Gal}(\overline{F}/F)$

Let $F$ be a $p$-adic field, $G = \operatorname{Gal}(\overline{F}/F)$ and $W$ the Weil group of $F$. The inclusion map $W \subset G$ is continuous with dense image, so $\rho \mapsto \rho|_W$ defines ...
0
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0answers
41 views

Linear polynomials with prescribed prime factors

Let $\mathcal{P}_1, \mathcal{P}_2$ be two subsets of primes with positive relative density (and not necessarily disjoint). Let $S_i$ be the subset of integers $n$ such that $p | n$ implies that $p \in ...
2
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0answers
241 views

Zagier's algebraicity of singular moduli

Let $\mathcal M_m\subset M(2,\mathbb Z)$ be the set matrices with determinant $n$. The modular group $\Gamma$ acts on $\mathcal M_m$ from the left and we have the following finite set as a set of ...
2
votes
0answers
53 views

What are the known results on the non-abelian Stark conjecture

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$, and let $\chi$ be a character of $G$. Let $c(\chi)$ be the leading coefficient of the Laurent series expansion of the ...
5
votes
1answer
273 views

A question on the injectivity of a canonical map between galois cohomology groups

I'm currently reading the book "Galois theory of $p$-extensions" by Helmut Koch. There, we calculate the cohomological dimension of the galois group $G(K/k)$ where $K$ is the maximal (normal) $p$-...
5
votes
0answers
95 views

Sign preserving Galois automorphisms

I have an algebraic number $\alpha \in \mathbb{Q}(\zeta)$, where $\zeta^n = 1$ is a root of unity (not primitive) given as a linear combination of powers of $\zeta$, i.e, $\alpha = \sum_{i=1}^k a_i \...
4
votes
1answer
129 views

Kummer congruences for totally real number fields

There is a generalization of the Kummer congruences to totally real number fields with characters due to Deligne-Ribet. For example, see the exposition here, more precisely see Theorem 2.1. What is ...