# Tagged Questions

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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### Property of Dirichlet character

Let $\chi_0$ be the unique Dirichlet character $\text{mod }1$ (i.e. $\chi_0(n) = 1$ for all $n$), $\zeta_p$ be a primitive $p$th root of $1$, and for any $a \in \mathbb{F}_p^\times$ and any Dirichlet ...
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### Kernel of the Artin map when dealing with S-ideles and S-divisors for function fields

Having understood that there is a strong correspondence between number fields and function fields, I am trying to work out some function field equivalents of class field theoretic invariants from Bost ...
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### Hecke characters and Conductors

Motivation: Let $\ell$ be an odd prime. There is a conductor-preserving correspondence between primitive Dirichlet characters of order $\ell$ and cyclic, degree $\ell$ number fields $K/\mathbb{Q}$. ...
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### Equation $x^2=y^p + 1$

can you help me please for solving this diophantine equation : $x^2=y^p+1$ and if you can give me a general method to studying such equation : $x^2=y^p+t$ Thanks
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### Even Galois representations “mod p”

Consider an irreducible $\mathrm{mod}$ $p$ representation: $$\rho: \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_2(\bar{\mathbb{F}}_p)$$ If $\rho$ is odd, it was conjectured by Serre in ...
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### Motivation for cyclotomic units

I am wondering the original motivation for considering cyclotomic units. Maybe one can rephrase the question as: Why did people initially consider such units in $\mathbb{Q}(\zeta_p)$ specially? ...
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In 1986, Don Zagier generalized Euler's theorem ($\zeta_\mathbb{Q}(2)=\pi ^2 /6$) to an arbitrary number field $K$: $$\zeta_K(2)=\frac{\pi^{2r+2s}}{\sqrt{|D|}}\times \sum_v c_v A(x_{v,1})...A(x_{v,s})... 1answer 209 views ### regulators of number fields Related question: Totally real number fields with bounded regulators Given a number field K with degree n and determinant D, what is the "best" upper bound for its regulator R, if any? I know ... 2answers 131 views ### Lubin-Tate modules and different uniformizers Suppose I have a local field \mathcal{O}_K and two different prime elements \pi and \overline{\pi}, i.e they differ by a unit \overline{\pi} = u \pi for some u \in \mathcal{O}_K^{\times} not ... 1answer 417 views ### Unramified extension of number fields Any finite field extension (in particular Galois extension) of \mathbb{Q} is ramified. Is there an intuitive geometric explanation of this fact? Suppose we have an number field K, is any Galois ... 3answers 313 views ### Argument of Zariski density to prove rationality of a regular map Question: I want to know if the following result is correct: Let k be a number field and k_v be a completion of k at some place v, denote K_v an algebraic closure of k_v. Proposition.(... 1answer 174 views ### Relation between ramification locus of a tower and of its constant field extension I am trying to understand Remark 7.2.22 (Page 256) of Algebraic Function Fields and Codes (Second Edition) by Henning Stichtenoth. In that remark he considers a tower \mathcal{F} = (F_0,F_1,F_2,\... 2answers 552 views ### What are the necessary conditions for a real number to be a cyclotomic integers？ The motivation of the question is that I try to test when a real number is not an cyclotomic integers. Or more specifically, when a positive real number is not a quantum dimension of a unitary fusion ... 0answers 131 views ### Residual Representation of a Motive Suppose we have M a hypergeometric motive, and \rho its associated Galois rep over \mathbb{Q}_{l}. Is there any easy/concrete way to find \bar{\rho}, the residual representation at a prime (in ... 1answer 172 views ### Compact hyperbolic 3-manifolds with prescribed quaternion algebra, quaternion parameters as ramification condition What is an interesting class of examples of hyperbolic 3-manifolds, each of which satisfies the following conditions? 1. It is compact 2. Its trace field contains a unique imaginary quadratic ... 2answers 384 views ### Dirichlet's approximation only using prime power as denominator I am not sure whether this is a suitable question for MO. We know the classical version of Dirichlet's approximation theorem that if x is a real number and Q>0 there exist p,q\in \mathbb{Z} ... 1answer 164 views ### number of generators of maximal ideals in an order of a number field let K be a number field of degree d over \mathbb{Q}), Let \mathcal{O}\subset K  be an order (i.e. a \mathbb{Z}-lattice of K contained in the integer ring \mathcal{O}_K of K). If  \... 0answers 129 views ### Class field theory for p-groups. I accidentally posted this question to math.stackexchange but think that it is more appropriate here (if not, please say so!): This question is from Neukirch's book "Algebraic number theory," page ... 1answer 133 views ### degree of Hecke field (number field of an eigenform) Let f\in S_k(\Gamma_1(N)) be an eigenform, and K_f be its number field, which is of finite degree over \mathbb{Q}. Consider the following statements. 1, [K_f:\mathbb{Q}]=\#\{Galois conjugates ... 1answer 188 views ### N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}? Assume that ab \neq 0. What is$$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}?$$I need this result, but unfortunately I am not a number theorist. Could ... 0answers 89 views ### On a theorem of Dwork and totally ramified extensions Suppose that K \subset L is a totally abelian ramified extension of local fields. Let \pi_L be a prime element of L^*. F \in Gal(\tilde{L}/L) is the Frobenius, where \tilde{L} is the maximal ... 2answers 994 views ### Motivating Lubin-Tate theory The Lubin-Tate theory gives an amazingly clean and streamlined way of constructing the subfield (usually denoted) F_\pi\subset F^\mathrm{ab} for a local field F fixed by the Artin map associated ... 1answer 669 views ### Are the algebraic numbers dense everywhere on the boundary of the Mandelbrot set? Let \mathcal{B} denote the boundary of the Mandelbrot set, and let \overline{\mathbb{Q}} denote the algebraic closure of the rationals. Further put \mathcal{B}_{\overline{\mathbb{Q}}} := \mathcal{... 2answers 414 views ### Frobenius elements in infinite extensions Let K be a number field, \bar K an algebraic closure and G the associated absolute Galois group. How can I define the Frobenius elements of G or at least their conjugacy class? I know how ... 1answer 170 views ### Ramification of prime ideal in Kummer extension Let \mu \in \mathbb{Q}(\zeta_n) lie above the rational prime p, and let the prime ideal \mathscr{P}\subset \mathbb{Z}[\zeta_n] have ramification index a over \mu. Why is it then true that ... 1answer 117 views ### Bibliography suggestion for Kummer theory I already posted a question about a sum involving the degree of a Kummer extension. Now I'm interested in a more specific fact about Kummer extensions. From Hooley's paper "On Artin's conjecture", we ... 1answer 143 views ### Doubt concerning a sum involving Kummer extension degrees I'd like to estimate the following sum$$ \sum_{n\leq x}\frac1{k_n}\;,\qquad x\rightarrow \infty\;, $$where k_n=[\mathbb{Q}(\zeta_n,a^{1/n}):\mathbb{Q}] is the degree of a Kummer extension for a ... 3answers 979 views ### Simple argument regarding sums of two units in a number field? I wonder if it is possible to show, without using the Schmidt subspace/Roth theorem/Baker's bounds on linear forms in logarithms or other very deep results, that, in a number field, not all integral ... 0answers 136 views ### Maximal abelian extension and tamely ramify extension Let K be a number field and v a finite place of K lying above a prime number p. Assume further that v is unramified over p. If K^{ab} is the maximal abelian extension let w be a ... 1answer 279 views ### Completion of a finite field extension is also finite? Let (L,w)/(K,v) be a finite extension of valuation fields, and let L_w, K_v be the respective completions of (L,w), (K,v). Is the field extension L_w/K_v finite? For nonarchimedean ... 0answers 191 views ### Geometric meaning of conductor Supppose L/K is a finite extension, choose \theta \in O_L such that L=K(\theta). We define the conductor of ring O_K[\theta] to be an ideal of O_L, namely: F=\{\alpha\in O_L|\alpha\cdot O_L\... 0answers 135 views ### calculation in a group ring I have some problems with the verification of the third equation in Lemma 1 in this paper. First of all, one has to notice that there is at least one Error in the Definition of a_{\kappa,\nu} above.... 1answer 608 views ### Elementary proof of a special case of Chebotarev's density theorem A special case of Cheboratev's density theorem states that, for K/\mathbb{Q} a Galois number field of degree n, then the rational primes that split completely in K have density 1/n. Is there ... 0answers 101 views ### On existence of rapid Arithmetic geometric procedure? We know that \pi can be computed by Arithmetic Geometric mean using Gauss-Legendre procedure which does provide fastest convergence rate as well with a guarantee of 2^n bits of \pi at nth ... 1answer 826 views ### A set of generators for \bar{\mathbb{Q}} Two questions: Does there exist a sequence \alpha_1,\alpha_2,... of algebraic numbers with degrees d_1,d_2,... s.t. for each i, d_i|d_{i+1} and \alpha_i= p_i(\alpha_{i+1}) with p_i a ... 0answers 209 views ### What is the relationship between the conductor of an order and the conductor of a number field extension? What is the relationship between the conductor \mathfrak{f}_{\mathcal{o}} of an order \mathcal{o}\subset \mathcal{O}_K and the conductor \mathfrak{f}_{L/K} of a field extension in the classical ... 2answers 520 views ### On bounds for idoneal integer What is the best known lower bound and upper bound known for such a number if it exists and have there been any attempts (computational including) to eliminate the existence of such a number in known ... 0answers 174 views ### The operator \left(q\frac{d}{dq}\right)^s and fractional derivatives of modular forms Recall the notion of a "nearly holomorphic modular form" introduced by Shimura: A function f : \mathfrak h \to \mathbb C is said to be nearly holomorphic of level \Gamma_1(N), weight k and ... 2answers 173 views ### Counting number of 2\times 2 unimodular matrices of particular type From set of numbers from \Bbb S=\{0,1,\dots,m\}, how many distinct 3\times 3 unimodular matrices parametrized by (a,b,c,d,e,f)\in\Bbb S^6 of following type can one form? \begin{bmatrix} a^2 &... 0answers 75 views ### Statements generalizing representability of integers by binary quadratic forms to n-variable higher homogeneous forms? Representing integers through the theory of binary quadratic forms is a well studied topic. We know that given a,b,c\in\Bbb N, based on discrimant b^2-4ac, we can study the representability of ... 0answers 159 views ### Writing integers in ring of integers of number fields Given a,b\in\Bbb N, we can write a=a_tb^t+a_{t-1}b^{t-1}+\dots+a_1b+a_0 where t=\lceil\log_ba\rceil and a_i<b<a. (1) Supposing if b\in\mathcal{O}_K where \mathcal{O}_K is ring of ... 0answers 154 views ### For K/E a number fields extension and F/E a finite Galois extension, how is the ramification in F\cdot K/K related to the one in F/E? Studying class field theory, I have come across the following Proposition: Proposition. Let K/E be an extension of number fields so that there is no nontrivial unramified subextension F/E with ... 2answers 558 views ### Does the Galois group of a Pisot polynomial contain the alternating group? Let n \in \mathbb{N}, and let p(X) \in \mathbb{Z}[X] be a monic polynomial of degree n. Suppose that exactly one complex root of p is of modulus > 1, and that the remaining n-1 roots of ... 2answers 217 views ### Decomposition and valuation rings I am reading Algebraic Number Theory by A. Fröhlich, M. J. Taylor, it first introduced the theorem: (K,u) be a field and its absolute value, (K_u,\bar u) be its completion and absolute value ... 1answer 186 views ### Can a product of conjugates be a Pisot number again? Let p(X) \in \mathbb{Z}[X] be an irreducible polynomial, and let \alpha_1 \dots, \alpha_n be its roots in \mathbb{C}. Suppose that \alpha_1 is a Pisot number, that is, \alpha_1 \in \mathbb{R},... 1answer 195 views ### On Pell's equation A post was made (Reduction from factoring to solving Pell equation) seeking clarification to solving$$x^2-Dy^2=1 to factoring when $D>0$. An answer was posted stating that to factor $N$, it ...
Let $(K,u)$ be a complete valued field, $u$ be its discrete absolute value (corresponds to a discrete valuation on $K$), then: ($\ast)$ Let $E/K$ is a finite separable field extension, then the ...
Set $h(x) = x^5+x^4+x^3+x^2+x-1$, let $L$ be the splitting field of $h$ over $\mathbb{Q}$, and let $p$ be a prime of $L$ lying over $2$. What is the isomorphism class of the inertia group $I_p$, ...