Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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 ...

2
votes
0answers
66 views

Local polynomials of Frobenius-semisimple Weil representations which are tensor products of an Artin representation and an unramified character

Let $K$ be a local field and $\rho: W_K \to \operatorname{GL}(V)$ be a Weil representation. The for any finite extension $F/K$, we define the local polynomial $$ P(\rho|_F,T) = \det{(1 - \operatorname{...
2
votes
0answers
70 views

Number of lattice points on spheres with center not at the origin

Let $k\ge1$. It is known that the number of lattice points on the $k$-sphere $S^k(0)$ (center at the origin, radius $R$), namely the size of $\mathbb{Z}^{k+1}\cap S^k(0)$, is bounded by $R^{k-1+\...
1
vote
0answers
58 views

Relate solutions to a polynomial system in complex numbers to solutions in a finite field

Suppose I have a system of polynomials which are homogeneous but of distinct degrees that I want to solve simultaneously: $$F_1(z_1,\ldots,z_n)=\cdots=F_m(z_1,\ldots,z_n)=0.$$ Let $X(\mathbb F)$ ...
3
votes
0answers
161 views

Is the intersection of two function fields over finite fields again a function field?

I am interested in the question above. I know that the answer is NO if the base field is for instance $\mathbb{Q}$ (the intersection of $\mathbb{Q}(x^2)$ and $\mathbb{Q}((x-1)^2)$ is $\mathbb{Q}$ ...
16
votes
1answer
437 views

Where's the best place for an algebraic geometer to learn some algebraic number theory?

There are lots of introductions to number theory out there, but typically they are streamlined to assume as little prerequisite knowledge as possible. I'm looking for a text which does the opposite -- ...
2
votes
0answers
62 views

Understanding an equality in the paper “Class groups, totally positive units, and squares”

This is an excerpt from the paper "Class groups, totally positive units, and squares" (page 36). I am struggling to understand the last equality $|K^{(1)}_{2}:K|=|\overline{O}_K^{+}|$, the bar ...
4
votes
0answers
151 views

Reference request: ramified and local geometric class field theory

There are lots of references on global unramified geometric class field theory (following Deligne's $\ell$-adic sheaves approach). There are also some notes talking about how to extend Deligne's ...
6
votes
0answers
138 views

A refinement of Faltings' lemma

In his proof of the Mordell conjecture, Faltings proved the following important result: Let $K$ be a number field and $S$ a finite set of primes in $K$. Then for any $g \geq 2$ there exists a number $...
-1
votes
0answers
127 views

Giving mod $p$ on $O_{\overline{\mathbb{Q}}}$

Let $O_{\overline{\mathbf Q}}$ be the ring of all algebraic integers. Can I give an equivalence relation $\bmod p$ on $O_{\overline{\mathbf{Q}}}$ satisfying the following conditions (where $p$ is a ...
4
votes
0answers
148 views

Is there some computational evidence of the $pq$ analog of Serre's conjecture?

The $pq$ analog of Serre's conjecture (see "Mod pq Galois representations and Serre's Conjecture"- Khare, Kiming) states that if $\bar{\rho}_1:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F}_p)$ is a ...
4
votes
0answers
66 views

Minimal index of number fields of small degree

Let $K$ be a number field and let $\mathcal{O}_K$ be its ring of integers. For $a \in \mathcal{O}_K$ not contained in any proper subfield of $K$, the ring $\mathbb{Z}[a]$ is contained in $\mathcal{O}...
23
votes
1answer
691 views

Intuitive reason why the $j$-invariant is a cube?

Let $\tau$ be a CM point of discriminant $D$. Assume that $D$ is not divisible by $3$. Then $j(\tau)$ is an algebraic integer of degree equal to the class number $h(D)$. Let $ \gamma_2(\tau)=j(\tau)^{...
3
votes
0answers
125 views

I want a elaboration of the sketch of proof given in the Serre's Galois Cohomology on the existence of the dualizing module

I've wanted to understand the concept of the Dualizing module in the theory of Galois Cohomology. There are many references on it and of them all Neukirch's Cohomology of Number Fields seems to be ...
1
vote
0answers
96 views

Point Counts on $G$-torsors over Finite Fields

Let's assume we have a $G$-torsor $X_{1} \to X_{2}$, where $G$ is a finite abelian group, and both $X_{1}$ and $X_{2}$ are defined over $\text{Spec}(\mathbb{Z})$. Is there an easy way to compute $\#...
6
votes
1answer
436 views

Are the ideles literally a picard group?

I understand that in the number field / function field analogy, the ideles $\mathbb I_K$ of a number field $K$ are supposed to be analogous to the Picard group of a function field. Question: Is this ...
1
vote
0answers
61 views

Roots of unity and coordinates of points in abelian varieties

We consider an abelian variety $A$ defined over the rational numbers $\mathbb{Q}$. For a torsion point $P\in A(\bar{\mathbb{Q}})$, consider the field $\mathbb{Q}(P)$ obtained by adjoining to $\mathbb{...
4
votes
0answers
99 views

Traces of Frobenius Endomorphism on Etale Cohomology and $G$-torsors

I have a smooth, projective, and rigid Calabi-Yau threefold $X$ defined over $\mathbb{Q}$. Such spaces always have integral models. Let's assume we have an action on $X$ by a finite abelian group $G$...
4
votes
2answers
120 views

Congruence subgroups in arithmetic lattices of $\mathrm{SO}(n,1)$

I am currently reading the paper Deformation Spaces Associated to Hyperbolic Manifolds by Johnson and Millson, Section 7, and the highlighted bit below has been giving me difficulty: Specifically, ...
4
votes
0answers
153 views

Lower bound for some sums of roots of unity

Let $n$ be a positive integer (assume $n$ is prime for simplicity), and let $x_k = \pm1$, for $k = 0,1,2,..., n-1$. Let $\rho$ be an $n-$th primitive root of unity, I am interested in a lower bound ...
3
votes
1answer
125 views

Conductor of Galois representation attached to newform

(Sorry for poor my english skill..) Let $k$ and $N$ be positive integers and $\chi$ be a Dirichlet character modulo $N$. Let $F$ be a newform with number field $K_{F}$. (All coefficients of $F$ in $...
3
votes
0answers
63 views

Independence of number fields generated by roots of Littlewood polynomials

Let $\mathcal{R}_d \subset \bar{\mathbb{Q}}$ be the set of all roots of degree $d$ polynomials with $\{-1,0,1\}$ coefficients and $$ c(d) := \min_{\substack{ \alpha, \beta \in \mathcal{R}_d \\ \alpha^...
8
votes
1answer
135 views

Index of the endomorphism ring of an abelian surface

For an abelian surface $A/\mathbb{Q}$ such that $R:=\mathrm{End}_{\mathbb{Q}}(A)$ is an order in a real quadratic field $K$ (so a $\mathrm{GL}_2$-type surface), is there a bound on the index $[O_K : R]...
2
votes
0answers
84 views

Uniform boundedness of integral points for Mordell's equation

Let $d$ be a non-zero integer. The equation $$\displaystyle y^2 = x^3 - d$$ is known as Mordell's equation. It is closely related to the "discriminant equation" for elliptic curves, in the sense ...
1
vote
0answers
78 views

Dirichlet series of Euler's totient function for Gaussian integers?

Define the Euler's totient function for Gaussian integers $f:\mathbb Z[i]_{\ne 0}\mapsto \mathbb Z_{>0}$: $$f(z):=\sum_{\substack{q\in\mathbb Z[i]_{\ne 0}\\|q|\le|z|, |\gcd(q,z)|=1}}1,$$ and the ...
1
vote
0answers
97 views

Is there a (Riemann) explicit formula for $\sum_{p\le x}\frac{1}{p}$ involving a sum over the non-trivial zeros ρ of the Riemann zeta function?

Let $f(x)=\sum_{p\le x}\frac{1}{p}$ and $f_0(x)=\frac{1}{2}(f(x+0)+f(x-0))$. Then is there a (Riemann) explicit formula for $f_0(x)$ involving a sum over the non-trivial zeros ρ of the Riemann zeta ...
4
votes
0answers
110 views

When can one expect that the $\mu$-invariant of a $\mathbb{Z}_p$-extension of a number field is zero?

What is special about $\mathbb{Z}_p$-extensions which are motivic to ensure that their $\mu$ invariant is zero? Is there a simple conceptual reason. Here are some examples. Let $F$ be a totally real ...
4
votes
1answer
154 views

When is this localization map injective, if at all?

Let $K$ be a number field and $E$ be an elliptic curve defined over $\mathbb{Q}$. Consider the localization map $$ E(K)\otimes \mathbb{Q}_p/ \mathbb{Z}_p \rightarrow \bigoplus_{v|p} E(K_v)\otimes \...
10
votes
1answer
192 views

Gauss sums for general number fields

There is a wide litterature for the classical Gauss sums. For $\chi$ a primitive Dirichlet character modulo $N$, it is given by $$\tau(\chi) = \sum_{n \text{ mod } N} \chi(n) \exp(2i\pi n/N).$$ An ...
2
votes
1answer
217 views

On triangular numbers modulo primes

Let $p$ be an odd prime. For $a\in\mathbb Z$ let $\{a\}_p$ denote the least nonnegative residue of $a$ modulo $p$. The list $\{1^2\}_p,\ldots,\{((p-1)/2)^2\}_p$ is a permutation of all the quadratic ...
4
votes
0answers
72 views

Some Questions regarding the fine Shafarevich-Tate group

The fine Shafarevich-Tate (ST) group appears to not have been studied in much depth except for one paper by Wuthrich (2007). However, it only talks about fine ST group for elliptic curves, which makes ...
31
votes
0answers
612 views

Is there a rigid analytic geometry proof of the functional equation for the Riemann zeta function?

The adèles $\mathbb A$ arise naturally when considering the Berkovich space $\mathcal M(\mathbb Z)$ of the integers. Namely, they are the stalk $\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$ ...
3
votes
1answer
97 views

Newform of Half-integral weight modular forms

(Sorry for my poor english..) Let $k$ and $N$ be integers. Let $f\in S_{k+\frac{1}{2}}(\Gamma_1(4N))$ be a half integral weight modular form. I know that if $g \in S_{k}(\Gamma_1(N))^{new}$ in ...
4
votes
0answers
128 views

The geometry and arithmetic of the intersection of a cubic and quadric threefold

Let $f,g \in \mathbb{Z}[x_0, \cdots, x_4]$ be a quadratic and cubic form (i.e., homogeneous polynomials) respectively, and let $V(f), V(g)$ denote their respective projective varieties; $V(f), V(g)$ ...
0
votes
0answers
184 views

Conjecture that relates matrix systems with some specific functions as solution sets

what is written below is a conjecture that I posed , and I ask for a proof or a disproof of it .I have checked the conjecture from $n$=$1$ up to $n$=$10$ using Matlab, and all results were in ...
7
votes
1answer
206 views

On $\prod^{(p-1)/2}_{i,j=1\atop p\nmid 2i+j}(2i+j)$ and $\prod^{(p-1)/2}_{i,j=1\atop p\nmid 2i-j}(2i-j)$ modulo a prime $p>3$

QUESTION: Is my following conjecture true? Conjecture. Let $p>3$ be a prime and let $h(-p)$ be the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p})$. Then $$\frac{p-1}2!!\prod^{...
2
votes
0answers
98 views

Recovering the covering curve in Parshin's construction

A key step in the proof of Mordell's conjecture by Faltings' is a construction due to Parshin, which allows one to show that there is a finite-to-one map between the sets $$\displaystyle \{K\text{-...
6
votes
2answers
220 views

Algebraic exponential values

Is there a non-zero real number $t$ for which there exist infinitely many prime numbers $p$ with $p^{it}$ an algebraic integer? I would even be surprised to find a real $t \neq 0$ with both $2^{it}$...
5
votes
1answer
326 views

Is the following variant of Shafarevich's theorem known?

Let $Q$ be a finite simple group which may be realized as the Galois group of some extension of $\mathbb{Q}$ (like for instance $PSL_2(\mathbb{F}_p)$ for $p\geq 5$, or the monster group) and let $G$ ...
3
votes
0answers
157 views

Class fields without class field theory

Is there an English reference for the analytic construction of the Hilbert class field of an imaginary quadratic field without using class field theory? I am in particular interested in a proof of the ...
8
votes
1answer
330 views

Gauss - Dirichlet class number formula

Let $p=8k+3$ be a prime. Then the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p})$ is given by $$h(-p)=\frac 13\sum_{k=1}^{\frac{p-1}{2}}\left(\frac kp \right).$$ While this is ...
7
votes
1answer
207 views

Do we need the Weber function to generate ray class fields of imaginary quadratic fields of class number one?

I'm a bit confused by the role of the Weber function in generating ray class fields of imaginary quadratic fields of class number one. More specifically, let $K$ be such a field and $E$ an elliptic ...
1
vote
0answers
57 views

Values of a quadratic polynomial with restricted prime factors

Let $a,c$ be non-zero, co-prime square-free integers. Let $P_a$ be the set of primes such that the Legendre symbol $\left(\frac{a}{p}\right) = 1$ and define $P_c$ likewise. Consider the quadratic ...
1
vote
0answers
42 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 $...
38
votes
3answers
1k views

The roots of unity in a tensor product of commutative rings

For $i\in\{1,2\}$ let $A_i$ be a commutative ring with unity whose additive group is free and finitely-generated. Assume that $A_i$ is connected in the sense that $0$ and $1$ are unique solutions of ...
1
vote
0answers
88 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
votes
1answer
182 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 ...
0
votes
1answer
78 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? [closed]

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{...
6
votes
1answer
247 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 ...
1
vote
0answers
70 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 ...
2
votes
1answer
66 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. ...