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 ...

**3**

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**0**answers

72 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

**0**answers

40 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 ...

**6**

votes

**1**answer

189 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

**0**answers

82 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{-...

**5**

votes

**2**answers

194 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 $2^{it}$...

**4**

votes

**0**answers

150 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

**0**answers

143 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 ...

**7**

votes

**1**answer

282 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 ...

**6**

votes

**1**answer

188 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

**0**answers

54 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

**0**answers

41 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 $...

**37**

votes

**2**answers

842 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

**0**answers

86 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

**1**answer

178 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

**1**answer

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? [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{...