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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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        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; $V(f), V(g)$ ...
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        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
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        1answer
        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^{...
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        votes
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        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
        2answers
        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 $2^{it}$...
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        votes
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        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 $G$ ...
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        0answers
        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 ...
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        votes
        1answer
        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
        1answer
        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 ...
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        vote
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        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
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        0answers
        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
        2answers
        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
        0answers
        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
        1answer
        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 ...
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        votes
        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? [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{...

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