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      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 cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces

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      Monogenic cubic rings and elliptic curves

      By an elliptic curve over $\mathbb{Q}$, we mean a genus 1 curve with a $\mathbb{Q}$-point. By a monogenic cubic order, we mean a unital cubic ring $R$ isomorphic to $\mathbb{Z}^3$ as a $\mathbb{Z}$-...
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      votes
      0answers
      120 views

      Newer versions of Mahler's Lemma

      I'm trying to find a way to numerically ensure that two constructible numbers are equal (this would be done by a computer). The idea is to find a polynomial $p(x)$ that contains both numbers as roots ...
      3
      votes
      2answers
      377 views

      On the product $\prod_{k=1}^{(p-1)/2}(x-e^{2\pi i k^2/p})$ with $x$ a root of unity

      Let $p$ be an odd prime. Dirichlet's class number formula for quadratic fields essentially determines the value of the product $\prod_{k=1}^{(p-1)/2}(1-e^{2\pi ik^2/p})$. I think it is interesting to ...
      3
      votes
      1answer
      107 views

      $L$-series and the $\zeta$-function of ideal classes modulo $f$

      Let $K$ be an imaginary quadratic field and let $\mathcal O_f$ be an order in $K$ with conductor $f$. Let $\chi$ be a proper character of $\operatorname{Cl}(\mathcal O_f)$ (non-principal if $f=1$). ...
      4
      votes
      1answer
      317 views

      The sign of an interesting sum involving a Dirichlet character

      Let $\chi_{q}$ be a primitive Dirichlet character with modulus $q$ (see definition at wikipedia ). For example for $q=5$ we have \begin{equation} \begin{aligned} \chi_{5,1}&=(1, 1, 1, 1, 0),\\ ...
      9
      votes
      1answer
      316 views

      A conjectural formula for the class number of the field $\mathbb Q(\sqrt{-p})$ with $p\equiv3\pmod8$

      Question. Is my following conjecture new? How to prove it? Conjecture. Let $p>3$ be a prime with $p\equiv3\pmod 8$, and let $h(-p)$ denote the class number of the imaginary quadratic field $\...
      4
      votes
      2answers
      196 views

      Order of Galois action of modular form

      (Sorry for my poor english...) Let $N$ be a positive integer and $f=\sum_{n=1}^{\infty} a(n)q^n\in S_{k}(\Gamma_0(N))\cap K[[q]]$ be a cusp form with number field $\mathbb{Q}(\xi_{N})\subset K$ ...
      2
      votes
      0answers
      51 views

      Vector extension for p-divisible group

      Background: I am trying to understand a proof of Messing's book at Page 120. My goal it to understand the universal vector extention. Reference: Messing, The crystals associated to Barsotti-Tate ...
      2
      votes
      0answers
      88 views

      Explicit example for Display Theory for p-divisible group

      Recently I am studying the display theory of formal p-div groups ([1] )by Zink. I would like to study by working on an example. As far as I understood, the display theory is a generalization of ...
      2
      votes
      1answer
      275 views

      Motivation to study the order theory (ring theory)

      I'm currently reading a paper of Georges Gras on the Reflection Principle. The paper uses some theorems about orders (ring theory) from the book "Maximal Orders" by Reiner. I find the book interesting,...
      6
      votes
      0answers
      161 views

      Diophantine applications of Paramodularity

      I’ve asked this question to quite a few people in person and so far haven’t seen a good answer...but I believe one should exist, so here goes! Ok, we all know how to (roughly) prove Fermat’s Last ...
      6
      votes
      1answer
      170 views

      Zariski closure of set of units in a number ring

      Let $\mathcal{O}$ be a number ring. Letting $r$ and $2s$ be the number of real and complex embeddings of $\mathcal{O}$, the number ring $\mathcal{O}$ is a lattice in $\mathcal{O} \otimes \mathbb{R} \...
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      votes
      2answers
      130 views

      Rank mod $p$ of a non-singular matrix with given determinant

      Let $A$ be a non-singular $n$-by-$n$ matrix with integer entries. Assume that $p^r\nmid \det(A)$. Does it follow that $A$ has an $(n-r+1)$-by-$(n-r+1)$ minor that is non-singular modulo $p$? If the ...
      3
      votes
      0answers
      36 views

      Specialization of an irreducible polynomial and monogeneous ring of integers

      Let $P\in\mathbb{Q}[X_1,\ldots,X_n][T]$ be an irreducible polynomial, which is monic in $T$ of degree $d\geq 1$. For $x=(x_1,\ldots,x_n)\in \mathbb{Q}^n$, let $P_x=P(x_1,\ldots,x_n,T)$. It is well-...
      7
      votes
      1answer
      239 views

      Is there any conditions on a finite abelian group so that it cannot be class group of any number field?

      The Cohen-Lenstra paper says the probability that the odd part of a class group being cyclic is close to 0.98. So I was thinking: can we find any conditions on a finite abelian group so that it cannot ...

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