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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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      98 views

      Extension of Erdos-Selfridge Theorem

      Erdos and Selfridge open their paper "The Product of Consecutive Integers is Never a Power" (1974) with the theorem $\text{Theorem 1:}$ The product of two or more consecutive positive integers is ...
      10
      votes
      1answer
      753 views

      Is there an elementary proof that there are infinitely many primes that are *not* completely split in an abelian extension?

      I'm currently in the middle of teaching the adelic algebraic proofs of global class field theory. One of the intermediate lemmas that one shows is the following: Lemma: if L/K is an abelian ...
      4
      votes
      0answers
      77 views

      Does the Gauss sum attached to $\chi$ ever belong to $\mathbb{Q}(\chi)$?

      Let $p$ be a prime number and $\chi$ be a primitive Dirichlet character of conductor $p$. We let $$g(\chi)=\sum_{a=1}^{p}{\chi(a)e^{2i\pi a/p}}$$ be the Gauss sum attached to $\chi$. Is this known ...
      1
      vote
      1answer
      129 views

      Bound on number of proper ideals of norm equal to n

      I have read in the paper by Einsiedler, Lindenstrauss, Michel and Venkatesh on Duke's Theorem the following bound that I don't understand: Let $d$ be a positive non-square interger and set let $K = \...
      4
      votes
      0answers
      82 views

      Hodge-Tate weights of cohomological cuspidal automorphic representation

      Let $\Pi$ be an algebraic cuspidal automorphic representation for $GL_{n}/\mathbb{Q}$ cohomological with respect to a dominant integral weight $\mu \in X^{*}(T)$ ($T \subset GL_{n}$ being the standard ...
      4
      votes
      1answer
      227 views

      Why do polynomials $x^n + 1 \bmod N$ close a shorter cycle when $n$ is even than when $n$ is odd?

      Polynomials $f(x) \bmod N$, where $f(x)$ is of integer coefficients and $N$ is a composite of two distinct primes $p, q$, form a cycle --- usually leaving a tail because the cycle tends to close not ...
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      votes
      0answers
      66 views

      Absolute convergence of the Fourier series of a smooth adelic function

      Let $f: \mathbb A/\mathbb Q \rightarrow \mathbb C$ be a smooth function. Smooth means that $f$ is continuous, smooth in the archimedean argument, for every $(x_0,y_0) \in \mathbb A = \mathbb R \...
      37
      votes
      1answer
      1k views

      Class field theory - a “dead end”?

      I found the claim in the title a bit astonishing when I first read it recently in an interview with Michael Rapoport in the German magazine Spiegel (8 February 2019). And I was wondering how he comes ...
      4
      votes
      0answers
      71 views

      Computation of Hochschild homology

      Let $A$ be a Dedekind domain. Let $n\geq 2$ be an integer. Is there a simple description of $HH_*(A, A/nA)$?
      4
      votes
      1answer
      243 views

      Confusion about topological Hochschild homology and $\mathbb{Z}_p$-topological Hochschild homology

      Let $R$ be the ring of integers in a perfectoid field of mixed characteristic $p$. Is $\pi_*THH(R)$ (as defined in Bhatt--Morrow--Scholze) $p$-complete (as an abelian group)?
      6
      votes
      1answer
      397 views

      The Hilbert symbols of quaternion algebras over a totally real field

      Let $k$ be a totally real number field. Every quaternion algebra $B$ over $k$ can be written as $$B = \left(\frac{a,b}{k}\right), $$ for some constants $a,b \in k^\times$. My question is, can I always ...
      9
      votes
      0answers
      169 views

      How small may the discriminant of an $S_d$-field be?

      In every degree $d$, the Galois closure of the typical number field has the maximal possible Galois group $S_d$. Denote by $f(d)$ the least absolute value of a discriminant of an $S_d$-field of degree ...
      5
      votes
      0answers
      160 views

      How did Gauss find the units of the cubic field $Q[n^{1/3}]$?

      Recently I read jstor article "Gauss and the Early Development of Algebraic Numbers", which gives a good description of the genesis of Gauss's ideas regarding the foundations of algebraic number ...
      0
      votes
      0answers
      59 views

      Fundamental System of Neighborhoods of the Identity in $G(\mathbb{A}_f)$

      Let $G$ be a linear algebraic $\mathbb{Q}$-group. Let $\mathbb{A}_f$ be the ring of finite adeles of $\mathbb{Q}$. It is known that the topology of $G(\mathbb{Q})$ induced from the inclusion $G(\...
      4
      votes
      1answer
      86 views

      Are there graphs with irrational eigenvalues which are all $>1$?

      The eigenvalues associated to a graph's adjacency matrix are necessarily algebraic integers, because the adjacency matrix itself is entirely integer. I'm curious as to whether it's possible to have ...

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