All Questions
Tagged with reference-request nt.number-theory
1,408 questions
5
votes
2
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813
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On Weil's characters of type (A)
In Weil's paper
"On a certain type of characters of the idele-class group of an algebraic number field",
Weil introduces a class of characters on the Idele class group (of not necessarily finite ...
- 7,609
11
votes
1
answer
1k
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Finiteness of Tate-Shafarevich
Does anyone happen to know who conjectured the finiteness of the Tate-Shafarevich group?
We recall the conjecture. Let $E/K$ be an elliptic curve where $K$ is a number field. Then $Ш(E/K)$ is finite.
- 1,458
8
votes
1
answer
516
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Lower bound on class number of binary quadratic forms of discriminant of the form $n^2+4$
While searching for a use for the "sum invariant" of indefinite binary quadratic forms of discriminant $D = n^2 + 4$ (see https://cs.uwaterloo.ca/journals/JIS/VOL17/Smith/smith5.html), I believe I ...
- 1,521
2
votes
3
answers
2k
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Algebraic extensions of p-adic closed fields
I have been working with p-adically closed fields and there are two results that are used time and times again in what I am reading, but I cannot find any references where they are proved...
The ...
5
votes
0
answers
328
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Definition of logarithm for universal vector extension
Let $S$ be a topological $\mathbb{Z}_p$-algebra and $R\to S$ a surjection (where $R$ has $p$ nilpotent) with topologically nilpotent kernel which has a PD structure.
We know that if $G/R$ is a $p$-...
- 843
1
vote
1
answer
311
views
automorphism group of a given period
Maxim Kontsevich and Don Zagier defined the algebra of periods and conjectured that one can pass from a representation of a given period to another one using only three rules. Assuming this conjecture,...
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7
votes
1
answer
573
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Euler's Triangular Number closure properties
Burton, in "Elementary Number Theory", states that the following problems are due to Euler 1775:
If $n$ is a triangular number, then so are $9n+1$, $25n+3$ and $49n + 6$.
R. F. Jordan in the J. of ...
- 233
5
votes
0
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169
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Where does the notation $\operatorname{Tr}(\cdot)\bmod \ell^\alpha$ implies isomorphism come from?
In J-P Serre's article on Faltings-Serre (Resume du Course 1984-1985) he states (without proof) that for two finite-dimensional $\ell$-adic Galois representations of $\operatorname{Gal}(\mathbb{Q})$, ...
- 2,429
3
votes
1
answer
334
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A database on Maass forms?
Is there somewhere a database on Maass forms that includes eigenvalues, Taylor coefficients, etc...?
I am mainly interested in classical forms on $\Gamma(1)\backslash H$.
- 6,891
5
votes
1
answer
202
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Sufficient conditions for $\sum_{n \ge 1} a_n e^{-(a_1+\cdots+a_n) s} \sim \frac{1}{s}$ as $s \to 0^+$
Let $(a_n)_{n \ge 1}$ be a sequence of non-negative real numbers such that $\sum_{n \ge 1} a_n = \infty$, and set $\lambda_n := a_1 + \cdots + a_n$ for each $n$. Then the (generalized Dirichlet) ...
- 10.5k
4
votes
1
answer
260
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Reference request: normalization of intertwining operators for GL(2, C)
Take $F$ a local field and $\chi_1, \chi_2$ two characters, write $M(\chi_1, \chi_2)$ for the standard intertwining integral
$$M(\chi_1. \chi_2).f(g) := \int_{F} f\left( \begin{pmatrix} 0&-1\\ 1&...
- 590
4
votes
2
answers
560
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Asymptotics of special square-free numbers
What is the asymptotic number of square-free numbers less than $x$ with exactly $k$ prime divisors?
- 432
8
votes
3
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977
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When is an extension of characters de Rham?
Let $G$ be the abolute Galois group of $\mathbb Q_p$, let $\delta_1, \delta_2: G\rightarrow L^{\times}$ be continuous characters, where $L$ is a finite extension of $\mathbb Q_p$. Assume that $\...
- 423
3
votes
1
answer
2k
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What are Santilli's isonumbers?
A friend of mine asked me yesterday about Santilli's isonumbers. I told him that it was quackery. As I based my answer only on the general reputation of the guy and had no knowledge of the subject, I ...
- 12.4k
1
vote
1
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81
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Reference request for multiple free sequences
Erdos usually named a sequence of integers no one of which is divisible by any other as an $M$- sequence (M stands for "multiple-free") or primitive sequence.
For example it is easy to see that $\...
- 3,866
2
votes
0
answers
146
views
English proof for decomposition of modular polynomials
Let $\Phi_n(X,Y)$ be the modular polynomial that are the canonical equations for the modular curve $X_0(n)$. They parameterise pairs of elliptic curves related by a cyclic isogeny of degree $n$.
I am ...
- 521
4
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0
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500
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Zeros of polynomials modulo a non-prime
Suppose I have a set $S$ and I want to find a polynomial $p$ such that $p(s) = 0 \mod n$ if $s \in S$, and that it is non-zero modulo $n$ otherwise.
In the literature such an $S$ is sometimes called ...
2
votes
1
answer
596
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On the upper Banach density of the set of positive integers whose base-$b$ representation misses at least one prescribed digit
Let $b$ be a fixed integer $\ge 2$ and $A$ a proper subset of $\{0, \ldots, b-1\}$. Then define $X$ to be the set of all positive integers whose base-$b$ representation consists only of digits from $A$...
- 10.5k
11
votes
1
answer
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The Bombieri Vinogradov Theorem restricted to moduli divisible by $k$
The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c}
a\...
- 11.4k
6
votes
2
answers
799
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the global m-th power reciprocity law and Quartic Reciprocity Law
I'm reading Cox "Primes of the form $x^2+ny^2$". And I read a chapter about the global m-th power reciprocity law. Now I'm not able to prove the quartic and cubic reciprocity laws. Where can i find ...
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2
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713
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Reference requested for $\lim_{n \rightarrow \infty} \frac{\sum_{i=1}^{n} \bar{s}(i)}{n^2} = \frac{\pi^2}{30}$
While analysing the average runtime of an algorithm, I came across the following identity, and would like to know if anybody knows of any references for it?
For $i \in \mathbb{N}$, let $\bar{s}(i)$ ...
- 347
2
votes
2
answers
1k
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Place stabilizers for the absolute Galois Group
Fix an algebraic closure, $\overline{\mathbb{Q}}$ for the rationals and consider the set, $B_p$, of all places of $\overline{\mathbb{Q}}$ over a fixed (possibly infinite) prime, $p$, of $\mathbb{Q}$. ...
- 1,049
8
votes
1
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621
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On the irrationality measure of $\sum_{n=1}^\infty a^{-b^n}$
Pick integers $a, b \ge 2$ and let $\xi_{a,b}$ be the sum of the series $\sum_{n=1}^\infty a^{-b^n}$. It is known that $\xi_{2,2}$ is transcendental: I learned a proof of this from notes by M. ...
- 10.5k
5
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2
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1k
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A generalized Möbius function?
There are a number of generalizations of the Möbius function out there, which can be found by Google. But I'd just like to know if anything has been said about this:
For $k \geq 2$, $k \in \mathbb{Z}$...
- 1,075
3
votes
1
answer
171
views
Does positive relative density imply asymptotic additive basis behaviour?
First definitions: let $A, B \ \subset \mathbb{Z_{>0}}$ and $1\in A, 1\in B$. We define the relative density of $A$ with respect to $B$ to be $$rel(A, B) = \inf_n \frac{|A \cap [1,n]|}{| B \cap [1,...
3
votes
0
answers
392
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Fermat-Wiles "first case" in extensions of cyclotomic fields
I fell on the following fact :
Let $p$ be an odd prime, let $K$ denote the $p$-th cyclotomic field, let $L$ be an extension of $K$ with finite degree not divisible by $p,$ and assume that the prime ...
- 1,215
-4
votes
1
answer
600
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Is SOC known to imply the Grand Riemann Hypothesis? [closed]
I'm currently working on a conditional proof of the Grand Riemann Hypothesis, which is based on the assumption that every field automorphism of $\mathbb{C}$ that commutes with an element of the ...
- 6,984
7
votes
0
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786
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"Forthcoming paper" of Goldston-Graham-Pintz-Yıldırım
The above-named authors of [1] and its (significantly different) published version [2] write:
In a forthcoming paper, we will show how the methods here can be extended to prove corresponding ...
- 9,114
0
votes
1
answer
101
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Any results on $\gcd(N^2, D(N^2))$ where $N^2$ is deficient and $D(N^2)$ is the deficiency of $N^2$?
Any results on $\gcd(N^2, D(N^2))$ where $N^2$ is deficient and
$$D(N^2)=2N^2 - \sigma(N^2)$$ is the deficiency of $N^2$?
I checked OEIS sequence A033879 and have so far been able to get hold of ...
4
votes
2
answers
507
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Empty lattice simplex or White's theorem
White has proved (White, G. K. Lattice tetrahedra -- Canad. J. Math. 16 1964 389–396.) the following theorem:
If $T$ is a closed tetrahedron and $\Lambda$ is a lattice which contains the vertices of $...
- 12.3k
1
vote
1
answer
247
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When is $a^{2^n}+1$ prime finitely often unconditionally?
Define generalized Fermat numbers following OEIS and mathworld.
For natural $a,n$ and $a$ even, the generalized Fermat number (GFN) is
$F_n(a)=a^{2^n}+1$.
Very large GFN primes are known (in the ...
- 25.4k
5
votes
1
answer
227
views
Unicity of additive, $(-1)$-homogeneous, and shift invariant probability measures on $\mathbf N^+$
Let $\mathcal D$ be the set of all (finitely) additive probability measures $\mu^\ast: \mathcal P(\mathbf N^+) \to [0,\infty[$ such that $\mu^\ast(k \cdot X + h) = \frac{1}{k} \mu^\ast(X)$ for all $X \...
- 10.5k
11
votes
1
answer
2k
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The Class Number One Problem for Real Quadratic Fields
An approach to the Gauß class number one problem for imaginary quadratic fields is to determine the integral points on the modular curve $Y_{nonsplit}(n)$ for a suitable $n$. Here follows a quick ...
9
votes
2
answers
587
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Number Fields Arising from Newforms
It is well-known that, given a normalized eigenform $f=\sum a_n q^n$, its coefficients $a_n$ generate a number field $K_f$.
In their 1995 paper "Fermat's Last Theorem", Darmon, Diamond, and Taylor ...
- 1,422
5
votes
1
answer
842
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Reference for $p$-adic Hodge theory with coefficients
Let $K$ be a $p$-adic field and $L$ be a finite or infinite extension (maybe algebraic ?) of $\mathbb{Q}_p$.
Is there a reference for $p$-Hodge theory for representations $\rho : Gal_K \rightarrow ...
- 527
10
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3
answers
1k
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p-adic representations of a quaternion algebra over a local field
How to determine a complete set of isomorphism class representatives of the irreducible algebraic representations of $D^{\times}/F$ (where $D$ is a quaternion algebra over a local field $F/\mathbb{Q} ...
6
votes
0
answers
218
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Structure theorem for modules over multi-variable Iwasawa algebras
It is well-known that if $\Lambda=Z_p[[X]]$ and $M$ a finitely generated $\Lambda$-module, then $M$ is pseudo-isomorphic to
$$
\Lambda^{\oplus r}\oplus\bigoplus_{i=1}^s\Lambda/(F_i)
$$
for some ...
- 621
6
votes
1
answer
752
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Optimal lower bounds for the sum of digits in base $b$
Let $b \geq 2$ be an integer and let $s_b(n)$ be the sum of the digits of the base-$b$ representation of the nonnegative integer $n$
(e.g., $s_{10}(726)=7+2+6$). From the weak law of large numbers, it ...
user40023
7
votes
1
answer
283
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On one class of Somos-like sequences
This question is motivated by integrability of the sequence mistakenly arisen in the question Does this sequence always give an integer?
Let $m_1,\ldots, m_{k-1}$ be positive integers and sequence $\{...
- 12.3k
2
votes
1
answer
322
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Reference request: Research done on whether the Euler prime can be the largest factor of an odd perfect number
(Note: This was cross-posted from MSE.) I posted the following reference request in MSE three (3) days ago, but was unable to elicit any responses. I am cross-posting it to MO, hoping that it is ...
-6
votes
1
answer
488
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Automorphisms of partitions [closed]
I would like to know whether the notion of automorphism of the set of partitions of a positive integer $n$ has been considered so far or not. To make things clearer, I say that a partition of $n$ in $...
- 6,984
23
votes
1
answer
4k
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Chapters 1--4 of the Artin-Tate notes on Class Field Theory
Emil Artin and John Tate held a seminar on class field theory at Princeton University in 1951--1952. Their notes were published in 1967 by Benjamin (New York), but the first four chapters covering (...
- 18.7k
5
votes
3
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652
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Reference for a theorem of Tate on the endomorphism rings of AVs over finite fields
Let $k$ be a finite field, and $A$, $B$ abelian varieties over $k$. Let $T_p(A)$, $T_p(B)$ be
the (contravariant) Dieudonn\'e modules associated to the p-divisible groups attached to $A$ and $B$, ...
- 2,406
6
votes
1
answer
414
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Buildings associated to generalized $BN$ pairs
I'll begin by asking a general question, and then specializing to the situation I really care about.
Let $G$ be a group and let $(B, N)$ be a $BN$-pair in $G$ (see, for instance, page 39 of Tits' "...
- 1,453
1
vote
2
answers
505
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A conjecture of Montgomery: reference request
In the answer to this question, engelbret mentions "a conjecture of H. L. Montgomery (not the one on pair correlations, another one), which implies both the GRH and the Elliott-Halberstam conjecture."....
- 26k
2
votes
1
answer
328
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Families of ordinary Siegel Modular Forms
I'm looking for references to constructions and treatments of Hida Families/Eigenvarieties for ordinary Siegel modular forms (In particular: genus 2).
So far I've been reading Richard Taylor's thesis ...
- 1,629
5
votes
3
answers
881
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A question about partial Euler products
Let $K/{\mathbb Q}$ be an extension of degree $d$. Let $S$ be the set of primes $p$ which split completely in $K$. What can one say about the analytic properties of
$$
\zeta_{K, S}(s) : = \prod_{p \...
- 1,362
0
votes
0
answers
197
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'Adelic torus' not arising from a rational torus
Let $G$ be a reductive group over a global field $F$, and $\gamma$ a strongly regular semi-simple element of $G(F)$. Then the centralizer $G_\gamma$ is defines an $F$-torus $T$, and hence by base ...
- 3,799
4
votes
1
answer
456
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References to proofs of upper and lower bounds on the number of coprimes in an interval?
On the first page of the article "When the sieve works", the authors present upper and lower bounds for $S(T,T+x;\mathcal{E})$; the number of integers in the interval $(T,T+x]$ that are coprime to all ...
- 965
3
votes
1
answer
1k
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Collatz conjecture— finite state machine transducer construction, origination?
wikipedia has an entry on the Collatz conjecture with a section on As an abstract machine that computes in base two. this apparently describes a construction of a FSM transducer computing sequential ...
- 529