Questions tagged [nt.number-theory]
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
4,673
questions with no upvoted or accepted answers
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Strong approximation, Chinese remainder theorem and surjectivity of reduction
I am trying to make sense of different interpretations of the strong approximation and related properties in algebraic groups. Already in the classical case, say for $SL(2)$, I would like to ...
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108
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Model for Shimura curves
There is a list of Shimura curves (upto genus 2) in the paper https://math.dartmouth.edu/~jvoight/articles/shimbound-mcom-fixed-errata.pdf. My question is can I construct corresponding models for them ...
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179
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Near Pochhammer symbols: the equation $(n)_m-(k)_l=2$ for integers greater than or equal to two
In this post I consider the following equation involving Pochhammer symbols,
$$(n)_m-(k)_l=2\tag{1}$$
for positive integers $n\geq 2$ and $k\geq 2$, and positive integers $m\geq 2$ and $l\geq 2$.
...
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407
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View Dirichlet character as a character of Galois group
In Jaclyn Lang's article "On the image of the Galois representation associated to non-CM Hida family" section 2, the Dirichlet character $\chi$ module $N$ is also viewed as a character $\chi\colon\...
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Why does $\theta: \mathbb{B}^+_{dr} \rightarrow \mathbb{C}_p$ have no continuous or equivariant section?
Fix a $p$-adic field $K$ with perfect residue field $k.$ Let $\mathbb{C}_K$ be the completion of the algebraic closure of $K,$ and let $$R = \varprojlim \mathbb{C}_K/p,$$ where the transition maps in ...
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219
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Extra line bundles from torsors
Another math.stackexchange question (here: $\mathbb{G}_m$-torsors and line bundles) goes over a way to construct a line bundle $L$ from a $\mathbb{G}_m$-torsor $T \to B$, by using a decomposition $\...
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Algebraic integers whose matrix representations have singular values in an interval
Let $K$ be a finite extension of $\mathbb{Q}$. Let $\mathcal{O}(K)$ be the ring of integers of $K$. Let $\omega_1,\ldots,\omega_n$ be an integral basis for $K$ over $\mathbb{Q}$.
For each $a \in K$...
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232
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On the values of $\prod_{k=1}^{(p-1)/2}(e^{2\pi i/12}-e^{2\pi i k^2/p})$ for primes $p>3$
In a recent preprint, I investigated
$$S_p(x):=\prod_{k=1}^{(p-1)/2}(x-e^{2\pi ik^2/p}),$$
where $p$ is an odd prime and $x$ is a root of unity.
Motivated by Question 337879 and Question 338325, ...
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90
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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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107
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Hecke equidistribution for $L^1$ functions
Put shortly, the question is, does the Hecke equidistribution hold for $L^1$ functions?
To be specific, the version of H.E. I need is
$T_N f \rightarrow \int f d\mu$ as $N \rightarrow \infty$,
...
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132
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Rational points on quotient of Fermat curve by symmetric group
One may take a quotient of the Fermat curve $x^n+y^n+z^n=0$ by the symmetric group $S_3$ permuting the coordinates. This should be a curve defined by a polynomial $p_n(e_1,e_2,e_3)=0$, where $e_1=x+y+...
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186
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How to find a CM point with the image in the elliptic curve under modular parametrization given
everyone! Let $E:y^2+y=x^3-61$ be the minimal model of the elliptic curve 243b. How can I find the CM point $\tau$ in $X_0(243)$ such that $\tau$ maps to the point $(3\sqrt[3]{3},4)$ under the modular ...
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Introduction to Hida Theory
I know that the obvious references to learn Hida theory are this own (difficult) papers. I`ve looked online for other sources but only found summaries which are too short.
Does anybody know of ...
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202
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Kronecker limit formula, modular curves, and the class number problem
Let
$$Q(x,y)=ax^2+bxy+cy^2$$
be a positive definite quadratic form with $a>0$ and $D=b^2-4ac<0$. Let
$$\zeta_Q(s)=\sideset{}{'}\sum_{m,n}Q(m,n)^{-s},$$
the accent indicating that $(0,0)$ is ...
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165
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Derived weight filtration on motivic Galois representations
Thanks to modern techniques (such as the pro-etale site), we can now understand etale (co)homology of varieties and motives as "genuinely" derived (e.g. DG) Galois-equivariant objects. I'm looking for ...
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Finite dimensional irreps of $p$-adic groups
What are some examples of finite dimensional irreducible complex representations of $SL_2(\mathbb{Q}_p)$?
One knows such a representations cannot be smooth, so probably the examples will be ...
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258
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Reference request for some result of de Bruijn on zeros of some holomorphic function
In a video lecture on Youtube, ''Vaporizing and freezing the Riemann zeta function'', Terry Tao states that ''de Bruijn proved that if for some $t_0$ the zeros of $H_{t_0}$ are contained in the strip $...
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240
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Numbers with a square sum arrangement
Informal version. For which $n>1$ can the numbers $1,\ldots, n^2$ be arranged in a square form such that the sums of the numbers in the little squares (consisting of $4$ numbers) are all equal?
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264
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Kaczorowski's Paper on Distribution of Primes
I am looking for a digital copy of the following paper by Jerzy Kaczorowski: ON THE DISTRIBUTION OF PRIMES (mod4)
https://www.degruyter.com/view/j/anly.1995.15.issue-2/anly.1995.15.2.159/anly.1995.15....
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499
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Four-square Conjecture
Lagrange's four-square theorem states that every nonnegative integer
can be written as the sum of four squares. My following conjecture is much stronger than this classical theorem.
Four-square ...
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157
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Limit of the real part of a geometric sequence
I came across the following problem, which turned out to be surprisingly hard:
Show that $\underset{n\rightarrow \infty}{\lim} \left| \mathrm{Re}((\frac{1+i\sqrt{7}}{2})^n)\right| = \infty.$
...
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270
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A positive irrational number $\alpha$ such that $\lfloor k^n \alpha \rfloor$ and $M$ are always coprime?
From my previous question:
Is it true that there always exists a positive integer $n$ such that $p | ⌊k^n⋅α⌋$
, I came up with a similar question:
Given a positive integer $k$ such that $k>2$, $...
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1
answer
759
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Hilbert class fields and transfer
Let $K/k$ be an extension of number fields and $H_k$, $H_K$ their respective Hilbert class fields. Is there a transfer map from $\text{Gal}(H_k/k)$ to $\text{Gal}(H_K/K)$?
4
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153
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The arithmetic meaning of opers (if any)
Let $G$ be a complex, connected semi-simple Lie group, $G'$ its Langlands dual
group, $\mathrm{Bun}_G$ the moduli stack of $G$-bundles on a smooth projective curve $\Sigma$ over complex numbers, $\...
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276
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Rough conjecture about eigenvalues of Maass forms
Basically, as I know, we know almost nothing about Maass forms. For example, Cohen constructed first (maybe not) example of a Maass cusp form by using one of Ramanujan's $q$-series, as a non-definite ...
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272
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What arithmetic would you do in parallel?
This is a post asking for references, and soliciting problems and people interested in accelerated computing. I will add the big-list tag and make it community-wiki. If this interests you strongly, ...
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60
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Average minimal index of cyclic cubic fields
It is known that the set of binary cubic forms
$$\displaystyle T_3 = \{F_{a,b}(x,y) = ax^3 + bx^2 y + (b - 3a)xy^2 - ay^3 : a,b \in \mathbb{Z}\}$$
parametrize the set of cyclic cubic fields, in the ...
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203
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Higher dimensional generalization of an identity between traces of Hecke operators and number of elliptic curves over finite fields?
In http://www.math.ubc.ca/~behrend/ladic.pdf, the author uses his generalization of Lefschetz trace formula to smooth algebraic stacks to prove an interesting identity (Proposition 6.4.11.):
$\sum_{k}...
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198
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Fourier coeffients of Cantor measure
For $0<\theta<\frac{1}{2}$, denote by $\mu_\theta$ the uniform Cantor measure with dissection ratio $\theta$. It is not hard to show that the Fourier–Stieltjes transform of $\mu_\theta$ is
$$
\...
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171
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On sums of minima and maxima
Let $h_1,\ldots,h_n$ be positive integers, and define
$$m(h_1,\ldots,h_n)=\sum_{r_1=0}^{h_1-1}\ldots\sum_{r_n=0}^{h_n-1}\min\left\{\frac{r_1}{h_1},\ldots,\frac{r_n}{h_n}\right\}$$
and
$$M(h_1,\ldots,...
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Local behaviour of fractions with bounded denominator / Was it already studied?
My question is about a point process that I feel it would be natural to study, but that I have never heard of… This point process would represent, morally, the local behaviour of the set of fractions ...
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362
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What is the analogy between the moduli of shtukas and Shimura varieties?
I have heard that moduli spaces of shtukas are supposed to be the analogue of Shimura varieties in the setting of function fields. Could someone more knowledgeable about these objects explain how this ...
4
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247
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Second derivative at 1 of L function of elliptic curve
Let $E$ be an elliptic curve over $\mathbb Q$ of conductor $N$ and rank $0$. It follows from the functional equation that
$$L'(E,1)=(\log(2\pi/\sqrt{N})+\gamma)L(E,1)$$
where $\gamma$ is Euler's ...
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193
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Geometric interpretation of the rationality of the $j$-invariant
Consider the modular curve $X_0(N)$. Let $\Phi_N(X,Y)$ be the modular equation. Then the curve $\Phi_N(X,Y) = 0$ can be interpreted as a model for $ X_0(N)$ because the function field of $X_0(N)$ is $\...
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134
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Growth of the number of fixed points of a $p$-adic group under natural filtrations
Let $G$ be a $p$-adic reductive group, so by definition as a locally profinite group it's the group of $\mathbb Q_p$ points of a connective reductive group over $\mathbb Q_p$, $K$ be a parahoric ...
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161
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Smoothed Weyl sum inequality
One version of Weyl's inequality states that for any $\alpha\in\mathbb{R}$ and $(a, q) = 1$ such that $|\alpha - a/q|\le 1/q^2$, we have that
$$\sum_{n\le X} e(n^k\alpha)\ll X^{1 + \varepsilon}(q^{-1}...
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How many exceptional conductors are there?
We say that a conductor $q$ is exceptional if there is a primitive quadratic character $\chi$ modulo $q$ such that $L(s,\chi)$ has a real zero $\beta$ such that $\beta > 1-c/\log q$ (where $c$ is ...
4
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457
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Galois representation of an elliptic curve with CM
Let $ E $ be an elliptic curve with complex multiplication by an order $ \mathcal O$ in an imaginary quadratic field $ K $. Suppose that $ E $ is defined over $\mathbb Q(j(\mathcal O))$. Let $n$ be an ...
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167
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Chen's theorem in which constituent primes are close together
Chen's theorem states that every sufficiently large even integer can be written as $n=p+q$, where $p$ is a prime and $q$ is a product of at most two primes.
I would like a representation $n=p_1+...
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lemma II.2.4 in Harris-Taylor (about drinfeld-katz-mazur level structure on 1-dimensional $p$-divisible groups)
Lemma II.2.4 on page 82 in Harris and Taylor's "The Geometry and Cohomology of Some Simple Shimura Varieties" (or lemma 3.2 here), says that given a Drinfeld(-Katz-Mazur) level structure $\alpha:(p^{-...
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420
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Index of the congruence subgroups of $PGL_2(\mathbb{Z}_p)$
Let $\Gamma_n$ be the $n$-th congruence subgroup of $GL(2,\mathbb{Z}_p)$. So $\Gamma_n$ consists of matrices in $GL(2,\mathbb{Z}_p)$ which are congruent to the identity matrix modulo $p^n$. Let $Z(\...
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219
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Calculating some Galois cohomology
Let $L/\mathbb{Q}$ be a Galois extension of degree $p$ and $E$ be an elliptic curve defined over $\mathbb{Q}$. Let $p$ be a fixed prime (of good ordinary reduction if required). We use $L_\infty, \...
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Are there "elementary" proofs of the openness of norm subgroups and of the norm limitation theorem?
Let $K$ be a local field and $L/K$ be a finite extension. Let $L^{ab}$ be the maximal abelian subextension of $K$ in $L$. Write $N_L$ (resp. $N_{L^{ab}}$) for the image of the norm map from $L$ (resp. ...
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104
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On a much weaker version of the Normal conjecture
I would like to ask you about the following question. It is conjectured that every algebraic irrational number is normal (absolutely normal). I know the result by Bugeaud and Adamczewski about the non-...
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186
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When does a continuous function's "Fourier series" converge pointwise almost everywhere to the function?
Let $G$ be a compact topological group. By the Peter-Weyl theorem, the complex Hilbert space $L^2(G)$ is the Hilbert space direct sum of the spaces of matrix coefficients of all the irreducible ...
4
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177
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Field K and integer n such that the etale fundamental group of (GL_n)_K is the profinite completion of GL_n(K)?
Prove or disprove:
Claim: there exists a field $K$ and an integer $n$ such that $\pi_1^{et}((GL_n)_K)$ is isomorphic to the profinite completion of the abstract group $GL_n(K)$?
Note that then $G_K \...
4
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133
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Plancherel measure and dimension
I would like to understand what the Plancherel measure is on the dual of a compact group, in more explicit terms than with an implicit definition. Indeed, to the extent of my knowledge it is defined ...
4
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78
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Minimal index of number fields of small degree
Let $K$ be a number field and let $\mathcal{O}_K$ be its ring of integers. For $a \in \mathcal{O}_K$ not contained in any proper subfield of $K$, the ring $\mathbb{Z}[a]$ is contained in $\mathcal{O}...
4
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115
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Abelian variety over Q with many roots of unity
Given an abelian variety $A$ over the rational integers $\mathbb{Q}$, and a prime $p$, we know that $\mathbb{Q}(\zeta_p)$ is contained in $\mathbb{Q}(A[p])$, the $p$-division field of $A$, and where $\...
4
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150
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Traces of Frobenius Endomorphism on Etale Cohomology and $G$-torsors
I have a smooth, projective, and rigid Calabi-Yau threefold $X$ defined over $\mathbb{Q}$. Such spaces always have integral models. Let's assume we have an action on $X$ by a finite abelian group $G$...