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

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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 ...
Gory's user avatar
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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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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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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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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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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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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 $\mathbb{Z}$-...
Stanley Yao Xiao's user avatar
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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 $N \rightarrow \infty$, ...
Seungki Kim's user avatar
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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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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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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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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 ...
Dmitry Vaintrob's user avatar
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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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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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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? ...
Dominic van der Zypen's user avatar
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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....
primefinder's user avatar
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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 ...
Zhi-Wei Sun's user avatar
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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 $\underset{n\rightarrow \infty}{\lim} \left| \mathrm{Re}((\frac{1+i\sqrt{7}}{2})^n)\right| = \infty.$ ...
Renaud Detcherry's user avatar
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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 $k>2$, $...
apple's user avatar
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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 $\text{Gal}(H_K/K)$?
Emmanuel Halberstadt's user avatar
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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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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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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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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 ...
Stanley Yao Xiao's user avatar
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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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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 $$ \...
user119197's user avatar
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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,...
Zhi-Wei Sun's user avatar
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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 ...
Rémi Peyre's user avatar
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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 ...
Kim's user avatar
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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 ...
Henri Cohen's user avatar
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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 $\...
Shimrod's user avatar
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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 ...
sawdada's user avatar
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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}...
Mayank Pandey's user avatar
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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 ...
Thomas Bloom's user avatar
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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 ...
Shimrod's user avatar
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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+...
Peter Dukes's user avatar
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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^{-...
aytio's user avatar
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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(\...
MathStudent's user avatar
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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. ...
Simon Pepin Lehalleur's user avatar
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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-...
Jean's user avatar
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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 ...
D_S's user avatar
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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 \...
David Corwin's user avatar
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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 ...
TheStudent's user avatar
4 votes
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78 views

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}...
Stanley Yao Xiao's user avatar
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115 views

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 $\...
A. GM's user avatar
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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 $G$...
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