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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

1
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1answer
109 views

Distance between primes that are quadratic residues modulo an other prime

Question: Is there an infinite sequence of primes $\{q_i\}_{i=1}^{\infty}$ that is not too sparse ( $q_n =O(poly(n))$ for a fixed polynomial) for which it is true that for every $k$ there is an $N(k)$ ...
3
votes
0answers
92 views

Weak Leopoldt Conjecture for the Split Prime $\mathbb{Z}_p$-extension

In his 1973 Annals paper, Iwasawa proved that the weak Leopoldt Conjecture holds for the cyclotomic $\mathbb{Z}_p$-extension of any number field. If $K$ is an imaginary quadratic field and $F/K$ is ...
1
vote
0answers
55 views

Prime generating polynomials

Continuation to this previous question. According to Lemke-Oliver, an irreducible polynomial $G$ of degree $g$ with positive leading coefficient and $\Gamma_G\neq0$ (with $\Gamma_G$ a certain factor ...
2
votes
1answer
122 views

On the function $f_m(p)=\left|\left\{1\leqslant k<\frac p2:\ \left\{\frac{k^m}p\right\}>\frac12\right\}\right|$

Let $m>1$ be an integer and let $p$ be an odd prime. Can we say something nontrivial about $$f_m(p):=\left|\left\{1\leqslant k<\frac p2:\ \left\{\frac{k^m}p\right\}>\frac12\right\}\right|$$ (...
51
votes
5answers
14k views

Have there been any updates on Mochizuki's proposed proof of the abc conjecture?

In August 2012, a proof of the abc conjecture was proposed by Shinichi Mochizuki. However, the proof was based on a "Inter-universal Teichmüller theory" which Mochizuki himself pioneered. It was known ...
-3
votes
0answers
65 views

Under which assumption exists the following number? [on hold]

Let $A=\{\alpha_1,\alpha_2,\ldots\}$ be a (at most countable) subset of the positive rational numbers $\mathbb{Q}_+$. Under which assumptions does there exist $m\in\mathbb{N}$ such that $\frac{m}{\...
4
votes
1answer
84 views

Bounds for the size of arrays with distinct subarray sums

Consider an array $A$ of length $n$ with $A_i \in \{1,\dots,s\}$ for some $s\geq 1$. For example take $s = 6$, $n = 5$ and $A = (2, 5, 6, 3, 1)$. Let us define $g(A)$ as the collection of sums of all ...
3
votes
1answer
203 views

Positive real root separation (v2)

(This is a follow-up question to Positive real root separation) Let $\beta\in(1,2)$ and $\gamma\in(1,2)$ be Galois conjugates of height 1. That is, there exists a polynomial $p$ with coefficients $-1,...
2
votes
0answers
120 views

Zeta function of the affine Grassmanian and Weil conjecture

Let $G$ be a split connective reductive group over $k=\mathbb F_q$, then the affine Grassmannian $X=Gr_G$ is representable by an ind-projective strict ind-scheme over $k$. (That is, there exists an ...
8
votes
1answer
607 views

Short divisor sum

Let $d(n)$ denote the number of positive divisors of the positive integer $n$. Pick some positive $X,h \in \mathbb{R}$ and consider the sum $$ S(X,h) := \sum_{X \leq n \leq X+ h} d(n).$$ In view of ...
4
votes
0answers
83 views

When can one expect that the $\mu$-invariant of a $\mathbb{Z}_p$-extension of a number field is zero?

What is special about $\mathbb{Z}_p$-extensions which are motivic to ensure that their $\mu$ invariant is zero? Is there a simple conceptual reason. Here are some examples. Let $F$ be a totally real ...
28
votes
2answers
898 views

Etale cohomology can not be computed by Cech

It can be proven that if in a quasicompact scheme $X$ any finite subset is contained in an affine open subset then for any sheaf $\mathcal{F}$ on $X$ its Cech cohomology $\hat{H_{et}^{\bullet}}(X,\...
114
votes
11answers
15k views

Why is the Gamma function shifted from the factorial by 1?

I've asked this question in every math class where the teacher has introduced the Gamma function, and never gotten a satisfactory answer. Not only does it seem more natural to extend the factorial ...
4
votes
0answers
88 views

Schoof's Algorithm for Hyperelliptic curves over $\mathbb{F}_q$ : Question regarding computation of resultant: Gaudry

I am new to StackExchange and I am currently going through Gaudry's paper on counting points on hyperelliptic curves (see https://hal.inria.fr/inria-00512403/document). As a part of the generalization ...
9
votes
0answers
129 views

Semi-primes represented by quadratic polynomials

According to Lemke-Oliver, irreducible quadratic polynomials $G$ with positive leading coefficient and $\rho(2)<2$, (where $\rho(m)$ denotes the number of incongruent solutions to the congruence $G(...
64
votes
13answers
21k views

Collecting proofs that finite multiplicative subgroups of fields are cyclic

I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic ...
3
votes
1answer
142 views

Does every index $p$ subgroup of $SL(2,\mathbb{Z}_p)$ contain $\Gamma(p)$?

Does every index $p$ subgroup of $SL(2,\mathbb{Z}_p)$ contain the principal congruence subgroup $\Gamma(p)$? Equivalently, must it be the preimage of an index $p$ subgroup of $SL(2,\mathbb{Z}/p\...
19
votes
1answer
1k views

Polynomials for addition in the Witt vectors

The addition of $p$-typical Witt vectors ($p$ a prime number) is given by universal polynomials $S_n=S_n(X_0,\dots,X_n;Y_0,\dots,Y_n)\in\mathbb{Z}[X_0,X_1,\dots;Y_0,Y_1,\dots]$ determined by the ...
16
votes
1answer
224 views

$GL_n(\Bbb Z_p)$ conjugacy classes in a $GL_n(\Bbb Q_p)$ conjugacy class

It is easy to classify conjugacy classes in $GL_n(\mathbb Q_p)$ by linear algebra. How to classify $GL_n(\Bbb Z_p)$ conjugacy classes in a $GL_n(\Bbb Q_p)$ conjugacy class? For example, for general ...
1
vote
0answers
75 views

Iwasawa Theoretic Interest in a certain type of result

This question is probably going to sound vague (since it does to me) and I wish I could make it more precise, but here goes. For $p\in \{107, 139, 271,379\}$ Ohtani and Blondeau (in separate papers) ...
2
votes
0answers
68 views

Non-cuspidal Hecke eigenforms and Eisenstein series

It's a direct check that ${\displaystyle E_{2k}(z )=\frac{\zeta(1-2k)}{2}+\sum _{n=1}^{\infty }\sigma _{2k-1}(n)q^{n}}$ is an eigenform for every Hecke operator $T_n$ with eigenvalue $\sigma_{2k-1}(n)$...
1
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3answers
247 views

Lower bounds for Petersson inner products of cuspforms with integral Fourier coefficients

Let $N\geq 1$ be an integer and let $S_2(\Gamma_0(N))$ be the cusp forms of weight 2 for the usual congruence subgroup $\Gamma_0(N)\subset SL_2(\mathbb Z)$. Let $a_n(f)$ denote the n-th Fourier ...
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votes
0answers
83 views

on second hardy-littlewood conjecture [on hold]

I am a self-taught , I discovered a result about second hardy-littlewood statement of conjecture is for $x\geq y \geq 2$ then Pi(x+y)<=Pi(x)+Pi(y) I PROVE that it holds for y>cx where c is cons&...
11
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1answer
1k views
2
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1answer
237 views

large image of galois representations

If one has a Galois representation $\overline{\rho}: G_{\mathbb{Q}} \rightarrow GL_2(\mathbb{Z}/p \mathbb{Z})$ where $\overline{\rho} = \left( \begin{array}{cc} \...
4
votes
1answer
165 views

Higher roots modulo prime complexity best algorithm

Given integers $a,\ell$ and prime $p$ we need to find the roots of the algebraic equation $x^\ell\equiv a\bmod p$. We know there are at most $\ell$ such $x$. What is the best method to find all such ...
7
votes
0answers
115 views

Comparison between Faltings height and Modular Height

Motivation/Context: In Faltings’ proof of the Mordell conjecture, there is a theorem that establishes a finiteness of abelian varieties with respect to the Faltings height under certain conditions. ...
9
votes
1answer
176 views

Gauss sums for general number fields

There is a wide litterature for the classical Gauss sums. For $\chi$ a primitive Dirichlet character modulo $N$, it is given by $$\tau(\chi) = \sum_{n \text{ mod } N} \chi(n) \exp(2i\pi n/N).$$ An ...
4
votes
1answer
170 views

On approximation of $\sum_{a,b=1}^n\gcd(a,b)$

Denote $g(n)=\sum_{a,b=1}^n\gcd(a,b)$, can we prove that $$g(n)=\frac6{\pi^2}n^2\ln n+Cn^2+O(n\ln n)$$, where $C=-\frac12+\frac{6}{\pi^2}(-\frac12+\gamma-\ln(2\pi)+12\ln A),$ where $\gamma$ denotes ...
2
votes
1answer
199 views

On triangular numbers modulo primes

Let $p$ be an odd prime. For $a\in\mathbb Z$ let $\{a\}_p$ denote the least nonnegative residue of $a$ modulo $p$. The list $\{1^2\}_p,\ldots,\{((p-1)/2)^2\}_p$ is a permutation of all the quadratic ...
4
votes
0answers
88 views

Projective modules over maximal orders of central simple algebras

In "Supersingular K3 surfaces" by TetsuJi Shioda, when proving Theorem 3.5 (Deligne) he considers a supersingular elliptic curve $C$ over an algebraic closed field of $\text{char}\ p>0$ and let $R ...
11
votes
0answers
231 views

Record analytic rank for an elliptic curve?

What is the current record (and reference) for the highest analytic rank of an elliptic curve over $\mathbb{Q}$? The highest algebraic rank is the Elkies curve with rank at least 28, but I cannot ...
1
vote
1answer
101 views

Homogeneous polynomials which ramify completely on a hypersurface

Let $F \in \mathbb{Z}[x_0, \cdots, x_n]$ be a homogeneous polynomial. Let $V \subset \mathbb{P}^n(\mathbb{C})$ be a hypersurface (defined over $\mathbb{Q}$ say), given by a homogeneous polynomial $G(...
16
votes
6answers
2k views

Advances and difficulties in effective version of Thue-Roth-Siegel Theorem

A fundamental result in Diophantine approximation, which was largely responsible for Klaus Roth being awarded the Fields Medal in 1958, is the following simple-to-state result: If $\alpha$ is a real ...
33
votes
2answers
6k views

Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?

Mathematica knows that: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)\;\;\;\;\;\;\;\;\;\;\;\; (1)$$ The von Mangoldt function should then be: $$\Lambda(n)=...
12
votes
0answers
154 views

Statistics for rational points on curves of genus $g$ over $\mathbb{F}_q$, $g\gg q$

Consider the distribution of the number of $\mathbb{F}_q$ points as I range over smooth projective curves of genus $g$ (defined over $\mathbb{F}_q$). If $q\gg g,$ the Hasse-Weil bounds give me a lot ...
0
votes
2answers
128 views

Analytical result of a combination like generating function

Here is the generating function I'm studying. $f=\prod^N_{j=1}\left(1+e^{i\cdot j\varphi}z\right)$. $\varphi$ is a phase related to a quantum optics problem. And I want to know the analytical ...
3
votes
0answers
68 views

$L$-functions for quadratic orders and Siegel's solution of the class number problem

Let $K$ be an imaginary quadratic field and $D_K$ its discriminant. Further let $\mathcal O$ be an order in $K$ with conductor $f$ and $$L(\chi,s)=\sum_{\mathfrak a}\chi(\mathfrak a)N(\mathfrak a)^{-...
5
votes
2answers
317 views

Is there a version of Serre's modularity conjecture for projective representations?

Serre's modularity conjecture asserts that a continuous odd irreducible representation $$\overline{\rho} : G_\mathbb{Q} \rightarrow \mathrm{GL}_2(\mathbb{F}_q)$$ must be modular, in the sense that ...
4
votes
1answer
167 views

Landau's theorem using nth roots

This question was asked earlier at MSE . Let $\omega$(n) denote the number of distinct primes dividing $n$. The Mobius function is defined as $\mu(n) = (-1)^{\omega(n)}$ if $n$ is squarefree and $\...
22
votes
4answers
2k views

Monic polynomial with integer coefficients with roots on unit circle, not roots of unity?

There are certainly non-monic polynomials of degree 4 with all roots on the unit circle, but no roots are roots of unity; $5 - 6 x^2 + 5 x^4$ for example. Now, for a monic polynomial of degree $n$, ...
4
votes
1answer
219 views

Positive real root separation

Let $\beta\in(1,2)$ and $\gamma\in(1,2)$ be Galois conjugates of height 1. That is, there exists a polynomial $p$ with coefficients $-1,0,1$ such that $p(\beta)=p(\gamma)=0$ (not necessarily minimal)....
28
votes
0answers
564 views

Is there a rigid analytic geometry proof of the functional equation for the Riemann zeta function?

The adèles $\mathbb A$ arise naturally when considering the Berkovich space $\mathcal M(\mathbb Z)$ of the integers. Namely, they are the stalk $\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$ ...
1
vote
0answers
59 views

Counting points on different hyperplane sections over a finite field

Let $X \hookrightarrow \mathbb P^n$ be a smooth projective variety over a finite field $k=\mathbb F_q$. How does the number of $\mathbb F_{q}$-points of $X_H:=X \cap H$ varies when $H$ varies over $\...
8
votes
1answer
172 views

Can discriminant polynomials become perfect powers on hyperplanes?

Let $$\displaystyle f(x) = a_d x^d + a_{d-1} x^{d-1} + \cdots + a_0.$$ Consider the discriminant of $f$, denoted by $\Delta(f)$, defined as $$\displaystyle \Delta(f) = a_d^{2d-2} \prod_{i < j} (\...
2
votes
3answers
299 views

A generalization of Landau's function

For a given $n > 0$ Landau's function is defined as $$g(n) := \max\{ \operatorname{lcm}(n_1, \ldots, n_k) \mid n = n_1 + \ldots + n_k \mbox{ for some $k$}\},$$ the least common multiple of all ...
7
votes
0answers
121 views

Determinant of a matrix filled with elements of the Thue–Morse sequence

Let $n$ be a positive integer. Suppose we fill a square matrix $n\times n$ row-by-row with the first $n^2$ elements of the Thue–Morse sequence (with indexes from $0$ to $n^2-1$). Let $\mathcal D_n$ be ...
3
votes
0answers
143 views

What is $\mathrm{O}_q/\mathrm{SO}_q$ if $q$ is a quadratic $\mathbb{Z}$-form which is degenerate?

Any binary quadratic $\mathbb{Z}$-form $q$ induces a symmetric bilinear form $$ B_q(u,v) = q(u+v) - q(u) -q(v) \ \ \forall u,v \ \in\mathbb{Z}^2 $$ and it is considered non-degenerate (over $\mathbb{...
0
votes
0answers
61 views

Is a tempered representation globally generic?

I know there is an general belief that globally generic representation is tempered. I am wondering whether the converse is known, that is tempered representation is generic? If it is not known, is ...
7
votes
0answers
112 views

Poisson summation formula for number fields

Poisson summation formula is widely used in many parts of the litterature, its classical formulation for sums over integers as well as its adelic version. What is its corresponding form for more ...