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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

3
votes
0answers
46 views

A refinement of Faltings' lemma

In his proof of the Mordell conjecture, Faltings proved the following important result: Let $K$ be a number field and $S$ a finite set of primes in $K$. Then for any $g \geq 2$ there exists a number $...
1
vote
0answers
106 views

Reference to a particular result of Scholl and Faltings

Let $f=\sum_{n\geq 1} a_n q^n$ be a normalized eigenform which is supersingular and crystalline at a prime $p$ and let $V_f$ be the associated crystalline representation, then it follows from the work ...
2
votes
0answers
41 views

How does the $\lambda$ invariant propagate with extra ramification?

Let $\mathbb{Q}^{cyc}$ denote the cyclotomic $\mathbb{Z}_p$ extension of $\mathbb{Q}$ and let $\Lambda$ denote the corresponding Iwasawa algebra. Let $p$ be a prime. Let $S$ denote a finite set of ...
-1
votes
1answer
213 views

Write the algebra closure of $F_p$ as union of finite fields [on hold]

This question follows Field theory by Steven Roman, Chapter 9, Exercise 20. Denote the algebraic closure of the finite field $F_q$ by $\Gamma(q)$, and let $a_n$ be any strictly increasing infinite ...
1
vote
0answers
102 views

A family of crystalline representations

Let $K$ be a number field and let $v$ be a finite place of $K$. Further, let $g \geq 1$ be a positive integer. Consider the family $F(K,v,g)$ consisting of abelian varieties $A$ of dimension $2g$, ...
1
vote
1answer
74 views

Local root numbers of the Hecke character associated with some specific CM elliptic curves, should they be some roots of unity?

TL;DR. Some local root numbers of the Hecke character associated with our specific CM elliptic curve by $\mathbf{Q}(i)$ seem to have value in $\mu_4$. But apparently our computation via Rohrlich's ...
15
votes
2answers
2k views

A Putnam problem with a twist

This question is motivated by one of the problem set from this year's Putnam Examination. That is, Problem. Let $S_1, S_2, \dots, S_{2^n-1}$ be the nonempty subsets of $\{1,2,\dots,n\}$ in some ...
4
votes
0answers
133 views

Is there some computational evidence of the $pq$ analog of Serre's conjecture?

The $pq$ analog of Serre's conjecture (see "Mod pq Galois representations and Serre's Conjecture"- Khare, Kiming) states that if $\bar{\rho}_1:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F}_p)$ is a ...
2
votes
1answer
89 views

When does the double coset representative for a congruence subgroup contain a $\text{SL}_2(\mathbb{Z})$-conjugacy class?

In the paper p-adic L-functions and p-adic periods of modular forms, Greenberg/Stevens assert that if $\sigma_l:=\begin{pmatrix}l&0\\0&1\end{pmatrix}$ is the usual Hecke operator at $l$ double ...
4
votes
0answers
64 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}...
-1
votes
1answer
101 views

summation of Euler totient function

Let $\phi(n)$ be the Euler totient function and let $2\leq k\in\mathbb{N}$. For $m\in\mathbb{N}$, are there any known results, upper bounds (tighter than just removing the coprimality) or ...
0
votes
0answers
60 views

How many pairs of integer numbers with bounded product?

Let $r\in (0,1)$ and denote by $A_r$ the set $A_r=\left\{ a,b\in\mathbb{N} ~:~ a,b\leq N, a\cdot b\leq rN^2 \right\}$. Is it possible to find a good estimation for $|A_r|$? It is known that $|A_r|= r(...
10
votes
0answers
147 views

sequences in non-abelian group cohomology

In general, if we have a (pro-)finite group $G$ and a sequence of (continuous) non-abelian $G$-modules $$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 0,$$ such that the image of $A$ lies in the ...
3
votes
1answer
107 views

Minimum planar bipartite graph to cover all perfect matching count

Given set $\mathcal T_n=\{0,1,\dots,2^n-1\}$ what is the minimum number of vertices $2m$ needed in a planar bipartite balanced graph such that at every $i\in\mathcal T_n$ there is a graph $G\in\...
5
votes
0answers
67 views

“middle” partial denominator in continued fraction expansion of square roots

Suppose $d$ is a positive integer that is not a perfect square such that the negative Pell equation, $x^{2}-dy^{2}=-1$ has no solution. Then we know the minimal period of the continued fraction ...
2
votes
0answers
78 views

Generalizing characterizing crystalline representations of dimension 2 to certain special classes of crystalline representations of higher dimension

Let $A$ be an abelian variety defined over a number field $K$, and let $v$ be a prime of $K$ such that $A$ has good reduction modulo $v$. Let $\rho$ be the representation of $G_K = \text{Gal}(\...
17
votes
0answers
332 views
+100

Intuitive reason why the $j$-invariant is a cube?

Let $\tau$ be a CM point of discriminant $D$. Assume that $D$ is not divisible by $3$. Then $j(\tau)$ is an algebraic integer of degree equal to the class number $h(D)$. Let $ \gamma_2(\tau)=j(\tau)^{...
7
votes
1answer
250 views

Products of Catalan numbers

Let $c(n)=\frac{1}{n+1}\binom{2n}{n}$ be the Catalan number. It seems that a product $\prod_{n\in I} c(n)$, where $I\subset\mathbb N_{>1}$, is never a Catalan number. Is this a (known) fact?
2
votes
0answers
67 views

Function equation over general number fields

Let $\chi$ be a Hecke character on a number field $k$, where could I find a precise reference for the function equation of the $GL(1)$ L-functions $$L(s, \chi)?$$ I only find references for the case ...
1
vote
1answer
206 views

Find the positive integers $x^3+y^3=3z^3$ [on hold]

By Fermat Last theorem, I don't know if that's been discussed. Find all positive integers $x,y,z$ such $$x^3+y^3=3z^3$$
3
votes
0answers
125 views

I want a elaboration of the sketch of proof given in the Serre's Galois Cohomology on the existence of the dualizing module

I've wanted to understand the concept of the Dualizing module in the theory of Galois Cohomology. There are many references on it and of them all Neukirch's Cohomology of Number Fields seems to be ...
3
votes
0answers
197 views

A new combinatorial problem for finite groups

In a recent preprint arXiv:1811.10503, I proved that if $a_1,\ldots,a_n$ are distinct elements of a torsion-free additive abelian group $G$, then there is a permutation $\pi\in S_n$ such that all ...
0
votes
0answers
32 views

get the function that passes through n k-tuples of values with rounded value [on hold]

well basically I need to get the function that passes through n k-tuples (in my particular case k is 4 and n is as big as needed) but the results of the function are rounded, I've already seen a way ...
4
votes
0answers
90 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 $\...
15
votes
1answer
289 views

Curves over number fields with everywhere good reduction

My question is the following:$\newcommand{\Q}{\Bbb Q} \newcommand{\Z}{\Bbb Z}$ What is known about number fields $K$ fulfilling the condition $C_{g,K}$ "there is a smooth projective curve of ...
5
votes
1answer
591 views

Is there a 2-power-twinless prime?

Call two primes 2-power-twins if their difference is (can you guess?) a power of 2. For example, 11 and 19 are 2-power-twins. Is there a 2-power-twinless prime? I would imagine that this is doable ...
9
votes
0answers
171 views

On the permanent $\text{per}[i^{j-1}]_{1\le i,j\le p-1}$ modulo $p^2$

Let $p$ be an odd prime. It is well-known that $$\det[i^{j-1}]_{1\le i,j\le p-1}=\prod_{1\le i<j\le p-1}(j-i)\not\equiv0\pmod p.$$ I'm curious about the behavior of the permanent $\text{per}[i^{j-...
2
votes
2answers
245 views

On the sum $\sum_{\pi\in S_{n}}e^{2\pi i\sum_{k=1}^{n}k\pi(k)/n}$

Motivated by Question 316142 of mine, I consider the new sum $$S(n):=\sum_{\pi\in S_{n}}e^{2\pi i\sum_{k=1}^{n}k\pi(k)/n}$$ for any positive integer $n$, where $S_n$ is the symmetric group of all the ...
2
votes
0answers
120 views

trivial solutions for Diophantine equations

Let $K$ be an odd degree number field. Consider the Diophantine equation: $$ X^4 + bY^4 =Z^2 $$ where $b\neq 0$. Say we know that the above equation has only trivial roots in $K$ (for some fixed ...
3
votes
1answer
76 views

Is coprimality in $NC$?

Following reference https://pdfs.semanticscholar.org/e86e/8d7a267a29b9ad4ca112828109adfec55e8b.pdf claims integer coprimality is in $NC$ and it also has one citation. Is this claim valid?
4
votes
0answers
67 views

Relation between Faltings height and height on moduli space

Let $E$ be an elliptic curve over a number field $K$. The difference between the semistable Faltings height $h_F(E)$ of $E$ and the height $h(j_E)$ of the $j$-invariant of $E$ can be bounded in terms ...
30
votes
1answer
3k views

Does the algorithm of the Greeks produce all prime numbers?

Let ${\cal P}$ be the set of prime numbers. Define a subset ${\cal P}'=\{p_1,p_2,p_3,\cdots\}$ of ${\cal P}$ by setting $p_1=2$ and defining $p_{n+1}$ to be the smallest element of ${\cal P}$ ...
6
votes
2answers
249 views

Reference Request for a result on divisors of $p-1$

I have seen this result in several places without an English reference: There exist infinitely many primes $p$ such that $p-1=2q_1q_2$ where $q_1$ and $q_2$ are prime numbers with $q_1,q_2>p^{1/4}$...
2
votes
0answers
151 views

A computation in Conrey's paper on Riemann zeta function

I am reading the Conrey's paper "More than two fifths of the zeros of the Riemann zeta function are on the critical line" (see here). I have question/doubt in a particular step: In P.10, it claimed ...
6
votes
0answers
164 views

A challenging problem on disjoint cosets

Motivated by my negative solutions (see this paper published in Chinese Ann. Math. 13A(1992)) to two open problems on disjoint residue calsses posed by A. P. Huhn and L. Megyesi [Discrete Math. 41(...
14
votes
0answers
534 views

What is the smallest unsolved diophantine equation?

If $P=\sum_{\bf i} a_{\bf i}x^{\bf i}\in {\mathbb Z}[x_1,\dots,x_d]$, let $|P|=\sum_{\bf i}|a_{\bf i}|x^{\bf i}$ and $h(P)=|P|(2,\dots,2)$, so that there is only a finite number of $P$ with $h(P)\leq ...
-2
votes
1answer
212 views

A curious relationship betwen $|\zeta(\sigma+it)|$ and $|\zeta(1-\sigma - it)|$

By use of the Riemann functional equation, it can be shown (see corollary 10.5 of Montgomery-Vaughan) that $$|\zeta(\sigma + it)| \asymp |t|^{\sigma-1/2}|\zeta(1-\sigma - it)|$$. where $\zeta$ ...
0
votes
2answers
183 views

Permutations of squares and finite fields

Let $S_n$ be the symmetric group of all permutations of $\{1,\ldots,n\}$, and let $$S(n)=\bigg\{\sum_{k=1}^nk^2\pi(k)^2:\ \pi\in S_n\}.$$ Motivated by Question 316142 of mine, here I ask the following ...
2
votes
1answer
84 views

Diophantine equation for generating computably enumerable set

By Matiyasevich's theorem, each member of computably enumerable set can be obtain from a diophantine equation system. For prime numbers, this system of diophantine equation is found. My question is: ...
3
votes
0answers
74 views

What is known about the Hopf map for quadratic field extensions?

This question is related to my previous post: Is this generalization of the Hopf map for quadratic field extensions surjective? I still would like to know more and, while that post got several votes,...
2
votes
3answers
192 views

Asymptotics for $\int_{0}^{T} \zeta(\sigma+ it) \mathrm{d}t$

Denote by $\zeta$ the Riemann zeta function. It is known that $$\int_{0}^{T} \zeta(1/2 + it) \mathrm{d}t = T + O(T^{1/2}).$$ But is a similar result for $\int_{0}^{T} \zeta(\sigma + it) \mathrm{d}...
6
votes
0answers
178 views

Best software to do big number calculations quickly [on hold]

I am trying to do some work on some math conjecture. I am testing the conjecture numbers using very large math numbers (100+ digits ). I am currently using python to test these numbers. In the ...
2
votes
0answers
85 views

What is the image of $-1$ by the local reciprocity map?

Consider the Weil group $W$ of $\mathbb{Q}_p$, that is, the subgroup of those elements of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ mapping to an integer power of Frobenius. Class field ...
5
votes
0answers
83 views

Reference Request: Cohomology and limits of coherent topoi. (Non-abelian case)

SGA 4 VI Discusses finiteness conditions one can impose on topoi to make limits behave correctly. I am not that familiar with SGA but it is my impression that this expose only discusses abelian ...
4
votes
0answers
204 views

Subsets $E$ of $\mathbb{F}_{p^k}$ with vanishing polynomial sums

The following question arose in some discussions recently as a misunderstanding of another problem. Question: Which subsets $E\subset \mathbb{F}_{p^k}$ satisfy the property that $ \sum\limits_{x\in E}...
7
votes
1answer
136 views

Limit of quotients of elements of special Fibonacci matrices

Let $F_n$ be the $n$-th Fibonacci number, started with $F_0=0,F_1=1$, and consider the matrices $$M_n=\pmatrix{F_{n+3} & F_{n+1} \\ F_{n+2} & F_{n}}.$$ Let $$\pmatrix{\alpha_n & \beta_n \\...
1
vote
0answers
102 views

$\mu=0$ for elliptic curves over number fields

Greenberg conjectured that given $E/K$, there always exists $E^\prime/K$ such that $E'$ is isogenous to $E$ and $\mu(E^\prime)=0$. Michael Drinen has shown that for an elliptic curve $E/K$, it is ...
2
votes
0answers
470 views

Can the ABC conjecture be expanded?

Has anyone considered expanding the range of terms $a$ and $b$ for each $c$? I have generated triples $(a, b, c)$ that form integer triangles including the degenerate case of $a + b = c$ such that $a ...
4
votes
0answers
42 views

On the orthogonal group of a lattice on a quadratic space over dyadic local field

Let $F$ be a local field with valuation ring $R$. $V$ is a n dimensional non-singular quadratic space over $F$ with bilinear form $B$ and quadratic map $Q$. As usual, $O(V)$ denotes the orthogonal ...
18
votes
0answers
262 views

Can we write each positive rational number as $\frac1{p_1-1}+\ldots+\frac1{p_k-1}$ with $p_1,\ldots,p_k$ distinct primes?

It is well-known that any positive rational number can be written as the sum of finitely many distinct unit fractions. This is easy since $$\frac1n=\frac1{n+1}+\frac1{n(n+1)}\quad\text{for all}\ n=1,2,...