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

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4
votes
1answer
134 views

Lifting of Frobenius on torsors over abelian varieties

This is related to my previous question Assume that $A$ is an abelian variety over a field $k$ of characteristic $p$, $\mathcal{L}$ is a line bundle on $A$. Assume that $A$ is ordinary and ...
2
votes
0answers
235 views

A problem on Hecke operators

I have a problem that is related to page 11 in http://www.cims.nyu.edu/~venkatesh/research/ml.pdf Fix $m\in\Bbb N$. Pick $(a,b)\in\Bbb Z_p^2$ randomly such that $m<a,b<2m$ with $a,b$ ...
3
votes
1answer
270 views

Generalizing a pattern for the Diophantine $m$-tuples problem?

A set of $m$ non-zero rationals {$a_1, a_2, ... , a_m$} is called a rational Diophantine $m$-tuple if $a_i a_j+1$ is a square. It turns out an $m$-tuple can be extended to $m+2$ if it has certain ...
2
votes
0answers
135 views

Bounds for an Egyptian Fraction Inequality

Question: If $A\geq B>0$ are rational and $x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ are integers such that $A\geq \sum_{j=1}^{n}\frac{1}{x_{j}}\geq B$, then what is an upper bound on $x_{j}$ in terms ...
5
votes
0answers
107 views

Factorization problem in Cyclic cubic field

Let K/$\mathbb{Q}$ be a cubic number field. Assume that K/Q be Galois with class number 1. Therefore Gal(K/Q) is cyclic cubic group and $\mathcal{O}_K$ is a PID. Let p be a rational prime, p ...
22
votes
1answer
457 views

Why is there a Parity Problem in Sieve Theory and not a Mod p problem for any other p?

The "parity problem" in sieve theory, so far as I understand it, is the fact that sieves can't distinguish between primes and $2$-almost primes, numbers with exactly two prime factors, and will always ...
3
votes
0answers
164 views

Repartition of 1's in the “Chacon word”

Consider the "Chacon words": $B_0=0$ and $B_{n+1} = B_nB_n1B_n$. The word $B_n$ has $\ell_n := \frac{3^{n+1}-1}{2}$ digits and the number of $1$'s in $B_n$ is $\ell_n - 3^n = \ell_{n-1} \sim ...
4
votes
0answers
198 views

Asymptotic estimate for a random model of primes

Question Let $$ \pi_{rm_c}(x) = \sum_{ \substack{ {n\leq x}\\{(n+a,P(\sqrt{n}))=1}}} 1-1, $$ where $P(x)$ is the product of all primes less or equal to $x$ and $a$ is a random integer constrained to ...
6
votes
2answers
325 views

Are the abelian absolute Galois groups of these local fields isomorphic?

For a field $F$ we denote by $F^{\mathrm{ab}}$ the compositum of all finite Galois abelian extensions of $F$. Is $\mathrm{Gal}(\mathbb{Q}_2(\sqrt[8]{3})^{\mathrm{ab}}/\mathbb{Q}_2(\sqrt[8]{3})) ...
18
votes
1answer
689 views

Concrete Applications of knowing $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$

I have very little experience with Galois representations, mostly as they relate to class field theory, elliptic curves, and modular forms, but they seem to have quite a reputation in number theory as ...
3
votes
1answer
160 views

Lifting of Frobenius on semi-abelian varieties

Let $A$ be a semi-abelian variety over a field $k$($char\, k=p$). Namely, there is an exact sequence of group schemes $$0\to T\to A\to B\to 0$$ where $T$ is a torus, $B$ an abelian variety. Assume ...
4
votes
2answers
201 views

Field cut out by a CM modular form is imaginary

Let $f=\sum_{n=1}^\infty a_nq^n$ be a newform of level $N$ and weight $k\ge 2$. Suppose that $f$ is a CM modular form in the sense of §3 of Ribet's paper Galois representations attached to eigenforms ...
5
votes
0answers
158 views

Solving a Laurent polynomial functional equation

I'm considering a set of functional equations: For a given $\phi(x)\in\mathbb {Z}[x,\frac {1}{x}] $ with $\phi(x)=\phi(\frac{1}{x}) $, $f(x)f(\frac {1}{x})+\phi (x)g(x)g(\frac {1}{x})=1, $ where ...
4
votes
1answer
123 views

Is Howe's construction of tame supercuspidal representations independent of additive character?

Let $F$ be a $p$-adic field. In "Tamely ramified supercuspidal representations of $Gl_n$" (Am. J. Math 73 (1977)), Howe constructs a supercuspidal representation $\pi_{\psi}$ of $GL_n(F)$ from the ...
3
votes
0answers
78 views

Universal vector extension of a p-divisible group and the exponential.

Given a coalgebra $U$ with a marked element $1\in U$, define its lie algebra to be the set of primitive elements ($u \in U|\Delta(u) = u \otimes 1 + 1\otimes u$, where $\Delta: U \rightarrow U\otimes ...
7
votes
0answers
124 views

The Fricke involution and expansions at infinity

Let $p$ be prime, and $f$ be a modular form for $\Gamma_0 (p)$ whose expansion at infinity has coefficients in ${\mathbb Z}\left[1/p\right]$. I'd like a down to earth proof that the same holds true ...
6
votes
3answers
409 views

Is there a rank for higher degree homogeneous forms analogous to that of quadratic forms?

Given a quadratic form $Q(x_1, ..., x_n)$, there is a natural notion of rank defined by looking at the rank of the unique symmetric matrix associated to the quadratic form, i.e. we consider the ...
1
vote
0answers
59 views

Behavior of partial Euler product in the critical strip (with Dirichlet Character)

Consider a primitive Dirichlet Character $\chi$ (non principal) and the partial Euler product attached to the L-function $L(\chi,s)$ ($p_i$ are the prime numbers) : $$P(\chi,N)=\prod_{i=1}^{N} ...
5
votes
2answers
630 views

Is the intersection of all p-adic fields equal to Q?

Fix an algebraic closure $\overline{\mathbb Q}$ of the field $\mathbb Q$ of rational numbers. For a prime $p$ let $K_p$ the field of all algebraic elements in ${\mathbb Q}_p$. Question: Is $K_p$ ...
12
votes
3answers
642 views

Profinite groups as absolute Galois groups

It is a well-known result that all profinite groups arise as the Galois group of some field extension. What profinite groups are the absolute Galois group $\mathrm{Gal}(\overline{K}|K)$ of some ...
6
votes
0answers
120 views

Is there a prime degree endomorphism on supersingular elliptic curves?

Let $E$ be a supersingular elliptic curve which is defined over field $\mathbb{F}_{p^2}$ and $l$ be a prime such that $gcd(l,p)=1$. Is there an endomorphism $\phi\in End(E)$ such that $deg(\phi)=l$? ...
4
votes
0answers
61 views

Binary quartic forms fixed by a 'large' matrix

Let $V_\mathbb{R}$ denote the $\mathbb{R}$-vector space of binary quartic forms. The group $\operatorname{GL}_2(\mathbb{R})$ acts on $V_\mathbb{R}$ via the standard substitution action. That is, if ...
3
votes
0answers
114 views

reduction mod $p$ of Weyl modules

Let $G$ be a reductive algebraic group defined over a non-Archimedean field $F$. Let $k_F$ be its residue field, of characteristic $p$. Assume $G$ is unramified over $F$, then it admits a hyperspecial ...
3
votes
1answer
193 views

Integers with small residues modulo all integers

Given a finite set $\Lambda=\{\lambda_1,\dots,\lambda_d\}\in \{1,2,\dots\}^d$ of $d$ strictly positive integers, we consider the real number ...
0
votes
0answers
43 views

Could the Hadamard product for $\xi(s)$ be split into two separate factors with $\rho$s and $\overline{\rho}$s, that each have a closed form?

With $\mu_n=0+2\,\pi\,n\,i$ and $\xi_{int}(s)=\frac{2\sinh(s)}{s}$, then: $$\displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\mu_n} \right) \left(1- \frac{s}{\overline{{\mu_n}}} \right) = ...
5
votes
1answer
282 views

About the logarithmic derivative of the Riemann zeta function

Let $\rho=\beta+i\gamma$ a non-trivial zeros of the Riemann zeta function and $s=\sigma+it$ a complex number. It is possible to prove that ...
14
votes
1answer
467 views

Growth of $\zeta_{\mathbf Q[\cos(\frac{\pi}{2^{n+1}})]}(2)$

Let $K_n$ be the field $\mathbf Q[\cos(\frac{\pi}{2^{n+1}})]$ (the real subfield of the cyclotomic field $\mathbf Q[e^{\frac{i\pi}{2^{n+1}}}]$). Is there anything known about the growth of the ...
9
votes
1answer
293 views

Parametrizing all cyclic extensions of the rational numbers of degree 5

Is there a polynomial $f(T,X) \in \mathbb{Q}(T)[X]$ in the indeterminate $X$ over the field $\mathbb{Q}(T)$ with $\mathrm{Gal}(f/\mathbb{Q}(T)) \cong \mathbb{Z}/5\mathbb{Z}$ such that for every Galois ...
3
votes
0answers
258 views

Are prime ideals of finite height in the powers series ring in infinitely variables finitely generated?

Let $A:= {\mathbb F}_p[[X_1,...,X_∞]]$ be the infinitely many variables formal power series ring over ${\mathbb F}_p$, which is UFD. Consider an arbitrary prime ideal $P$ of $A$ such that the height ...
1
vote
0answers
107 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} ...
15
votes
1answer
359 views

Is the number of representations as the sum of two elements of a polynomial sequence always small?

Let $f(x) \in \mathbb{Z}[x]$ be a degree $d>1$ polynomial with integer coefficients. Define $$r(n) := | \{x,y \in \mathbb{Z} : f(x)+f(y) = n \}|. $$ My question is: Is it true that ...
1
vote
0answers
61 views

summability and analytic continuation

Let $d_n=LCM(1,\cdots,n)$. It is well-known that $d_n=e^{\Psi(n)}$ where $\Psi$ est the second Chebyshev function. One knows that $\Psi(x)=\sum_{k\le x}\Lambda(k)$ where $\Lambda$ is the Von Mangold ...
9
votes
2answers
168 views

Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?

I came across a problem concerning about the convergence of products. I wonder if the complex series $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converges to zero when $\alpha$ is irrational. Of course, ...
12
votes
1answer
332 views

Weak Mordell-Weil over number fields

I have a question regarding the Mordell Weil theorem a number field $K$. I read the proof of the Mordell Weil theorem in "rational points on elliptic curves" by Tate and Silverman. They presented a ...
1
vote
1answer
181 views

Does there exist a sum of two squares which is written in MORE than 4 distinct way? [closed]

This follows from "Enumerating ways to decompose an integer into the sum of two squares" , and my investigation on the 3x3 magic square of squares problem. In addition to following "Fermat's theorem ...
1
vote
1answer
182 views

Is it proved that for every integer $p>0$ there exists an integer $k>0$ such that every integer $n>0$ can be expressed as $j_1^p+\dots+j_k^p$?

It has been shown, by elementary methods, that every positive integer can be expressed as the sum of $4$ squares. This type of result has been proven for many different powers $p$, for example, when ...
1
vote
0answers
152 views

Is the following claim true about systems of quadratic congruences modulo consecutive prime numbers

Is the following true? Choose any value for $y : y \in \mathbb{N}$ If $N(y)$ is the smallest natural number that satisfies the following system of quadratic congruences: $N(y)^2 \not\equiv 1$ ...
1
vote
0answers
106 views

Asymptotic of a sequence related to $LCM(1,\cdots,n)$ [closed]

Let $d_n=LCM(1,2,\cdots, n)$ and $u_n$ be a sequence such that $u_n=o(d_n)$. Some testing in maple suggests the following asymptotic: ...
4
votes
2answers
151 views

Evolution of partial sum of a sequence of induced Dirichlet characters

Let's consider the Dirichlet Character $\chi_3(n)$ modulo 3 given by $\chi_3(1)=1$, $\chi_3(2)=-1$ and $\chi_3(3)=\chi_3(0)=0$. Lets consider the sequence of induced characters $\chi^{P_N} $ obtained ...
11
votes
1answer
330 views

Reference request: seminar report of Serre from late 60s on possibility of Galois representations attached to modular forms?

See here for a comment of Matt Emerton. There are also various seminar reports of Serre, e.g. his report on mod p modular forms, but also his report from the late 60s on the possibility of Galois ...
12
votes
1answer
347 views

Counting lattice points inside a three-dimensional ellipsoid

I want to answer the following simple question: Given a three-dimensional ellipsoid defined by $Q(x, y, z) \leq Z$ for a positive definite quadratic form $Q$, how many lattice points in ...
16
votes
2answers
829 views

Can the Dedekind zeta function distinguish between real and imaginary quadratic number fields?

Suppose I am given a machine that gives me the coefficients $a_1$, $a_2$, $a_3$, ... of a Dirichlet series $$\sum_1^{\infty} \frac{a_n}{n^s} $$ and assume that I know that this Dirichlet series is the ...
5
votes
0answers
84 views

Gaussian Hypergeometric Functions and Legendre Character

I was hoping somebody might be able to point me to a good reference on Gaussian hypergeometric functions defined over a finite field. the reason I'm interested is that I've encountered sums of the ...
6
votes
0answers
153 views

drinfeld shtukas over higher dimensional spaces

Everytime I encounter Drinfeld Shtukas, the definition begins with vector bundles over a curve $X$ over a finite field. My question is: why the restriction to curves? Is there any interest or results ...
3
votes
1answer
173 views

On partial sum of non-primitive Dirichlet characters

Consider a Dirichlet character, $\chi(n)$, and the partial sum : $$S(\chi,x)=\bigg |\sum_{n=1}^{x} \chi(n)\bigg|$$ There are many works to bound this sum when $\chi$ is a primitive character, but ...
0
votes
2answers
351 views

Graphs determined by sets of consecutive integers

Given a set of positive integers, its P-graph is the graph whose vertex set consists of those integers, two of which are joined by an edge if they have a common divisor greater than 1, that is, they ...
9
votes
0answers
254 views

Independence between the number of prime factors of $n$ and $n+2$

I am interested in having an upper bound for the cardinality of $\#\left\{n\leq x\,:\quad\omega(n)=k, \omega(n+2)=\ell\right\}$ for $k,\ell\geq 1$, where $\omega(n)=\sum_{p\vert n}1$ counts the ...
10
votes
1answer
593 views

The origin of the Ramanujan's $\pi^4\approx 2143/22$ identity

What is the origin of the Ramanujan's approximate identity $$\pi^4\approx 2143/22,\;\;\tag 1$$ which is valid with $10^{-9}$ relative accuracy? For comparison, the relative accuracy of the well known ...
7
votes
3answers
432 views

Which groups are Galois over some p-adic field?

Suppose I have some finite $p$-group $G$, or a little extension of it. How do I know if there exists a prime $l$ and a finite extension $K$ of $\mathbb{Q}_l$ such that $G$ is the Galois group of ...
29
votes
2answers
2k views

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

In April 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 ...