# Tagged Questions

**2**

votes

**2**answers

313 views

### How many integer solutions of $a^2+b^2=c^2+d^2+n$ are there?

Are there any references to study the integer solutions (existence and how many) of Diophantine equations like $a^2+b^2=c^2+d^2+2$, $a^2+b^2=c^2+d^2+3$, $a^2+b^2=c^2+d^2+5$...? Actually, I can prove ...

**6**

votes

**0**answers

92 views

### Is the analogue of the Simple Continued Fraction in p-adic number fields useful?

Is there an analogue of the Simple Continued Fraction in p-adic number fields? Is it useful and does it have relations to best rational approximation in the p-adic sense? In the analytic case there is ...

**7**

votes

**4**answers

844 views

### Clarification on the weak BSD conjecture

It is usually told that Birch and Swinnerton-Dyer developped their famous conjecture after studying the growth of the function
$$
f_E(x) = \prod_{p \le x}\frac{|E(\mathbb{F}_p)|}{p}
$$
as $x$ tends to ...

**2**

votes

**1**answer

121 views

### Clarification request on sign changes of Hecke eigenvalues

In their paper 'Sign changes of Hecke eigenvalues', Matomaki and Radziwill established in Lemma 6.2 the following result: There exists absolute positive constants $c$ and $\eta$ such that uniformly in ...

**3**

votes

**1**answer

513 views

### On progress towards inverse Galois problem over rationals

I have heard that Progress towards the Inverse Galois problem over $\mathbb{Q}$
is very well documented for sporadic groups ($M_{23}$ is the only case open) and for $PSL_n(q)$.
From where I can read ...

**2**

votes

**1**answer

124 views

### Simultaneous $\pmod{p}$ congruences of two ternary quadratic forms

Let
$$\displaystyle f(x_1,x_2,x_3) = a_1 x_1^2 + a_2 x_2^2 + a_3 x_3^2,$$
$$\displaystyle g(x_1, x_2, x_3) = b_1 x_1^2 + b_2 x_2^2 + b_3 x_3^2$$
be two integral ternary quadratic forms with $f$ ...

**3**

votes

**0**answers

178 views

### Effective prime number theorem

The prime number theorem implies that for every $ϵ>0$, there is $n_\epsilon$ such that for all $n≥n_\epsilon$ the number of primes in $[n,cn]$ is at least $\frac{(c−1−\epsilon)n}{\log n}$ and at ...

**3**

votes

**1**answer

286 views

### Frobenius at ramified primes

Let $E$ be an elliptic curve defined over $\mathbf{Q}$, fix an odd prime $p>3$, let $T_p$ denote the $p$-adic Tate module of $E$, and let $V_p = T_p \otimes \mathbf{Q}_p$.
If the action of $G_\...

**3**

votes

**0**answers

184 views

### Number of solutions to $x_1x_2=x_3x_4\bmod n$

In https://www.math.ksu.edu/~cochrane/research/xyuvmodp.pdf it is shown $x_1x_2=x_3x_4\bmod p$ where $p$ is a prime has $\frac{|\mathcal B|}p+O(\sqrt{|\mathcal B|}\log^2p)$ solutions $(x_1,x_2,x_3,...

**4**

votes

**0**answers

283 views

### Proof for new deterministic primality test possible?

Conjecture:
Let $n \in \mathbb{N}$ and $n$ odd.
Then the number $N=2^2 + n^2$ is prime, if and only if $N$ divides $2^{(N-1)/2} + 1$.
Thanks.

**13**

votes

**3**answers

459 views

### Infiniteness of the Galois cohomology over a number field with coefficients in a finite Galois module

Let $k$ be a number field and $M$ be a nonzero finite discrete $\mathrm{Gal}(\bar k/k)$-module. Is it true that $H^1(k,M)$ is infinite?
This would complete the answer of Daniel Loughran. There is a ...

**5**

votes

**0**answers

115 views

### Primitive element for a number field, and ramification

Let $K=\mathbb Q(\theta)$ be a number field with integral primitive element $\theta$, and let $f(x)$ be the minimal polynomial of $\theta$. Let $p$ be a rational prime. It's well known that if $p$ ...

**3**

votes

**0**answers

103 views

### Globalizing local field extensions with controlled ramification

Let $K_1/k_1, \ldots, K_r/k_r$ be "separable cyclic" extensions of degree $n$ where each $k_i$ is a local field of characteristic 0 (archimedean or not). By separable cyclic of degree $n$, I mean the ...

**1**

vote

**0**answers

112 views

### What interesting information can be deduced from knowledge of how deep a geodesic ventures into the cusp

First of all I have to apologise as I am not a geometer and my knowledge of geometry is poor. Let $M$ be the modular surface and $\gamma$ to denote a geodesic in $M$. In the the following paper by ...

**8**

votes

**1**answer

200 views

### Algebraic points of uniformly bounded degree on an algebraic variety

Let $k$ be a perfect field, and let $\bar k$ be a fixed algebraic closure of $k$.
Let $\overline{X}$ be a nonempty smooth algebraic variety over $\bar k$.
Does there exist a natural number $d=d(\...

**4**

votes

**1**answer

144 views

### Irreducible monic polynomials

I am looking for criteria for the irreducibility of monic polynomials with constant term $\pm1$ over $\mathbb Q$. Eisenstein's criterion clearly doesn't apply here.
For instance, for the family of ...

**4**

votes

**1**answer

214 views

### Reference for: Every local field can be realized as the completion of a global field

It is well known that every local field (i.e. nondiscrete topological field locally compact with respect to the topology) is the completion of some global field. I know the argument, a nice ...

**2**

votes

**0**answers

78 views

### Is $\sum_{\rho} \frac{1}{(\rho - s)^{1 + \delta}}$ bounded as Im(s) goes to infinity?

Let $\rho$ denote the zeros of the Riemann zeta-function
and $\delta > 0$.
Is the function
$f(s) = \sum_{\rho} \frac{1}{(\rho - s)^{1 + \delta}}$
bounded as Im(s) goes to infinity?(the real part ...

**17**

votes

**2**answers

926 views

### What is the smallest positive integer for which the congruent number problem is unsolved?

The congruent number problem is the problem of figuring out whether a given positive integer $N$ is the area of a right-angled triangle with all side lengths rational. According to Dickson's "History ...

**4**

votes

**2**answers

201 views

### Group cohomology question, trivial Galois action on discrete Galois module means we can say what about kernel of map

Say we have a number field $K$. Let $G_K = \text{Gal}(\overline{K}/K)$. Let $M$ be a discrete $G_K$-module. We know that $H^1(K, M) := H^1(G_K, M)$, i.e. profinite group cohomology. For each place $v$ ...

**15**

votes

**1**answer

432 views

### Pop's proof that $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})=\mathrm{Aut}\underline\pi^{alg}_{\overline{\mathbb{Q}}}$

I've heard of this result in a paper on which Yves André proves the p-adic analogue (that is, $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)=\mathrm{Aut}\underline\pi^{temp}_{{\mathbb{C}}_p}$), ...

**2**

votes

**0**answers

75 views

### Conditions to check whether a prime ramifies in a certain way

Let $p$ be a prime number and let $f = a_n x^n + \cdots + a_0$ be an irreducible polynomial of degree $n \geq 3$ with a root $\alpha$. We say that $p$ ramifies in a powerful way over $K = \mathbb{Q}(\...

**7**

votes

**1**answer

197 views

### Characters of simply connected semsimple algebraic groups over local fields

Let $G$ be a semisimple algebraic group over $\mathbb{Q}_p$. Then by definition $G$ admits no non-trivial algebraic characters, i.e. homomorphisms $G \to \mathbb{G}_m$.
However, it is quite possible ...

**2**

votes

**0**answers

128 views

### Uniform bound on the Mordell-Weil rank of elliptic curves

I want to know that if there is an uniform upper bound for the rank of elliptic curves over $\mathbb{C}(t)$, the rational function field over complex numbers, and generaly over the function field of ...

**0**

votes

**0**answers

51 views

### Estimating exponential sum of the form $\sum e( \alpha_1 f + \alpha_2 L)$, where $f$ is quadratic and $L$ is linear, on the minor arcs

Let $f(x_1, x_2, x_3)$ be a degree two homogeneous polynomial with coefficients in $\mathbb{Z}$. Let $L(x_1, x_2, x_3)$ be a linear homogeneous polynomial with with coefficients in $\mathbb{Z}$. Let ...

**1**

vote

**0**answers

72 views

### “Algebrazing” canonical subgroups of elliptic curves

I'm puzzled by a part of the construction of the canonical subgroup of a "not too supersingular" elliptic curve. In Katz's paper, one produces a subgroup of the formal group of the elliptic curve but ...

**2**

votes

**0**answers

380 views

### Analytically continuing the limit of this series?

Main Question
I believe the following formula gives the right answer:
$$ \lim_{k \to \infty} \lim_{n h \to k} \left( \sum_{r=1}^n c_r f(hr) h\right) = \int_0^\infty f(x) \, dx \times \sum_{r=1}^\...

**1**

vote

**2**answers

313 views

### Expository articles on Algebraic Number Theory

I am about to start learning Algebraic Number Theory and thus was looking for some expository articles on this subject. So far I have found two such articles:
Dickson, L. E.. (1917). Fermat's Last ...

**2**

votes

**1**answer

174 views

### Fundamental Units in Totally Real Cubic Fields

How much is known about the fundamental units in totally real cubic fields? For example, Daniel Shanks has a family of totally real cubic fields for which the fundamental units are known; those with ...

**1**

vote

**1**answer

190 views

### Repdigit numbers, which are sum of consecutive squares

Following up on this question,
http://math.stackexchange.com/questions/1788015/is-112122132142152162-1111-special/1788102?noredirect=1#comment3649733_1788102
is anything known about the sequence of ...

**2**

votes

**1**answer

152 views

### Character group of the multiplicative rationals

I was reading some stuff on Hecke characters and came across an issue I have not been able to resolve. I posted it here on math stack exchange first.
Let $\mathbb{Q}^{\times}$ be the multiplicative ...

**6**

votes

**2**answers

278 views

### Noncoprime polynomial values

Let $p_1, \ldots, p_n$ be a finite sequence of nonconstant polynomials with integer coefficients. Does there exist a finite sequence of integers $x_1, \ldots, x_n$ such that the integers $p_1(x_1), \...

**4**

votes

**2**answers

240 views

### Adjoint semi-simple algebraic groups over non-algebraically closed fields

Let $k$ be a field of characteristic zero and let $G$ be an adjoint semi-simple algebraic group over $k$.
On p34 of the paper "Sansuc - Groupe de Brauer et arithmétique des groupes
algébriques ...

**12**

votes

**1**answer

484 views

### A combinatorial identity involving harmonic numbers

The harmonic numbers are given by $$H_n=\sum_{k=1}^n\frac{1}{k}.$$
Numerical calculation suggests
$$
\sum_{k=1}^{n}(-1)^k{n\choose k}{n+k\choose k}\sum_{i=1}^{k}\frac{1}{n+i}=(-1)^nH_n.
$$
I can not ...

**2**

votes

**1**answer

141 views

### Want more details about the image of a Maass form in the AIM press release concerning LMFDB

Actually I came upon this through MO a couple of days ago: in here
(http://aimath.org/aimnews/lmfdb/) there is a mesmerizing image
The caption reads
A Maass form, one of the 20 different types ...

**1**

vote

**1**answer

105 views

### Linear functions

Let $(f_1, f_2, \ldots, f_n)$ be an $n$-tuple of functions mapping non-negative integers to non-negative integers. Let $m$ be a positive integer.Suppose there exists a function $f$ apping non-negative ...

**1**

vote

**2**answers

193 views

### Implicit constant in Tenenbaum's result

In his famous book 'Introduction to Analytic and Probabilistic Number Theory', Gérald Tenenbaum established the following result (Theorem III.3.5):
Let $g$ be a positive multiplicative function and ...

**15**

votes

**1**answer

336 views

### Combinatorics problem about sum of natural numbers

Following combinatorics problem is claimed to be an open problem in "The Princeton Companion to Mathematics" (pp. 6)
Let $a_1,a_2,a_3,...$ be a sequence of positive integers, and suppose that each ...

**1**

vote

**0**answers

104 views

### Ramanujan sum type

I try to show
$$\sum _{k=1}^{\infty } \frac{e^{-2 k} k}{e^{-2 k}+1}=\frac{\pi ^2}{48}-\frac{\pi ^2-6 \left(\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}(1)+\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}\left(\frac{-2 i+\pi }...

**2**

votes

**1**answer

93 views

### On attempting a proof for $r > 1$, if $M = {2^r}{b^2}$ is an even almost perfect number which is not a power of two

(Preamble: I first thought that this question might be more appropriate for MSE. However, I posted it here nonetheless in the hope that someone with that brilliant idea can help with answering my ...

**6**

votes

**0**answers

130 views

### Is every Polish ring topology on $\mathbb{C}$ defined by an absolute value?

There is a unique up to isomorphism algebraically closed field of characteristic 0 and cardinality of the continuum. Let's call it $K$.
We usually call it $\mathbb{C}$, but by this we impose a ...

**1**

vote

**1**answer

250 views

### Is this function $\mathbb{Q}$ periodic?

Considering a function $f$ exponentially decreasing at infinity, is the following function $\mathbb{Q}$ periodic ?
$$F(x)= \sum\limits_{q =1}^{\infty} \; \sum\limits_{n =1}^{\infty} \; \sum\...

**9**

votes

**2**answers

311 views

### Neron models and ramification

I encountered this result while reading a few things, and it was stated without reference. I am having a hard time finding a reference for it (or a simple proof), so maybe you can help me:
let $E$ be ...

**1**

vote

**1**answer

340 views

### Integer Polynomial solutions to functional equation

Recently I came across a functional equation which always has a polynomial with integer coefficients solution.
Let
$$
L_n(x)=(2 x+1)^2f(x+1)-4x(x+n+1)f(x)-((2 n+1)!!)^2\prod_{i=1}^n(x+i).
$$
Problem:
...

**5**

votes

**2**answers

846 views

### The number of prime factors of a natural number

It seems to be true that OEIS sequences A001222 and A257091 are closely related. First one is the number of prime divisors of n counted with multiplicity. The second one is the logarithms to the base ...

**3**

votes

**1**answer

189 views

### Connected-étale sequence for ordinary CM elliptic curves

Let $E/k$ be an elliptic curve over algebraically closed field of characteristic $p$ with CM, for simplicity, by the maximal order of a quadratic imaginary field $K/\mathbb{Q}$.
Suppose that $p$ is ...

**3**

votes

**0**answers

63 views

### Can we express the degree 10 and degree 15 Galois resolvents of sextic binary forms in terms of its basic invariants?

Let $V_6$ denote the 7 dimensional $\mathbb{C}$-vector space of binary sextic forms. For $T = \begin{pmatrix} t_1 & t_2 \\ t_3 & t_4 \end{pmatrix} \in \operatorname{GL}_2(\mathbb{C})$, $T$ ...

**1**

vote

**2**answers

301 views

### Is the singular integral that come up in circle method independnet of the representatin of the equations?

Let $F_1(\mathbf{x}), F_2(\mathbf{x}) \in \mathbb{Z}[x_1, ..., x_n]$ be a degree $d$ homogeneous polynomial.
For the system of equations $$F_1(\mathbf{x})= F_2(\mathbf{x}) =0,$$
we have the ...

**1**

vote

**0**answers

118 views

### Class field theory, Ideles class

Let $H$ be a totally complex Galois extension of $\mathbb{Q}$ and $g:G_H \rightarrow \bar{\mathbb{Q}}_p$ be a continuous morphism. By class field thoery we have $\mathrm{Hom}(G_H, \bar{\mathbb{Q}}_p)\...

**10**

votes

**1**answer

237 views

### On $\eta(6z)\eta(18z)$ and the splitting / modularity of $x^3 - 2$

Consider one of the simplest non-abelian examples of modularity. Let $$\eta(6z)\eta(18z) = q\prod_{n=1}^\infty (1 - q^{6n})(1 - q^{18n}) = q - q^7 - q^{13} -q^{19} + q^{25} + 2q^{31} - q^{37} + 2q^{43}...