**5**

votes

**0**answers

228 views

### Sporadic and Exceptional

I have been reading this recent paper of J.McKay and YH. He (they've written a number of papers recently, including a fun and joking one on 42 which overflow commented on) called "Sporadic and ...

**2**

votes

**2**answers

492 views

### Abelian varieties with CM

In this site, I looked at a paper of Kazuma Morita claiming the BSD conjecture for the CM case
posted on his homepage (he made a mistake three years ago for full BSD).
But, I am interested in this ...

**1**

vote

**0**answers

64 views

### A limsup representation for the upper Buck density

The upper Buck density, introduced by R.C.Buck in 1946, is defined to be the function
$$
\mathfrak{b}^\star: \mathcal{P}(\mathbf{N}^+) \to \mathbf{R}: X\mapsto \inf_{X\subseteq A:A\in ...

**2**

votes

**0**answers

196 views

### Elliptic curve over $\mathbb{Q}$ vs over $K$ [duplicate]

I saw the paper claiming the proof of the BSD conjecture for the CM case.
Apart from the truth and falsehood of this paper, I noticed that the author
says that the congruent number problem can be ...

**1**

vote

**1**answer

160 views

### A certain invariant of non-singular algebraic surfaces

Let $X \subset \mathbb{P}^3$ be a non-singular surface defined over $\mathbb{Q}$ of degree $d \geq 3$. It is a theorem of Colliot-Thelene (see the appendix to this paper: ...

**20**

votes

**2**answers

705 views

### Describing the crystalline extension of $\mathbb{Q}_p$ by $\mathbb{Q}_p$

Let $K$ be a finite extension of $\mathbb{Q_p}$. The group $\ker H^1(G_K, \mathbb{Q}_p) \rightarrow H^1(G_K, B_{crys})$ is one-dimensional, which tells us that among all extensions of Galois modules
...

**3**

votes

**1**answer

202 views

### An exponential sum over squares

I have the following exponential sum:
$\sum _{M<n\leq N}e\left (x/n^2\right )=\sum f(n),$
say, where $M$ and $N$ are something like $x^{1/4}$ and $x^{1/2}$.
My question is basically, how do I ...

**3**

votes

**0**answers

277 views

### Lifting a real quadratic twist of an Elliptic Curve to the modular curve

Let $E$ be an elliptic curve of conductor $N\cdot p^2$ over $\mathbb{Q}$, defined by the equation
$$y^2=x^3+p^2b\cdot x + p^3\cdot c$$
and parametrized by a map
$$X_{0}(N\cdot {p}^{2})\rightarrow E$$
...

**0**

votes

**1**answer

104 views

### Number of turning points on a nondecreasing $n^2 \times n^2$ matrix

Given a integer value $n$, we generate a $n^2 \times n^2$ integer matrix $M$ in the following way.
Each ceil has value range $[1~n]$
In each row, the value is nondecreasing. E.g. $M[i, j] \leq M[i, ...

**2**

votes

**0**answers

84 views

### Logarithmic bound for Diophantine equation

Let $a_1 \geq a_2 \geq a_3$ be given positive integers and let $N(a_1,a_2,a_3)$ be the number of solutions $(x_1,x_2,x_3)$ of the equation $$\dfrac{a_1}{x_1}+\dfrac{a_2}{x_2}+\dfrac{a_3}{x_3} = ...

**3**

votes

**1**answer

486 views

### Euler product for sum of multiplicative function times log

(Cross-posted from StackExchange). Let $g$ be a multiplicative function which satisfies $0 \le g(p) \ll 1/p$ and
$$ \sum_{p\le x} g(p) = \log \log x + C + O((\log x)^{-10}). $$
Iwaniec and ...

**17**

votes

**1**answer

466 views

### Why do the $2$-Selmer ranks of $y^2 = x^3 + p^3 $ and $y^2 = x^3 - p^3 $ agree?

I was playing around with sage, when I found that the ranks (over $\mathbf{Q}$) of the elliptic curves $y^2=x^3+p^3$ and $y^2=x^3-p^3 $ almost always agree, for $p$ prime. The first few exceptions ...

**5**

votes

**1**answer

90 views

### Reference request: normalization of intertwining operators for GL(2, C)

Take $F$ a local field and $\chi_1, \chi_2$ two characters, write $M(\chi_1, \chi_2)$ for the standard intertwining integral
$$M(\chi_1. \chi_2).f(g) := \int_{F} f\left( \begin{pmatrix} 0&-1\\ ...

**6**

votes

**2**answers

377 views

### Adding coprimes to get a coprime

These questions arose in my research. Let $n$ be odd and let $\mathcal{C}_n$ be the set of integers less than $n$ which are coprime with $n$.
Question 1: For each integer $\ell: 0 \leq \ell < n$ ...

**5**

votes

**1**answer

301 views

### Local Langlands correspondence and Galois equivariance

The local Langlands correspondence $\text{rec}$ for $\text{GL}_{n}$ itself is not Galois equivariant (i.e. invariant under automorphisms of its field of definition) but rather its twist by ...

**0**

votes

**0**answers

50 views

### Cubic modular equations solutions when decomposition field is not a HCF

I was interested in counting (and more generally having somehow an interesting expression) the numbers of solution of cubic equations modulo a prime $p$.
So here are my thoughts.
Let take a cubic ...

**2**

votes

**0**answers

170 views

### Short Kloosterman sum

Let $K(m,n,c)=\sum_{p\in A}\exp(2\pi i(pm+np^{-1}))$, where $m,n\in\mathbb{Z}$, $c\in\mathbb{N}$, and $A\subset (\mathbb{Z}/c\mathbb{Z})^\times$.
Does anybody know any nontrivial bound for this kind ...

**1**

vote

**0**answers

212 views

### Is this a proof of the Hardy-Littlewood inequality? [closed]

V.V. Miasoyedov posted a paper to the arXiv claiming a proof of the Hardy-Littlewood conjecture $\pi(x+y) \le \pi(x)+\pi(y)$. It seems a bit off, and not only because the conjecture is widely believed ...

**4**

votes

**1**answer

161 views

### Improvement of a bound on divisor distributions from “Divisors” (Hall and Tenenbaum)?

In the classic text referred to in the title of this question, the bound
$$
H(x,y,2y) \ll \frac{x}{(\log y)^{\delta}\sqrt{\log \log y}},\quad (3\leq y\leq \sqrt{x})
$$
is given, where ...

**1**

vote

**1**answer

169 views

### A quantitative Kronecker theorem

I encounter the following question.
$\textbf{Problem}$: For almost all Matrix $M\in\mathcal M_{m\times n}(\mathbb R),$ all $y\in \mathbb R^m$ and any $N$, small $\epsilon>0$, there exists a ...

**7**

votes

**1**answer

219 views

### Is this system always solvable in radicals by quartics, octics, $12$-ics, etc?

While considering this post, it made me wonder about its generalization in another direction and from the perspective of Galois theory.
Question: Is it true that, given four constants ...

**-1**

votes

**4**answers

180 views

### Finding integer zeroes for a particular family of equations [closed]

Given $p,q\in\mathbb Z^+$, and a vector $v=(x_1,\dots,x_{p+q})$ we consider the function $\chi(v)$:
$$\chi(v)=x_1^2+\dots+x_p^2-x_{p+1}^2-\dots-x_{p+q}^2$$
We wish to find solutions to $\chi(v)=0$ ...

**6**

votes

**1**answer

184 views

### Bounds for maximum determinant of circulant matrices

The Hadamard circulant conjecture states that there do not exist circulant Hadamard matrices with more than $4$ columns.
An $n$ by $n$ Hadamard matrix where the entries are chosen from $\{-1,1\}$ ...

**1**

vote

**0**answers

112 views

### Maximizing sum of reciprocals of the elements of a subset of {1,2,..,n} subject to LCM(a,a')>n

In an answer to a previous MO question here, MO user @fedja showed (by way of an upper bound) that if $A\subseteq \{1,\ldots,n\}$ has the property $LCM(a,a')\geq n+1$ for all $a\neq a' \in A$ then the ...

**2**

votes

**0**answers

141 views

### Number of solutions to pentagonal-pentagonal numbers

Continuing the investigation from this question on CGSE about pentagonal-pentagonal numbers:
Defining $p(n)$ as the $n$th pentagonal number (a positive integer of the form $n(3n−1)/2,\ n\geq 1$), and ...

**30**

votes

**2**answers

1k views

### Estimating the size of solutions of a diophantine equation

A. Is there natural numbers $a,b,c$ such that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an odd natural number ?
(I do not know any such numbers).
B. Suppose that $\frac{a}{b+c} + ...

**6**

votes

**1**answer

445 views

### Why is $(A^\perp)^\perp = A$?

On page 52 of this paper, Iwasawa considered the bilinear symmetric non-degenerate pairing $\Phi_n \times \Phi_n \rightarrow \mathbb{Q}_p/\mathbb{Z}_p$ defined by
$$\langle \alpha, \beta \rangle_n := ...

**6**

votes

**1**answer

210 views

### Adelic and classical modular forms on quaternion algebras

Let $R$ be an Eichler order of an indefinite quaternion algebra $B/\mathbb{Q}$ (suppose B is not the collection of $2\times 2$ matrices) and $S$ the corresponding Shimura Curve. Modular forms of ...

**5**

votes

**1**answer

199 views

### Estimating size of greatest prime divisor of a sequence of integers

Consider the numbers of the form: $$A_n = \prod_{\pm}\left(\pm 1\pm \sqrt{2} \pm \cdots \pm \sqrt{n}\right)$$
where, the product in taken oven all $2^n$ terms with variations in sign. We know such ...

**4**

votes

**1**answer

113 views

### Mean value of a function associated with continued fractions

Suppose that an irrational $x$ in $(0,1)$ has convergents $c(k,x)$, and let
$$d(x) = \sum_{k=0}^{\infty} \mid x - c(k,x)\mid.$$
What is the mean value of $d$?

**8**

votes

**1**answer

117 views

### The scope of correspondence principle in quantum chaos

My understanding of the so-called correspondence principle in quantum chaos, is that it is a connection between the behaviour of a classical Hamiltonian system (chaotic/completely integrable) and the ...

**9**

votes

**2**answers

711 views

### Algebraic independance of exponentials

First of all, a happy new year. Be it better than 2015,
healthy, wealthy, fruitful and cross-fertilizing
for you, familly and friends.
In order to cope with families of solutions of evolution ...

**3**

votes

**1**answer

144 views

### Confusion regarding Riesz Function Definition

According to wikipedia:
'In mathematics, the Riesz function is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series
$$
{\rm ...

**1**

vote

**1**answer

185 views

### Hecke character and CM elliptic curve

The L-function of CM elliptic curve $E$ over an imaginary quadratic field can be
written as the product of the Hecke L-functions (for simplicity, I assume that the base field of the elliptic curve ...

**6**

votes

**1**answer

343 views

### Factorial-based constant

Am looking for a name for:
$$\prod\dfrac{1}{1-\dfrac{1}{n!}}$$
$$=2.529477472079152648180116154253954242$$
Wolfram|Alpha
Expanding the formula gives:
...

**3**

votes

**0**answers

185 views

### Colmez conjecture and endomorphism rings

It is given by the Colmez conjecture that if $A$ has a CM-type $(k, \phi)$ and $\text{End}(A) = \mathcal{O_k}$ $$\displaystyle h_{\text{fal}}(A) = \sum_{\text{irr} \hspace{1 mm}\rho: ...

**0**

votes

**1**answer

134 views

### Pruning primitive sequences but still attaining Pillai's lower bound on sum of reciprocals

Background: The answer to a previous question I asked here specified a construction to achieve Pillai's bound on reciprocal sums of primitive sequences. A primitive sequence ...

**6**

votes

**1**answer

470 views

### Is $n=6$ the only integer satisfies ${\sigma}_x(n) \equiv 0\bmod{n}$ for every odd integer $x > 0$ and $2 (\bmod n)$ if $x$ is even integer?

After a few computations in wolfram alpha about the divisor function for some values of $n$ to look the behavior of $\sigma_x(n)\bmod n$ for $\,n=6,\,$ i got this result : $\sigma_x(6)=0 \bmod 6$ for ...

**7**

votes

**0**answers

341 views

### Lemma in Scholze-Weinstein

In the paper "Moduli of $p$-divisible groups" by Scholze and Weinstein (see http://math.bu.edu/people/jsweinst/Moduli/Moduli.pdf), one finds the following claim in Lemma 5.2.7:
Lemma: Let $K$ be a ...

**4**

votes

**1**answer

177 views

### Well-spacing of the roots of a quadratic congruence

On pages 956-957 of this paper, it is established that for any two $v_1, v_2$ satisfying $v_1^2 + 1 \equiv 0\operatorname{(mod} d_1), v_2^2 + 1\equiv 0\operatorname{(mod} d_2)$, $$\left\lVert ...

**5**

votes

**0**answers

211 views

### Showing the positivity of $p$-adic density of zeroes of a polynomial

Let $f \in \mathbb{Z}[x_1, \ldots, x_n]$ and $p$ be a prime. Let $\nu_t(p)$ denote the number of solutions $\mathbf{x} \in ((\mathbb{Z}/p^t \mathbb{Z}))^*)^n$
to the congruence
$$
f( \mathbf{x} ) ...

**7**

votes

**0**answers

205 views

### Quadratic twists of 1-motives

Quadratic twists of elliptic curves (or, more generally, abelian varieties) are familiar objects in arithmetic geometry. I would like to extend that definition to the category of 1-motives over global ...

**0**

votes

**0**answers

119 views

### Deeply ramified implies non discrete valuation - Almost ring theory

In their book "Almost Ring Theory" (http://arxiv.org/abs/math/0201175), Ofer Gabber and Lorenzo Ramero define a valued field $K$ to be "deeply ramified" if the module of Kähler differentials ...

**4**

votes

**1**answer

114 views

### Superadditivity of the lower density

Let $\mu^\star$ be a real-valued function defined on the power set of the positive integers $\mathbf{N}^+$ such that for all $X,Y\subseteq \mathbf{N}^+$ the following axioms hold:
(F1) ...

**2**

votes

**1**answer

193 views

### Generalization of a theorem of Mahler to higher dimensions

A seminal theorem of Kurt Mahler, in his papers Zur Approximation algebraischer Zahlen. I-III., is the following:
Let $F(x,y) \in \mathbb{Z}[x,y]$ be a binary form of degree $d \geq 3$ and non-zero ...

**2**

votes

**0**answers

60 views

### On mid common divisor

Given $a,b\in\Bbb N$ of $n$ bits each and $c,d\in\Bbb N$ of $m$ bits each with $\frac{n}\beta<m<n$ with $\beta>1$ is there a better way to decide if $\exists e\in[c,d]\cap\Bbb N$ such that ...

**7**

votes

**1**answer

333 views

### On (a generalization of) the Gauss Circle Problem

Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r_2(n)$, the ...

**1**

vote

**1**answer

163 views

### Reduced resultant of monic polynomials

Let $f(x)$ and $g(x)$ be coprime monic polynomials in $\mathbf{Z}[X]$ of positive degrees $m$ and $n$ respectively. It seems that in this case their reduced resultant can be obtained from the ...

**2**

votes

**2**answers

317 views

### Is $\liminf \frac{\sigma_{k}(({2}^{m-1})({2^m-1}))}{\phi_{k}(({2}^{m-1})({2^m-1}))}$ finite for every $k$?

I would like to check if this limit :$$\liminf \frac{\sigma_{k}(({2}^{m-1})({2^m-1}))}{\phi_{k}(({2}^{m-1})({2^m-1}))}$$ finite for every $k$?
where :$\phi_{k}$ is iterating Euler - totient function ...

**0**

votes

**1**answer

396 views

### The $\zeta-$word [closed]

I was wondering about classical notations in number theory. I will not ask here about special functions in general but about the more ubiquitous number theory functions. That which made me wonder ...