# Tagged Questions

**3**

votes

**0**answers

114 views

### Iwasawa theory, $\mathbb{Z}_p^{2}$-extension, Greenberg module

Take $H\subset \bar{\mathbb{Q}}$ be a quartic imaginary number field such that $\operatorname{Gal}(H/\mathbb{Q})=\mathbb{Z}_2 \times \mathbb{Z}_2$. Denote by $F$ the quadratic real subfield of $H$ and ...

**0**

votes

**2**answers

161 views

### Is there an example of integers ($x,p, q ,y$ ) which satisfies the below conditions in this claim? [closed]

Edit 01:In order to look divisibility among power divisor function where i would like to know if there a such integer $n>1 $ with y coprime to $x$ then we have: :$\sigma_y(n)\bmod \sigma_x(n)=0$, ...

**0**

votes

**1**answer

113 views

### Find the rational cases where ${t}^{2} - 4$ is a perfect square with height bound $|t| \le N$ for positive integer $N \ge 1$

Find the unique cases when ${t}^{2} - 4$ is a perfect square say, ${n}^{2}$, with height bound $|t| \le N$ for positive integer $N \ge 1$, when $t$ is a rational where $t = p/q$ and integers $p$ an ...

**2**

votes

**2**answers

365 views

### Closed formula for the generating function of the sequence of powers

Does anyone know of a closed formula for the function
$f_k(x)=\sum_{n=1}^{\infty}{n^k x^n}$ ? That is, the generating function of the sequence $1^k,2^k,3^k...$.
It is not hard to see that ...

**6**

votes

**2**answers

191 views

### Variation on the Subset Sum Problem

Given a nonempty set of integers, and given that there exists a subset of this set whose elements sum to zero, is finding the smallest such subset NP-complete?
Disclaimer: The above question ...

**0**

votes

**0**answers

90 views

### What are the elements of endomorphism ring of an elliptic curve? [on hold]

Let $E$ be an elliptic curve which is defined over a finite field $\mathbb{F}_{p^r}$. It is well known that $E$ has complex multiplication which is an order in a quadratic imaginary field or an order ...

**1**

vote

**0**answers

280 views

### The Sato-Tate conjecture (Frobenius eigenvalues)

Another question crossed my mind.
In the statement of the Sato-Tate conjecture, one usually assumes that
the elliptic curve has no CM. But, I read the Morita's paper for the BSD in the CM
case and ...

**-3**

votes

**0**answers

173 views

### estimate sum of $(\log \log p)^2/p$ [closed]

Edited: It is well known that $$\sum_{p\leq x} \frac{\log p}{p}=\log x+c.$$
From a very nice previous answer to this question, it is also known that $$\sum_{p\leq x} \frac{\log \log p}{p} = ...

**5**

votes

**1**answer

175 views

### Lattice points near a curve

Bombieri and Pila had a well known bound for the count of lattice points on an algebraic curve in the plane. Does it generalize to a bound for the count of lattice points near (say within a distance ...

**3**

votes

**0**answers

96 views

### Distribution of the inbetween prime

Let $\ \mathbb J_n\,:=\,\{1\ \ldots\ n\}\ $ be the initial interval of natural numbers, and
$$2=p_0<p_1<\ldots$$
be the increasing sequence of all primes. Let
$$ \forall_{n=1\ 2\ \ldots}\ \ ...

**0**

votes

**0**answers

147 views

### Is there any significance in Heegner numbers (or class number 1) representation symmetry?

$\mathrm{A003173}(n) = 1+((1 + \sqrt{3})^{n-1} - (1 - \sqrt{3})^{n-1})/(2\sqrt{3})$
for n = 1,2,3,4.
$\mathrm{A003173}(n) = 19+24((1 + \sqrt{3})^{n-6} - (1 - \sqrt{3})^{n-6)})/(2\sqrt{3})$
for n = ...

**2**

votes

**2**answers

385 views

### Curves of higher genus

I saw the question:
Abelian varieties with CM
and though I know that there are rare CM elliptic curves, I wonder
what kind of curves with higher genus have the CM Jacobians?

**1**

vote

**0**answers

47 views

### On the sum of digits of primes in binary form [duplicate]

Let $s_2(m)$ be the sum of digits of $m$ in binary form.
I would like to ask the following question:
Is it true that for every $n\in \mathbb{N}$ there is at least one
prime $p$ which has ...

**1**

vote

**0**answers

118 views

### Anticyclotomic character

Let $E$ be a CM field and $\mathfrak{p}$ a prime in $E$ non-split in the extension $E/E^{+}$. Does there exist an interger $r_{E}$ such that for any integer $r \geq r_{E}$, there exists a finite order ...

**-5**

votes

**1**answer

99 views

### Automorphisms of partitions [closed]

I would like to know whether the notion of automorphism of the set of partitions of a positive integer $n$ has been considered so far or not. To make things clearer, I say that a partition of $n$ in ...

**2**

votes

**1**answer

207 views

### Question related to Fermat curve: Does the equation $A x^n + By^n = C z^n$ have any solution in $\mathbb{N}$?

Let $A, B, C \in \mathbb{N}$ be such that $\gcd(A,B,C)=1$. Is it known if the equation $A x^n + By^n = C z^n$ has any non-trivial solutions $x,y,z \in \mathbb{N}$? I know there are no such solutions ...

**2**

votes

**1**answer

145 views

### Bounds on $\pi(x)$ vs. bounds on $\vartheta(x)$

If $\pi(x) > \operatorname{Li(x)},$ is $\vartheta(x) > x$? Are the two inequalities (solutions to both of which are known to exist but not known exactly) equivalent, similar, or mostly ...

**1**

vote

**1**answer

82 views

### Ternary cyclotomic polynomials with $n=15r$

Let $n=15r$ where $r>5$ is an odd prime number. If $r\!\!\! \mod 15 \equiv w$ then is it true that $\Phi_{n}(x)$ is not flat whenever $2<w<13$ ? In other words, are the flat ones necessarily ...

**4**

votes

**0**answers

107 views

### Characteristic zero lifts of a mod 4 cusp form

Let $E$ be an non-CM elliptic curve over $\mathbb{Q}$ of conductor $N$ with a cyclic rational $4$-isogeny and let $f$ denote the corresponding cuspidal eigenform of level $N$. Let $E_4$ denote the ...

**1**

vote

**0**answers

78 views

### Inequality between two sums over numbers of divisors

(cross-posted from Math SE)
For a given integer $b \geq 1$, let $D_b(m)$ be the number of positive divisors of $m$ that are less than $b$, and let $d_b(m)$ be the number of divisors of $m$ that are ...

**0**

votes

**1**answer

169 views

### For which $x$ and $y$ does $\sigma_x(n) $ divide $\sigma_y(n)$ for all $n$?

I would like to know more about divisibility among power-divisor functions. Put $\sigma_k(n) = \sum_{d \mid n} d^k$ for all positive integers $k$ and $n$.
My question here is : for which positive ...

**3**

votes

**3**answers

241 views

### Curve with given Frobenius polynomial

Does there exist a prime $p$ and a smooth genus 2 curve $C / \mathbf{F}_p$ such that the characteristic polynomial of Frobenius on the Tate module of $J(C)$ is given by $(T^2 - p)^2$?
More generally, ...

**4**

votes

**2**answers

352 views

### Are there multiplicative functions which are not rational?

Vaidyanathaswamy calls an arithmetic function rational if it is the convolution of some finite collection of functions which are either completely multiplicative or inverse to a completely ...

**4**

votes

**1**answer

312 views

### Birch's conjecture from Representation Theory

Birch has a conjecture about which automorphic forms on $PGL(2)$ are the lifts from nonsplit $O(3)$. Temporarily ignore global issues, and focus on the local nonarchimedian picture. The automorphic ...

**9**

votes

**0**answers

314 views

### Riemann hypothesis in Zilber's field

Question. What is known about the situation (truth or falsity) of Riemann hypothesis in the Zilber's field?

**5**

votes

**1**answer

105 views

### lower bound for Perron-Frobenius degree of a Perron number

A Perron number is an algebraic number which is greater than one in absolute value and is greater than all of its Galois conjugates in absolute value as well. Lind's theorem states that any Perron ...

**3**

votes

**1**answer

106 views

### How different can characters be for a sum of modular forms to still be in Gamma_0?

I have a modular form I am constructing out of sums and products of various dissected divisor-sum series, namely forms of the type $$f_i = \sum_{j=0}^\infty \sigma_1(36j+i) q^{36j+i}.$$
Each of these ...

**4**

votes

**1**answer

612 views

### When does this interesting sum diverge?

For $x \gt 0,$ what is the greatest $y$ such that $$\sum_ {1\le h^x \le k^y} \frac{1}{h^x k^y}= \infty ?$$
I don't know of any references or methods for this -- not even for $x=1$, for which the ...

**13**

votes

**3**answers

446 views

### Infinitely many $k$ such that $[a_k,a_{k+1}]>ck^2$

Let $a_n\in \mathbf{N}$ be an infinite sequence such that $\forall i\neq j, a_i\neq a_j$.
I have the following theorem:
For $0<c<\frac{3}{2}$, there are infinitely many $k$ for which ...

**11**

votes

**0**answers

139 views

### Is special value of Epstein zeta function in 3 variables a period?

Kontsevich-Zagier's article "Periods" contains the following question
Is $\displaystyle \sum_{x,y,z \in \mathbb{Z}}' \frac{1}{(x^2+y^2+z^2)^2}$ an extended period?
($\sum'$ means we do not sum ...

**1**

vote

**0**answers

187 views

### Is it known that the factorization of a discriminant of a classical modular polynomial $disc(\Phi_p(X,Y))$ in $\mathbb Z[X]$?

Let $\Phi_p(X,Y)$ be a classical modular polynomial, namely, $\Phi_p(j(z),j(pz)) = 0$ for the elliptic modular function $j(z)$.
I study the factorization of a discriminant of a classical modular ...

**1**

vote

**1**answer

170 views

### Convergence of a double sum involving prime numbers

This has been moved from math.stackexchange;
I am attempting to prove/disprove convergence of the following sum
$$ \lim_{n \to \infty} \frac{1}{n} \sum_{p \leq n} \sum_{k=0}^\infty \ln p ...

**2**

votes

**1**answer

229 views

### Estimate of the number of rabbit integers with a given congruence

Consider the Fibonacci words $B_n$:
$B_1 = 1$
$B_2 = 10$
$B_3 = 101$
$B_4 = 10110$
$B_5 = 10110101$
(start with $B_1=1$, and go from $B_n$ to $B_{n+1}$ by replacing every occurence of $1$ in ...

**11**

votes

**1**answer

230 views

### Analogue of Tate curve for $g>1$

Is there any analogue of the Tate curve for (principally polarized) abelian varieties of dimension $g$ ?

**2**

votes

**0**answers

136 views

### Polynomials representing locally constant functions

Let $K$ be a finite field with $p$ elements.
(a) Let $f\in K\lbrack x\rbrack$ be such that (i) $\deg(f)<p$ and (ii) $f(2x) = f(x)$ for $\geq (1-\epsilon) p$ values of $x$ in $K$. What can we say ...

**2**

votes

**1**answer

197 views

### On the divisibility of a certain power sum

Does $1^n + 2^n + \cdots + m^n$ divide $(1+2+ \cdots +m)^n$ for any even integers $m, n\geq 2$ ?.
For $n\leq 4$, the solution easily follows from the relevant identities. For $n\geq 6$, i suspect ...

**0**

votes

**0**answers

82 views

### How do divisors of smooth numbers clump together?

Simply put, let $n$ have $p=n^r$ be its largest prime factor, with $r\lt 1/j$ for some integer $j$. For real $t \in [0,1-r]$, let $f(t)$ be the fraction of divisors $d$ of $n$ with $d \in ...

**4**

votes

**1**answer

98 views

### Calculation of one constant similar to MZV

The series arose in the calculation of Mean value of a function associated with continued fractions:
$$C=\sum_{1\le b\le d<\infty}\frac{1}{b(b+d)d^2}.$$
Obviously
$C=C_1-C_2,$
where
...

**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

484 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

61 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

195 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

158 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

698 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

197 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

276 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

480 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

464 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 ...