# Tagged Questions

**10**

votes

**1**answer

411 views

### Examples of elliptic curves over $\mathbb{Q}$

I need examples of two non-isogenous elliptic curves $E_{1}, E_{2}$ over $\mathbb{Q}$ having the following 2 properties -
1) $E_{1}, E_{2}$ have no rational torsion points.
2) $E_1[9] \cong E_2[9]$ ...

**5**

votes

**1**answer

274 views

### Integer numbers of the form $m = x^n + y^n$

First of all, I am no number theorist, so this question may be a little dummy.
The two squares theorem imply that $m = x^2 + y^2$ for some (possible zero) integer numbers $x,y$ iff $m$ factors as $m ...

**1**

vote

**1**answer

107 views

### A special curve with points of order 3(or 6)

Could you please tell me what is the points of order 3 (or 6) Hon the elliptic curve $y^2=x^3+sx^2-x$ where
$$s = -\frac{1}{432}\frac{(81k^8-2592k^4-6912)}{k^6}$$ and $k$ is rational?

**20**

votes

**3**answers

627 views

### Consecutive square values of cubic polynomials

Let $P(x)$ be a cubic polynomial with integer coefficients. Does there exist a constant $c$ such that at least one of the following values $P(0),P(1),...,P(c)$ is not a square?
It is known that the ...

**6**

votes

**0**answers

81 views

### Joint representation of the semi-direct product of the metaplectic group and Heisenberg group

Given a symplectic space $W$ over a local field $F$ and a additive character $\psi$ of $F$, we can construct the Weil representation $\omega_\psi$, which can be viewed as a representation of the ...

**10**

votes

**0**answers

262 views

### Unramified Galois cohomology of number fields

I'm trying to understand unramified Galois cohomology of number fields a bit better.
Set-up: Let $k$ be a number field and $S$ a finite set of places of $k$ which contains all the archimedean ...

**0**

votes

**1**answer

140 views

### On complex exponential sum estimation

Let $c>0$ a real number, let $N$ a large natural number and let $e\left(x\right):=e^{2\pi ix}$. Is it true that $\forall k\in\left[1,\,2N\right]$, $k$ natural number, that ...

**0**

votes

**0**answers

46 views

### Volume of toric conjugacy class

Let $p$ be a prime and $M/\mathbb{Q}$ an imaginary quadratic extension. Let $T_{M,p}$ be the torus in $GL_{2}(\mathbb{Z}_{p})$ arising from $M$ and $C_{M,p}$ an open subset in $GL_{2}(\mathbb{Z}_p)$ ...

**27**

votes

**3**answers

1k views

### A translation of the Cantor set contained in the irrationals

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here.
What would ...

**1**

vote

**1**answer

166 views

### Simplifying a sum in terms of divisor function Cauchy products

I'm trying to simplify this combinatorial looking sum:
$$\sum_{ax+by=n}_{(a,x,b,y)\in \mathbb{N^4}}\max{\{a,b\}}$$
In terms of possibly some scaled divisor functions plus a Cauchy product/convolution ...

**1**

vote

**2**answers

213 views

### prime zeta function when $0<s<1$

I will not be surprised if this question seems trivial in MO but i asked it first in MathSE and i did not get an answer.
So, here it is:
I would like to know if there is a good estimate for the sum ...

**3**

votes

**3**answers

381 views

### Modular form on $\Gamma_0(N)$

I recently asked this question on Math.StackExchange with no answer so far. So I thought maybe I can find an answer here.
Let $M(k,\Gamma_0(N))$ be a space of modular forms of weight $k$ on ...

**0**

votes

**1**answer

154 views

### Elliptic curves over $\mathbb{Q}$ with no rational torsion and $\mu$-invariant equal to 1 at $p=3$

How to find out examples over elliptic curves over $\mathbb{Q}$ with no rational torsion and $\mu$-invariant equal to 1 at $p=3$ $?$

**3**

votes

**1**answer

248 views

### Possibly an easy application of Tate-Nakayama duality and local CFT

Let $F$ be a non-archimedian local field and $F^{ur}$ the maximal unramified extension. Let $T$ be a torus defined over $F$ such that $T$ splits over a tamely ramified extension and is anisotropic ...

**4**

votes

**4**answers

553 views

### Normalizer of SL_2(Z) in GL_2(R)

What is the normalizer of ${\mathrm{SL}}_2({\Bbb Z})$ in ${\mathrm{GL}}_2({\Bbb R})$? Namely
${\mathrm{N}}_{{\mathrm{GL}}_2({\Bbb R})}({\mathrm{SL}}_2({\Bbb Z}))$?

**1**

vote

**0**answers

36 views

### Modular form on $\Gamma_0(N)$ [duplicate]

Let $M(k,\Gamma_0(N))$ be a space of modular forms of weight $k$ on $\Gamma_0(N)$.
Each $f \in M(k,\Gamma_0(N))$ has a Fourier expansion of the form
$$f(\tau)=\sum_{n\in ...

**4**

votes

**1**answer

159 views

### What is the interpretation of this galois cohomology set?

Let $K$ be a field of characteristic zero. Let $G_K:=Gal(\bar{K}/K)$
The nontrivial elements of the set $H^1(G_K,PGL_2)$ correspond to $\bar{K}/K$-forms of $\mathbb{P}^1$; i.e. curves that are ...

**3**

votes

**0**answers

122 views

### What is the function like when its Mobius inversion is $\sum_{w|r, (w,t)=1}\mu(w)q^{r/w}$?

Everyone, I am now reading a paper named The Irreducible Factors of $(cx+d)x^{q^m}-(ax+b)$ over $GF(q)$, http://qjmath.oxfordjournals.org/content/14/1/61.extract. And I’m confused with one of its ...

**21**

votes

**0**answers

640 views

### Monstrous Moonshine for Thompson group $Th$?

I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2,
$$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, ...

**9**

votes

**1**answer

410 views

### Integer Solutions of $x+y^n = y + x^m$ for $n < m$

I found 8 of them and believe there is no more:
$$2+3^2=3+2^3$$
$$2+6^2=6+2^5$$
$$6+15^2=15+6^3$$
$$3+16^2=16+3^5$$
$$3+13^3=13+3^7$$
$$2+91^2=91+2^{13}$$
$$5+280^2=280+5^7$$
$$30+4930^2=4930+30^5$$
...

**6**

votes

**3**answers

801 views

### Objections to and arguments for the simplicity of all Riemann zeros

It seems to be that the simplicity of all the zeros is quite widely accepted as a working hypotheses, and it is known that a positive proportion are as such.
Titchmarsh explains in the last chapter ...

**0**

votes

**0**answers

109 views

### Invariants of an L-function

For any element $F$ of the Selberg class, one can define invariants of $F$ such as the degree, the conductor, the h- invariants, etc. My question is: these any known set of invariants such that a ...

**1**

vote

**0**answers

129 views

### Average rank of elliptic curves over function fields

De Jong showed in 2002 if the finite field $\mathbb{F}_q$ has characteristic not equal to 3, then the limsup of the average of 3-Selmer rank is bounded above, where the average is taken over the ...

**1**

vote

**0**answers

60 views

### Unbranched cover of a curve of CM type

Let $C$ be a curve of genus $g\geq 2$ over complex number. Assume that $C$ has complex multiplication (CM).
Does there exist such a curve $C$ such that $C'$ is also of CM type for any unbranched ...

**3**

votes

**1**answer

115 views

### On Universal Abelian surfaces over a Shimura curve.

Let ${\cal O}, {\cal O}'$ be two order in ${\mathrm M}_2({\Bbb R})$ that are sets of all $2 \times 2$ matrices over real number ${\Bbb R}$. Assume that we have the relation ${\cal O}' = a{\cal ...

**2**

votes

**2**answers

254 views

### Connection between quadratic forms and ideal class group

I'm studying the classic results on binary (integer) quadratic forms and I'm looking for a reference on the following result (maybe a book that contains a proof):
Let $O_k$ be the ring of algebraic ...

**4**

votes

**1**answer

400 views

### Lines in image; are they significant to prime numbers if so how?

Amateur math question. I was playing around generating some 2D images, and wondered what it would look like if placed $P_{i}$ dots on a circle with diameter of $i$ for increasing values of $i$, where ...

**2**

votes

**0**answers

74 views

### The number of different lattice triangles

Two convex lattice polygons are equivalent if there is a lattice-preserving affine transformation mapping one of them to the other. Equivalent polygons have the same area. Let $H(A)$ denote the number ...

**5**

votes

**1**answer

258 views

### Number of partitions whose blocks form arithmetic progressions

As is known, the set $\{1,\ldots,n\}$ has $2^n$ many subsets and $B_n$ (the $n$th Bell number) many partitions, where clearly $B_n<2^{2^n}$ and it is actually known that $B_n<n^n$ for large $n$. ...

**3**

votes

**1**answer

303 views

### Giuga's Conjecture: Central or Peripheral?

An earlier MO question
highlighted
Giuga's Conjecture:
A positive integer $n>1$ is prime if and only if
$$\sum_{k=1}^{n-1} k^{n-1} \equiv -1 \pmod{n}$$
For example, for the prime $n=5$, ...

**3**

votes

**0**answers

89 views

### Integers in a given box that can be represented by a polynomial

Suppose that $F(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$ is a polynomial of degree $d$, and examine the quantity
$$\displaystyle N(F;X, B) = \# \{(x_1, \cdots, x_n) \in \mathbb{Z}^n | -X ...

**6**

votes

**1**answer

281 views

### Misunderstanding in the hypotheses of Schlessinger's criterion

Good day to everyone.
In studying deformation theory of Galois representations, I've come surely to an error, relating Schlessinger's criterion.
Let's fix a representation $\bar{\rho}$ of a group ...

**2**

votes

**0**answers

92 views

### A $GL_1$ Voronoi formula

I want a functional equation for the function defined by the Dirichlet series,
$$ D(s,a/q)= \sum_{n=1}^\infty \frac{e^{2\pi i n a/q}}{n^s}. $$
which sends $s$ to $1-s$ and preferably sends $a$ to ...

**2**

votes

**1**answer

183 views

### An upper bound to sum of ratios of gcd's and products

It seems like the sum $S(n)$ should be possible to upperbound by an expression of the form ${\mathcal O}(n^a\cdot \log^b(n))$ as $n \rightarrow \infty$:
$$
S(n)\stackrel{\triangle}{=}\sum_{1\leq x ...

**11**

votes

**2**answers

314 views

### The intersection of a circle and a rank 3 subgroup of the plane

Let $A$ be a rank 3 subgroup of the Euclidean plane, i.e. $A = \mathbb{Z} v_1 + \mathbb{Z} v_2 + \mathbb{Z} v_3$, where $v_1, v_2, v_3 \in \mathbb{R}^2$ are three $\mathbb{Q}$-linearly independent ...

**1**

vote

**0**answers

124 views

### Proofs for almost prime limits

A number $n$ with prime factorization $$n=\prod_{i=1}^rp_i^{a_i}$$
is a k-almost prime if it has a sum of exponents $$\sum_{i=1}^{r}a_i=k$$ i.e., when the prime factor (multiprimality) function ...

**0**

votes

**0**answers

69 views

### On the local theta correspondence between U(1) and U(2)

Let $E/F$ be a quadratic extension of number fields and $V,W$ hermitian and skew-hermitian vector spaces of dimension 1,2 respectively. Let $v$ be a place of $F$ and $\chi$ be a fixed character of ...

**4**

votes

**1**answer

183 views

### N^2 and two counter machines

I asked this question on cstheory a few months ago, but I didn't receive an answer, so I'm posting it here to see if there are original ideas from the "math world" to solve it. The original question ...

**7**

votes

**0**answers

399 views

### Looking for a paper of Hartshorne

In a famous paper
Hartshorne - Varieties of small codimension,
Hartshorne formulates a conjecture, which roughly says that varieties of small codimension in projective space are complete ...

**3**

votes

**0**answers

173 views

### Logarithms of ratios of squarefree numbers

Let $M \geq 1$, and $N=2^M$. Let $a_1, \dots, a_N$ be the set of all the numbers that you get when forming all square-free products of the first $M$ primes.
For example, for $M=2$ and $N=4$ you get ...

**1**

vote

**0**answers

98 views

### Does the global theta lift performed twice yield the identity when it doesn't vanish?

Let $E/F$ be a quadratic extension of number fields and $V,W$ are hermitian and skew hermitian vector space over $E$ whose dimension is $n,m$ respectively.
Let $\pi$ be a irreducible tempered ...

**1**

vote

**0**answers

346 views

### A question from Number Theory

I'm trying to solve the following question.
Let $(b_1, b_{2}, b_3) \in \mathbb{Z}^3$ such that
$0 < b_1 \leq b_{2} < b_3$ and $\gcd\{b_3, b_i\}= 1$ for $i=1, 2$.
Let $Q(b_1, b_{2}, b_3)$ be ...

**0**

votes

**0**answers

81 views

### Right angle parallelepipeds with the same greatest diagonal (the sides and greatest diagonal positive integers with g.c.d=1)

Let’s have the equation
$$(r_1\times p+p)^4-(4r_1^3+6r_1^2+4r_1+1)\times p^4=(r_1\times p)^4$$
And if $p=2$ and $r_1=n/2$ where $n$ a positive integer the above equation is written as follows:
...

**6**

votes

**4**answers

229 views

### Number of solutions of linear homogenous Diophantine equation inside a box

Let $a_1, ..., a_d$ be positive reals and consider the linear Diophantine equation
$$
\sum_i a_in_i = 0.
$$
I am interested in estimating the number of integer solutions of this equation inside a ...

**10**

votes

**1**answer

387 views

### Chebotarev density theorem for $k$-almost primes

Consider a finite Galois extension $L$ of $\mathbb Q$, of Galois group $G$. Let $k \geq 1$ be a fixed integer. Let $D$ be a subset of $G^k$ invariant by conjugation and by the natural action of the ...

**8**

votes

**1**answer

250 views

### Distribution of $n$-th roots modulo a smooth number

Let $n \ge 2$. Let $p_1,\dots,p_m$ be distinct primes $\equiv 1 \pmod{n}$. Let $N=p_1 p_2 \cdots p_m$. If $\gcd(a,N)=1$ and the equation $x^n \equiv a \pmod{N}$ has a solution then it has $n^m$ ...

**0**

votes

**1**answer

262 views

### Number field of degree 5

I am interested in field extensions of the rationals. About degree 3 extensions there are many refrences including the famous paper of Shanks "The Simplest Cubic Fields". In particular he gave ...

**0**

votes

**0**answers

47 views

### A lower bound on the number of matrices whose image contains all multiples of $p^e$

Let $0\leq e<e^\prime$ be integers. Now suppose $N$ is the number of $n\times n$ matrices over the ring $R:=\mathbb{Z}/p^{e^\prime}\mathbb{Z}$ (where $p$ is prime) such that ...

**4**

votes

**1**answer

373 views

### Point of order 5 over an elliptic curve

For this curve $y^2=x^3+b^2x^2-a^2b^2x$ where $a \neq b$ and $a,b$ are rational. I can prove that if $b^2+4a^2$ is square then torsion group of curve is $\mathbb Z2 \times \mathbb Z2$,
and when ...

**1**

vote

**0**answers

115 views

### How decomposable a modular representation can you get by reducing a given p-adic representation?

General Background
Take $G$ to be a finite group, and say $V$ is s $G$-representation over $\mathbb{Q}_p$. By picking a $G$-invariant lattice $L\subset V$ we can get an $\mathbb{F}_p$ representation ...