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

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10
votes
1answer
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
1answer
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
1answer
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
3answers
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
0answers
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
0answers
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
1answer
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
0answers
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
3answers
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
1answer
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
2answers
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
3answers
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
1answer
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
1answer
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
4answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
3answers
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
0answers
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
0answers
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
0answers
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
1answer
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
2answers
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
2answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
4answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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 ...