# Tagged Questions

**7**

votes

**2**answers

503 views

### Isomorphism problem for two radical extensions

Let $n\geq 2$ and let $a,b\in{\mathbb Q}$. Suppose that both the
polynomials $A=X^n-a$ and $B=X^n-b$ are irreducible. We want to know whether
( * ) there is a root $\alpha$ of $A$ and a root $\beta$ ...

**28**

votes

**4**answers

2k views

### Fermat's last theorem over larger fields

Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite.
Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite?
Here ...

**0**

votes

**1**answer

79 views

### Are the natural numbers a disjoint union of infinite sets of zero asymptotic density? [closed]

Suppose $\mathbb{N}=\bigsqcup_{i\in\mathbb{N}}E_i$ with $\#E_i=\infty$ for each $i$.
Is it possible that $\limsup_{N\to\infty}\frac{1}{N}\#(E_i\cap\{1,\ldots,N\})=0$ for all $i$, which would mean ...

**0**

votes

**0**answers

77 views

### Adelic integral factorization

In order to calculate Tamagawa numbers, I need to justify that for a nice (say Schwartz-Bruhat) function, the following identity holds :
$$\int_{\mathbf{A}^2} f(x)dx = \int_{SL_2(\mathbf{A})/SL_2(K)} ...

**2**

votes

**0**answers

102 views

### Siegel Walfisz Theorem for algebraic number fields

Is there a generalization of the Siegel Walfisz to algebraic number fields? This has been done for the prime number theorem in the prime ideal theorem.

**5**

votes

**1**answer

157 views

### Biquadratic reciprocity for $p\equiv 1\pmod 4$ and $q\equiv 3\pmod 4$

For a prime $p\equiv 1\pmod 4$, let $\left(\frac{\cdot}{p}\right)_4$ denote the rational biquadratic residue symbol; that is,
$$ \left(\frac{a}{p}\right)_4 =
\begin{cases}
\ \ \ ...

**22**

votes

**1**answer

796 views

### Underlying idea for (automorphic) L-function?

To preface, I am a student of automorphic representation theory, and I know full well the definition of the L-function attached to an automorphic representation.
I am intending to give a talk on the ...

**4**

votes

**2**answers

260 views

### Decomposing adelic points using torsors

Let $k$ be a number field and $X$ be a $k$-scheme. Let $G$ be a linear algebraic group over $k$ and let $f: Z \to X$ be a $G_X$-torsor ($G_X = G \times_k X)$. We can twist the torsor $f$ by 1-cocycles ...

**13**

votes

**3**answers

603 views

### Conjecture regarding closest point inside a discrete ball to a line

I'm a PhD student in image processing, where I've stumbled into a problem that seems to be essentially number theory. I've hunted around online and while I've found many results on similar problems, ...

**5**

votes

**1**answer

295 views

### Many integral points on quartic models of elliptic curve via differences of squares

Pick fourth power free integer $n$ ($p^4$ doesn't divide $n$).
Represent $n$ as difference of possibly negative integer squares
$n=v_i^2-u_i^2$.
The goal is to find quadratic polynomial with integer ...

**0**

votes

**0**answers

97 views

### E- and A-algorithms for finite arithmetic prime progressions and other sets

There is certain Eratosthenes spirit to my problem (See below). First of all I'd like to stress the mathematical aspect of my question. Also, my question does not amount to the divide and conquer ...

**3**

votes

**0**answers

84 views

### Partitions with each part dividing the original number

I have a question on partitions that I have not seen being discussed. It deals with those related to divisors.
My definition of partitions I am working with is as follow: a sequence of weakly ...

**9**

votes

**1**answer

329 views

### Eichler-Shimura congruence

I'm trying to understand the Eichler-Shimura congruence which relates the Hecke operator $T_p$ to Frobenius at $p$ in characteristic $p$.
Two possible ways to compute $T_p$ mod $p$ seem to be:
A) ...

**2**

votes

**2**answers

183 views

### Bound on exponential sum with weights

Let $e(z)$ denote $e^{2 \pi i z}$ and let $f(z)$ a smooth real function.
I know one can bound sums of the form
$$
\sum_{x \leq X} e(f(x))
$$
via for example Van der Corputs's result, provided we make ...

**1**

vote

**0**answers

75 views

### Is there a general connection between value distribution and zero distribution for functions representable by Dirichlet series?

Some time ago I read part of a book in which the author made some conjectures outlining what kind of zero distribution is expected for functions representable by Dirichlet series with completely ...

**1**

vote

**0**answers

117 views

### Questions on prime integral ideal congruences

Suppose that we are given a fixed pair $a_1, a_2$ of non-zero irrational algebraic integers in some number field $K$ which are independent over $\mathbb{Q}$. Suppose that $\mathcal{P}$ is a prime ...

**1**

vote

**0**answers

92 views

### Functoriality for non-split orthogonal groups

I am trying to understand the functoriality conjectures of Langlands. We know that the functoriality conjectures imply that automorphic $L$-functions of a connected reductive group are equal to ...

**4**

votes

**1**answer

199 views

### Isomorphisms between spaces of test functions and sequence spaces

I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...

**4**

votes

**0**answers

181 views

### Generating Function of distinct way of partitioned square sums of positive integers

Let's define a function $p_2(n)$ that it is total distinct way to write $n$ positive integer as sum of square of positive integers. For example, $9$ can be partitioned as sum of squares in 4 distinct ...

**0**

votes

**0**answers

69 views

### Nilpotent differential operators

I am reading Dwork's book an Introduction to G-Funcions and confronted with a problem. In section 2, chapter III (page 81), he assumes that $\mathscr F=K(X)$ the field of rational functions with ...

**3**

votes

**1**answer

250 views

### Sum of two squares and implication of Bunyakovsky conjecture

Bunyakovsky conjecture states that a polynomial with integer
coefficients takes infinitely many prime values at integers,
unless this is impossible for trivial reasons.
Let $a_1(x), a_2(x), a_3(x), ...

**13**

votes

**1**answer

438 views

### References for $K_{4k}(\mathbb{Z})$

Weibel's "Algebraic K-theory of rings of integers in local and global fields" says $K_{4k}(\mathbb{Z})$ are known to have odd order, with no prime factors less than $10^7$, but are conjectured to be ...

**2**

votes

**2**answers

319 views

### Rate of convergence of an irrational rotation

Let $\alpha, \beta \in \mathbb{R}$. Let $\{x\}$ denote the fractional part of $x$ and let $\|x\| = \min(\{x\}, 1-\{x\})$.
If we assume that $\alpha$ is irrational, then there exists an increasing ...

**3**

votes

**1**answer

179 views

### Thin sets that are well-distributed over arithmetic progressions?

The primes do a nice job of intersecting an arithmetic progression $\{a+dn\}_{n=0}^\infty$ when $a$ and $d$ are coprime (see Dirichlet's theorem).
I would like a set of integers $S$ such that
the ...

**0**

votes

**1**answer

191 views

### The number of solutions of a Diophantine equation [closed]

Is $\lim_{n \rightarrow \infty} |\{(x,y) \in \mathbb{Q}(\zeta_n)^2 : y^3 = x^3 + x + 1\}| < \infty ?$ where $\zeta_n$ is a primitive $n$-th root of unity.
That is, I am asking whether the number ...

**2**

votes

**0**answers

169 views

### Diophantine equations over cyclotomic fields

Let $\mathbb{Q}^{\text{ab}}$ be the compositum of all finite abelian extensions of $\mathbb{Q}$. Explicitly, $\mathbb{Q}^{\text{ab}}$ is the field obtained from $\mathbb{Q}$ by adjoining all roots of ...

**3**

votes

**0**answers

80 views

### Oscillatory integral moments of $L(\frac{1}{2} + it, f \times f)$

Understanding moments and subconvexity bounds for $L$-functions is a big topic with a lot of activity. I'm currently looking at a related problem, bounding
$$
\int_0^T L\left(\tfrac{1}{2} + it, f ...

**11**

votes

**0**answers

288 views

### What is the expected dimension of the Zariski closure of the rational points on the moduli space of curves?

For each genus $g$, there are many curves of genus $g$ defined over $\mathbb Q$. How many? We might study this question by considering the rational points of the Deligne-Mumford moduli space of curves ...

**5**

votes

**1**answer

428 views

### Disjoint images of polynomials

Are there any $f,g \in \mathbb{Q}[x]$ such that for every root of unity $\zeta$, and every $a,b \in \mathbb{Q}(\zeta)$, $f(a) \neq g(b)?$

**1**

vote

**1**answer

182 views

### every arithmetic progression contains a sequence of $k$ “consecutive” primes for possibly all natural numbers $k$?

I ask the same question here:http://math.stackexchange.com/q/1019404/192097
writing a little better the previous question: it´s true that if we let $a$ and $b$ be coprime integers, then the ...

**4**

votes

**1**answer

146 views

### Voronoi formula and twists by additive characters

I was wondering if there are any references for the error term in the problem
$$\sum_{n\leq x} r(n) \exp(2\pi i\frac{a}{q}n)$$
where $r(n)$ is the number of representations of $n$ as a sum of two ...

**11**

votes

**3**answers

2k views

### How many Pythagorean triples are there in which every member is triangular?

How many Pythagorean triples $(a,b,c)$ are there such that $a, b$ and $c$ are triangular?
Any two solutions with only $a$ and $b$ interchanged are considered equivalent.
The question of existence ...

**4**

votes

**0**answers

140 views

### How to check whether a scheme of finite type over Spec Z is regular or not [duplicate]

Let $f_1, f_2, \ldots f_k$ be a set of polynomials in $n$ variables, with integer coefficients. These define an affine scheme $X$ of finite type over $Spec \mathbb{Z}$. (We could also consider ...

**2**

votes

**1**answer

153 views

### Bound for sums of bounded multiplicative functions that are zero at primes

Let $h:\mathbb{N}\rightarrow\mathbb{C}$ be a bounded multiplicative function with $h(p)=0$. The motivation for this question is just a general enquiry and, since I suppose it has already been ...

**0**

votes

**0**answers

123 views

### The Diophantine equation $x^2 + bxy + cy^2 = p^z_1 \cdots p^{z_k}$

Let $b,c \in \mathbb{Z}$ and let $p_1,\ldots,p_k$ be given primes. Is there an effective algorithm to find all the solutions of the Diophantine equation $$x^2 + bxy + cy^2 = p_1^{z_1} \cdots ...

**1**

vote

**1**answer

146 views

### Existence of arithmetic function satisfying a certain property

I was interested in an arithmetic function satisfying a certain property, I am not sure at the moment if such thing even exists or not. But I was wondering maybe I could get some hint or idea or input ...

**5**

votes

**1**answer

386 views

### Are there infinitely many primes p such that both p-1 and p+1 have at most 3 prime factors, counted with multiplicity?

This is a subsequence of primes $p$ for which $p^2-1$ has at most 6 prime divisors counted with multiplicity.
This sequence described in the question is the sequence A079153 in OEIS.
I could not ...

**0**

votes

**0**answers

45 views

### Arguments of Dirichlet coefficients of prime index of primitive elements of the Selberg class

Let $F$ and $G$ be two primitive elements of the Selberg class such that for $\Re(s)>1$, $\displaystyle{F(s)=\sum_{n>0}\dfrac{a_{F}(n)}{n^s}}$ and ...

**12**

votes

**3**answers

455 views

### For an elliptic curve $E/\mathbb{Q}$ can the cohomology group $H^1(\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q}), E[p])$ be nontrivial?

Suppose that $E$ an elliptic curve defined over $\mathbb{Q}$ and $p$ an odd prime. Let $G=\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q})$. I am wondering whether the cohomology group $H^1(G, E[p])$ can be ...

**3**

votes

**0**answers

137 views

### A Diophantine equation revisited

No integer solution of this Diophantine equation $$x^4+y^4+1=z^2$$ is known other than the trivial ones.
While I was reading a paper of Don Zagier, I realized that his idea on the Euler's sum of ...

**14**

votes

**2**answers

889 views

### Images of polynomials

Let $f,g \in \mathbb{Q}[x]$ be polynomials such that $\{f(a) : a \in \mathbb{Q}\} \subseteq \{g(a) : a \in \mathbb{Q} \}$. Must there be some $h \in \mathbb{Q}[x]$ such that $f(x) = g(h(x))$ for all ...

**6**

votes

**0**answers

155 views

### Is ${\rm lcm}\{{\rm ord}_p(q)\colon q\mid p-1,\ q>2\}>\sqrt p\ \,$?

The following question is "ideologically related" to the one I recently asked here.
For a prime $p$, let $M_p$ denotes the least common multiple of the orders modulo $p$ of all odd prime divisors of ...

**9**

votes

**2**answers

264 views

### A back and forth Euclidean algorithm over the integers--does it have bounded length?

cLet $a,b,c,d\in \mathbb{Z}$ and suppose we have the equation $ac+bd=1$. One way of thinking about this equation is it expresses the fact $\gcd(c,d)=1$. It is well-known that all other similar ...

**3**

votes

**4**answers

397 views

### Pairs of quadratic polynomials taking values pairs of consecutive squares

Let $f,g \in \mathbb{Z}[x]$ be quadratic and neither square.
For $x,y,z \in \mathbb{Z}$ what is the maximal number
of solutions to $f(x)=z^2,g(y)=(z+1)^2$?
Solutions are integral points on the genus ...

**6**

votes

**0**answers

185 views

### A conjecture of Erdos on consecutive differences of primes

Let $d_k = p_{k + 1} - p_k$ be the difference between consecutive primes and define
\begin{equation}
e_k = \left\{\begin{array}{c l} 1 &, d_{k + 1} > d_k \\ 0 &, \text{otherwise} ...

**0**

votes

**0**answers

120 views

### Estimating the number of twin primes of given natural configuration order

This question is a follow-up from About Goldbach's conjecture. Let
$$\mathrm{Co}_{k}(x):=\{n\le x:\mathrm{ord}_{c}(n):=\pi(\sqrt{2n-3})=k\},~~~\mathrm{co}_{k}(x):=\vert\mathrm{Co}_{k}(x)\vert$$
...

**7**

votes

**0**answers

240 views

### Lindelof Hypothesis implying Selberg Eigenvalue Conjecture?

(General) Lindelof Hypothesis which says for any $L$-function we have $$L(1/2+it)\ll Q(t)^{\epsilon}$$ for any $\epsilon>0$ where $Q(t)$ is the conductor of $L(s)$ at $t$.
For a Maass form $\phi$ ...

**11**

votes

**0**answers

179 views

### What's the dimension of the space of CM cusp forms?

I would guess that the following is very well known, but I don't know the answer and I couldn't find anything with some googling.
Let $\Gamma \subset \mathrm{SL}(2,\mathbf Z)$ be a congruence ...

**-1**

votes

**1**answer

110 views

### CD - continuous development

Let K/Q be a galois extension, p an odd prime and L/K a Z_p extension, different from the cyclotomic one. Let H/K be finite abelian and linearly disjoint from L.
a) Are there infinitely many primes ...

**3**

votes

**1**answer

142 views

### Is $p$ is square modulo $F_p$ when $p=4k+1 > 5$?

$F_n$ are the Fibonacci numbers.
In On computing factors of cyclotomic polynomials p.1 for odd square-free $n>1$ the cyclotomic polynomial $\Phi_n(x)$
satisfies:
$$ 4 \Phi_n(x)=A_n(x)^2 - ...