**2**

votes

**0**answers

130 views

### Distribution of Fourier coefficients of Maass forms

In the sense of Maass an automorphic function $\phi$ with Laplace-Beltrami eigenvalue $\frac{(d-1)^2}{4}+t^2$ on $d$-dimensional hyperbolic space which can be thought as ...

**2**

votes

**1**answer

163 views

### extensions of crystalline representations

Denote by $G_p$ a choice of an absolute Galois group of $Q_p$, the field of $p$-adic numbers. Consider a continuous representations of $G_p$ on a $3$-dimensional $Q_p$ vector space that is a ...

**6**

votes

**0**answers

224 views

### What is miraculous about the mirabolic subgroup?

I recently asked this question about Euler subgroups and generalizing the automorphic theory of $\mathrm{GL}_n$ to a more general setting. My question here is more specific.
As mentioned there, the ...

**10**

votes

**0**answers

289 views

### Word complexity of primes mod 4

For an infinite binary word $w$, the word complexity $f_w(n)$ is defined as the number of different subwords of length $n$. The asymptotic behavior of this function is an important parameter of the ...

**3**

votes

**1**answer

225 views

### Non-principal ideals in cyclotomic fields

Let $K=\mathbb Q(\xi_{39})$ be the 39-th cyclotomic field. Pari-GP told me that the prime ideals above $3$ and $13$ are not principal. Is there a way to prove that by hand (no computation made by ...

**16**

votes

**1**answer

2k views

### Mazur secret Bourbaki report “Analyse p-adique”

Does anyone happen to know if a scan of Mazur's report exists, and, if so, where to find it? It appears in the references for Katz's "Higher congruences" and "Eisenstein measure" papers.

**3**

votes

**2**answers

535 views

### Automorphisms of $\mathbb C_p$

I am looking for a non-trivial automorphism $\sigma$ of $\mathbb C_p$ such that $\sigma(\mathbb Q_p)\subset\mathbb Q_p$.
If $\mathbb C_p$ were spherically complete, then by Hahn-Banach theorem, that ...

**5**

votes

**0**answers

140 views

### Euler Subgroups and Automorphic L-functions

Recently, I have read about the Whittaker expansion for $\mathrm{GL}_n$ and was struck by the utility of the mirabolic subgroup, $\mathrm{P}_n\subset \mathrm{GL}_n$ of matrices with bottom row $(0\; 0 ...

**8**

votes

**3**answers

282 views

### Lower bound for spectral radius on $\operatorname{GL}(n,\mathbb{Z})$

Consider the group of matrices $G =\operatorname{GL}(n,\mathbb{Z})$ with integer entries and determinant $\pm 1$. For each matrix $D \in G$, the product of the eigenvalues of $D$ is equal to $\det D ...

**1**

vote

**0**answers

133 views

### Explicit description/calculation of norm group of ideles of characteristic $p$ global field

I posted the same question earlier in stack exchange,
(http://math.stackexchange.com/questions/1130391/algebraic-proof-of-2nd-inequality-of-global-class-field)
thinking it is most definitely not a ...

**14**

votes

**1**answer

1k views

### How important is Weil's decomposition theorem today?

Andre Weil's Apprenticeship of a Mathematician (p. 46) tells how he as a student realized that all of Fermat's uses of descent are unified in one principle: "If $P(x,y)$ and $Q(x,y)$ are homogeneous ...

**1**

vote

**1**answer

176 views

### Heegner points on elliptic curves

I want to know about Heegner point computations for a CM elliptic curve. What is the best reference book/paper for reading?

**3**

votes

**0**answers

93 views

### ramified principal prime ideals in Artin-Schreier extension

Let $q$ be a power of $2$ and $K$ be the quadratic extension of $\mathbb F_q(T)$ defined by $K:=\mathbb F_q(T)[y]$ where $y^2+y=f(T)\in\mathbb F_q(T)$.
Put $\mathcal O_K$ the integral closure of ...

**1**

vote

**1**answer

134 views

### Is there a Poisson Summation formula for imprimitive Dirichlet characters?

I was wondering if there exists a Poisson Summation formula (like the one existing with primitive character) for imprimitive Dirichlet characters ?
For a primitive Dirichlet character $\chi$ we have:
...

**5**

votes

**0**answers

194 views

### Legendre polynomials and formal groups

Let $P_n(x)$ be Legendre polynomials:
$$\frac{1}{\sqrt{1-2tx+t^2}}=\sum\limits_{n=0}^{\infty}P_n(x)t^n.$$
Usual arguments from the theory of formal groups allow to
prove that for any $n$
...

**1**

vote

**0**answers

144 views

### Fourier expansions at the cusps of $\Gamma_0(N)$

My question may be basic but I can't find any answer. Let $N$ be a positive integer. I need to find the constant term (of the Fourier series) at each cusps of a modular form
...

**14**

votes

**0**answers

377 views

### Can there be arbitrarily many cubic fields unramified outside $\{p,\infty\}$?

Observe, trivially, that since quadratic fields correspond to rational integers modulo squares (viz. discriminants), there are (roughly about, but certainly at most) $2^{|S|+1}$ quadratic fields ...

**2**

votes

**0**answers

91 views

### Linear forms with best approximation vectors lying in a subspace

Setup: For $u \in \mathbb{R}^n$, let $\rho(u)$ be the Euclidean length, $\sqrt{u_1^2 + \ldots + u_n^2}$. For $x \in \mathbb{R}$ let $\|x\| = \min_{k \in \mathbb{Z}} |x - k|$, and for $x \in ...

**2**

votes

**2**answers

116 views

### asymptotic for a sequence coming from multiplicative sets

Let $a,b$ two multiplicatively independant positive integers with $1<a<b$
that is $a^mb^n\ne1$ for all $m,n\in\mathbb N\setminus\{0\}$
We sort out the set $E=\{a^mb^n\mid m,n\in \mathbb ...

**2**

votes

**0**answers

132 views

### Consequences failure of $\tau$ conjecture

A question Bounds for constructing $n!$ with additions, subtractions, and multiplications starting from $1$ was asked on constructing $ak!$ with ring operations.
$\tau$ conjecture states if $\exists$ ...

**0**

votes

**0**answers

129 views

### Ring of integers in Artin-Schreier extension

Question put in mathstackexchange but received no answer.
It is well-known( see Goldschmidt book: Algebraic Functions and Projective Curves)
that for $q$ a power of $2$ a quadratic separable ...

**96**

votes

**1**answer

4k views

### What is a Frobenioid?

Since there will be a long digression in a moment, let me start by reassuring you that my intention really is to ask the question in the title.
Recently, there has been a flurry of new discussion ...

**12**

votes

**2**answers

583 views

### Formal group law over $\mathbb{F}_p$

Let $p$ be a prime. For each $n > 0$ there is a unique 1-dimensional commutative formal group law $F$ over $\mathbf{Z}$, $F(X, Y) = X + Y + \dots \in \mathbf{Z}[[X, Y]]$, whose logarithm function ...

**4**

votes

**2**answers

598 views

### How to calculate the sum of remainders of N?

I'm trying to sum the remainders when dividing N by numbers from 1 up to N
$$\sum_{i = 1}^{N} N \bmod i$$
It's easy to write a program to evaluate the sum if N is small in O(N) but what if N is large ...

**1**

vote

**0**answers

53 views

### Question regarding a theorem of Erdos and Renyi on $B_2(g)$ sequence

Let $S \subset \mathbb{N}$. We say $S$ is of type $B_2(g)$ if the number of representation
of the form $n = s_1 + s_2 \ (s_1 \leq s_2)$ is bounded by $g$ for every $n \in \mathbb{N}$.
Let $S(n)$ be ...

**1**

vote

**1**answer

215 views

### Sum of two squares - Number of steps in Fermat descent

If a prime $p$ can be written as the sum of two squares, then one can construct this representation via Fermat descent if we know an $x$ such that $x^2 \equiv −1 \mod p$. Is there a possibility to say ...

**10**

votes

**0**answers

706 views

### Primes and Parity

This problem is motivated by the polymath4 project. There, the aim was to find an efficient deterministic algorithm for finding a prime larger than $N$. The hope was to find a polynomial algorithm in ...

**17**

votes

**2**answers

766 views

### A possibly surprising appearance of Lucas numbers

Let $S$ be the set of polynomials defined as follows: $0$ is in $S$, and if $p$ is in $S$, then $p + 1$ is in $S$ and $x \cdot p$ is in $S$, so that $S$ "grows" in generations: $g(0)=\{0\}$, ...

**1**

vote

**0**answers

97 views

### A question related to metric Diophantine approximation

In metric Diophantine approximation you are often interested in finding conditions on $(\phi(q))_{q \geq 1}$ which guarantee that
$$
\left| \alpha - \frac{p}{q} \right| < \frac{\phi(q)}{q}
$$
has ...

**7**

votes

**2**answers

821 views

### Surveys of the items of Erdős' “toolbox”

Could you point out some survey papers and monographs that highlight the kernel of tricks, techniques, and tools that Paul Erdős employed the most in his research work (in particular in graph theory, ...

**7**

votes

**1**answer

176 views

### Orders in Central Simple Algebras. Applications

It is known that orders in quaternion algebras (over a number field) are used for constructing geometric objects like hyperbolic orbifolds and Shimura curves. Moreover, if one knows embedding ...

**4**

votes

**0**answers

259 views

### Is there a hidden symmetry in the prime numbers distribution?

Under Goldbach's conjecture, let's consider once again the map $r_0\colon n\mapsto r_{0}(n)$ such that $r_{0}(n)$ is the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime.
...

**2**

votes

**0**answers

106 views

### Number of common solutions of polynomial system

Let $ \mathbb{F}_p$ be a finite field and $\{f_j\}_{j=1}^{j=r} \subseteq \mathbb{F}_p[X_1,...,X_n]$ be a set of polynomials.
Let consider the system of equations:
$f_j(x_1,...,x_n)=0$ for $j = ...

**22**

votes

**5**answers

1k views

### Parametric solutions of Pell's equation

Given a positive integer $n$ which is not a perfect square, it is well-known that
Pell's equation $a^2 - nb^2 = 1$ is always solvable in non-zero integers $a$ and $b$.
Question: Let $n$ be a ...

**7**

votes

**1**answer

170 views

### A positivity problem involving the number of ways of expressing $n$ as a product of $k$ factors

Let $d_k(n)$ denote the number of ways of expressing $n$ as a product of $k$ factors, and let $$D_k(x)=\sum_{n\leq x}d_k(n)$$ be the summatory function. During a study of Mertens' function I was lead ...

**0**

votes

**0**answers

103 views

### Coefficients of Hilbert polynomials

Recall that we can define the Hilbert series of a graded commutative algebra
$$\displaystyle S = \bigoplus_{n \geq 0} S_n$$
over a field $K$ by
$$\displaystyle \mathcal{H}_S(t) = \sum_{n=0}^\infty ...

**4**

votes

**2**answers

271 views

### References for general Hasse-Weil zeta function

Most research on the Hasse-Weil zeta function focuses on some particular type of algebraic variety, and general surveys usually deal mostly with the better understood elliptic curve case.
I am ...

**2**

votes

**0**answers

96 views

### Comparing the size of two sums

Let $q$ be a prime power and $\mathbb{F}_q$ be the field of $q$ elements.
Let $\mathbb{F}_q[t]$ be the polynomial ring over $\mathbb{F}_q$.
I am working on a research project, where I bounded a ...

**2**

votes

**1**answer

114 views

### Asymptotic expansion for the Bell numbers

The Bell numbers $B(n)$ (that is, the numbers that count the set partitions of a set, and have exponential generating function $\exp(e^x -1)$ ) admit the asymptotic expansion
$$\frac{\log B(n)}{n} = ...

**2**

votes

**1**answer

164 views

### Rankin-Selberg convolution and product of degrees

As I'm kinda obsessed with the Selberg class and because of the general converse conjecture, I'm still trying to get a rough idea of what automorphic representations and their L-functions as well as ...

**11**

votes

**5**answers

1k views

### Bounds for constructing $n!$ with additions, subtractions, and multiplications starting from $1$

Found this on Complexity Zoo warning expired certificate
check NP Over The Complex Numbers.
[BCS+97] show the following striking result. For a positive integer $n$, let $t(n)$ denote the minimum ...

**12**

votes

**1**answer

554 views

### Elliptic curves and connected components

Are there elliptic curves of positive rank with two real connected components
in which all the rational points lie only on one component?
Concrete examples are really appreciated.

**18**

votes

**1**answer

650 views

### What are the products $\prod_{A\subset{\mathbb F}_p\colon |A|=n} \sum_{a\in A} \zeta^a$ equal to?

This is a somewhat more explicit version of a question I have recently asked.
Let $p$ be an odd prime, and write $\zeta:=\exp(2\pi i/p)$ (any other primitive $p$th root of unity will do as well). For ...

**8**

votes

**1**answer

422 views

### Open access journals in number theory

I'm a phd student in number theory, and I'm required (by the funding council) to publish any article I write in open access journals. The problem is that all journals I can find are either out of my ...

**4**

votes

**1**answer

136 views

### Length of the binary representation of a primorial

Let $p_n\# = \prod_{k=1}^n p_k$ be the $n$-th primorial.
Q1. Given $n$ (in binary) is there an efficient way (polynomial time) to calculate the exact number of digits of the binary representation ...

**1**

vote

**2**answers

580 views

### Has this formula about prime gaps already been conjectured and/or proven?

While playing around with prime gaps, I found out that the following formula seems to be a rather good approximation of the ratio $\dfrac{p_{b}-p_{a}}{b-a}$ where $a<b$ are positive integers:
...

**9**

votes

**0**answers

235 views

### Transcendence of products of certain real algebraic numbers

Let
\begin{equation}
z := \prod_p p^{1/p^2},
\end{equation}
where the product is over all prime numbers $p$, and we always take the positive real root. Is $z$ transcendental or algebraic, or (as I ...

**0**

votes

**1**answer

116 views

### Fixed field of the Nebentypus of a newform for $\Gamma_1(N)$

Let $f=\sum_{n\geq 1}\in S_2(\Gamma_1(N),\varepsilon)$ be a normalized newform without CM and with Nebentypus $\varepsilon$. Let $L=\mathbb Q(a_n\colon n\in \mathbb N)$ be the number field generated ...

**-8**

votes

**1**answer

185 views

### Proof of a cubic equation problem [closed]

Well I was doing some questions and i found something. This equation
$x^3+y^3+z^3=w^3$
has only one solution which is
$x=3,y=4,z=5,w=6$.
And what I have have proposed is that there is not other ...

**3**

votes

**1**answer

109 views

### Density of tuples of conjugate algebraic numbers

One can see that algebraic numbers are dense in the complex plane by just looking at quadratic polynomials. I am interested in a "higher order" density of algebraic numbers.
More specifically: is it ...