## New answers tagged nt.number-theory

3
votes

### $\pi$-adic Galois representations of attached to newforms at $p \nmid N$ are crystalline

Blasius and Rogawski's paper "Motives for Hilbert modular forms" (1993) proves a more general result for Hilbert modular forms over any totally-real field, which includes this as a special ...

6
votes

Accepted

### Calculating the centralizer of a subgroup of $\mathrm{GL}(n, \mathbb{Z})$

There is an algorithm to do this (and also to test two matrices in ${\rm GL}(n,{\mathbb Z})$ for conjugacy) described in the paper:
The conjugacy problem in ${\rm GL}(n,{\mathbb Z})$
Bettina Eick, ...

5
votes

### Splitting fields of degree 4 irreducible polynomials containing a fixed quadratic extension

This works even for any symmetric group and any quadratic number field, and can be done very explicitly.
When $n$ is even, let $f(t,X) = X^{n-1}((n-1)X-n) + t$, and when $n$ is odd, let $f(t,X) = X^n-...

3
votes

Accepted

### Will this "tree" cover all rational numbers in a range?

The main Question is answered in the negative by @Saúl RM in the comments.
We answer the sub-question
"can there even be branches that have the same value?":
not for any rational $r$ other ...

6
votes

### Splitting fields of degree 4 irreducible polynomials containing a fixed quadratic extension

Yes, this can be done. Let $K$ be a $S_4$-quartic field, $C$ its cubic resolvent field, and $L$ the Galois closure of $K$. By Galois correspondence, $L$ contains a unique quadratic subfield $Q$ which ...

9
votes

Accepted

### Polynomial that is not always a square over $\mathbb{Z}_p$

Yes, this is true.
We have $(1+x^2)^3-1 = x^2 (x^4 + 3x^2 + 3)$, so we're asking for
nonzero x such that $x^4 + 3x^2 + 3$ is not a square. If none exist then
the genus-1 curve $y^2 = x^4 + 3x^2 + 3$ ...

2
votes

### Sum of $\frac{1}{(\delta_1,\delta_2)}$ with congruence restrictions

Define $f(n)=\frac{1}{n}\prod_{p\mid n}(1-p)$ now $\sum_{d\mid n}f(d)=\frac{1}{n}$ thus $\sum_{d=1}^{\infty}f(d)[d\mid \delta_1][d\mid \delta_2]=\frac{1}{(\delta_1,\delta_2)}$ which means $\sum_{d=1}^{...

9
votes

### Cubic polynomial over $\mathbb{Z}_p$

A monic polynomial of degree $3$ has this form if and only if its value at $1$ is $u$. So this is just the number of products of three monic linear factors whose value at $1$ is $u$. For each $\alpha \...

1
vote

### Why do Chern forms show up in Arakelov geometry?

I apologize for answering late.
I think the 1D case has been discussed multiple times in the forum already. The high dimensional case you suggested was first defined by Bost. See page 63 in below:
...

3
votes

### Exponential sum with weight in bottom

Under your assumption that no $n$ from $1$ to $X$ has $|1 -e(c_1n)|<\epsilon$, we have an upper bound for your sum of the form $2 \epsilon^{-1} \log X + O(X)$.
This is the "trivial bound" ...

5
votes

### Is the value of the power series at 0.1 transcendental?

For the UPDATE, allowing coefficients $a_n < M$ for a fixed $M$. Then there are examples with $f(1/10)$ rational.
Let's do this. Define a sequence $(a_n)$ of coefficients as follows: Start with ...

8
votes

Accepted

### Is the value of the power series at 0.1 transcendental?

This question is likely open.
We can tell whether $f(1/10)$ is rational or irrational (by asking whether $a_n$ is eventually periodic); in this case, definitely irrational.
Can we have an algebraic ...

3
votes

### Number of solutions of $am \equiv bn \pmod{q}$

I made a fatal mistake in my attempt at obtaining a simple asymptotic formula for the number $J$ of solutions to the congruence with the given restrictions. Undaunted, I come back with the proof of a ...

7
votes

Accepted

### Faithful representations of integral models

Yes, there exists a closed immersion $\mathcal{G}\to \mathrm{GL}_n$ over
$\mathbb{Z}$. This is folklore. For a proof, see for example Proposition 3 of arXiv:2012.05708v3

10
votes

### Kissing number lower bound vs. upper bound - precise meanings?

Lots of questions here, I'll see how many I can address. To be clear, $K_L$ and $K_U$ are summaries of our current knowledge. There is some true kissing number in each dimension, and hopefully we'll ...

2
votes

### Leech lattice shortest vector vs other 23 cases and E8 cases

Leech lattice $Λ24$ has larger distance between centers of the two balls (namely $D=2$), in contrast with other 23 classes of 24-dimensional lattice which as $D=√2$, also in contrast with the E8 ...

3
votes

### The theta function of an odd Dirichlet character

The point is that $\theta_\chi(t)$ is a modular form whose coefficients are essentially $\chi(n)$. If we twist $\chi(n)$ by some random odd smooth function of $n$, we don't get a modular form. (...

10
votes

### Integer solutions of an exponential equation

Here's a proof that the $x=1$ solution is unique
using only facts about "Pell equations" that were already known to Fermat
(if not centuries earlier to Bhaskara II et al.)
and should ...

3
votes

### On four Ramanujan-type "Legendrian" sequences with a 3-term recurrence?

Stupid of me. As O. Gorodetsky mentions, these are classical:
$$F_1=(91\zeta(3)-2\pi^3\sqrt{3})/432$$
$$F_2=(28\zeta(3)-\pi^3)/64$$
$$F_3=(117\zeta(3)-2\pi^3\sqrt{3})/243$$
In addition, note that ...

3
votes

Accepted

### Lang's remark on Lindemann-Weierstrass theorem

For brevity of notation, if $I = (i_1,\ldots,i_n) \in \mathbf N^n$, write $x^I = x_1^{i_1}\cdots x_n^{i_n}$. Write $\boldsymbol \alpha$ for $(\alpha_1,\ldots,\alpha_n)$, and set $I \cdot \boldsymbol \...

18
votes

Accepted

### Integer solutions of an exponential equation

A general method, not necessarily best for this particular equation, is to split into three cases by writing $x=3u+v$ with $v\in\{0,1,2\}$. Then rewrite your equation as
$$ A(7^u)^3 + 2 = y^2\quad\...

1
vote

Accepted

### Iterated exponential sums

This sum is equal to
$$
\frac{\left( \sum_{n \leq x} e(f(n)) \right)^2 - \sum_{n \leq x} e(2 f(n))}{2}
$$
In most situations, we do not expect there to be more than square root cancellation, and thus ...

1
vote

### On Zagier's missing continued fraction with multiple limits?

To complete the 12 cfracs in this post and the 4 in the next, all associated with 16 "sporadic sequences", then 13 of them have closed-forms, 1 has six limits (also with closed-forms but one ...

4
votes

### Sum of $\frac{1}{(\delta_1,\delta_2)}$ with congruence restrictions

Your sum is
\begin{align*}
L(Q)&=\sum _{d\leq Q}\frac {1}{d}\sum _{\delta _1,\delta _2\leq Q/d\atop {a|d\delta _1\atop {b|d\delta _2\atop {(\delta _1,\delta _2)=1}}}}1=\sum _{dh\leq Q}\frac {\mu (...

0
votes

Accepted

### On the continued fractions using Cooper's sequences $s_7,\, s_{10},\, s_{18}$ and the Zudilin-Cohen sequence

(This answers Question 2.)
Thanks to Cohen's 2022 paper, turns out there is a deg-$4$ and one can find polynomials $Q_k(n)$ for general deg-$k$ such that,
$$(n+1)^k s_{n+1} = Q_k (n)\, s_n - n^k s_{n-...

3
votes

### Geometric interpretation of Iwasawa algebras: $\mathbb{Z}_p[[T]]$ as a disk?

I give this answer just to make David's answer a little more general. Precisely, your $\Lambda$ should be thought as functions with norm less than $1$ in some Banach $\mathbb{Q}_p$-algebras. We fix a ...

7
votes

Accepted

### Geometric interpretation of Iwasawa algebras: $\mathbb{Z}_p[[T]]$ as a disk?

The correct viewpoint is not "$\Lambda$ is like a disc", but "$\Lambda$ is like the functions on a disc".
To see this, ask yourself: given an element $f \in \mathbb{Z}_p[[T]]$, ...

71
votes

Accepted

### For a finite set A of positive reals, prove that the set A + A - A contains at least as many positive as negative elements

Here is a counterexample. We first need a "more sums than differences" construction:
Lemma. For any $\varepsilon>0$ there exists a cyclic group ${\bf Z}/N{\bf Z}$ and a non-empty subset ...

3
votes

Accepted

### Special configurations on a circle from a homological algebra problem

There is a simple characterization of interesting configurations:
Lemma. A configuration $x_0=0< x_1 < x_2 < ... <x_r$ of Gorenstein dimension $g$ is interesting if and only if there exist ...

3
votes

### Asymptotic density of an infinite union of subgroups

It's false. Up to an (irrelevant) factor of $2$, I work with $\mathbb{N}$ instead of $\mathbb{Z}$.
The only ingredient needed is that $c_D \to 0$ as $D \to \infty$, where $c_D > 0$ is such that $$\...

15
votes

### Does this conic have a rational point?

The answer is no. If there was, we could assume that $X,Y,Z$ are in $\mathbb Q[u,v]$ and are coprime (since that ring is a UFD). Setting $v=0$ we get $X(u,0)^2+uY(u,0)^2=0$ in $\mathbb Q[u]$, which (...

5
votes

### For a finite set A of positive reals, prove that the set A + A - A contains at least as many positive as negative elements

The following idea may work, the last step looks heuristic though.
Let $A=\{a_1, \ldots, a_n \}$ and let
$$C:= \{0 , \pm \alpha_1, \ldots \pm \alpha_k \}$$ with $\alpha_1, \ldots, \alpha_k >0$.
Let ...

3
votes

Accepted

### Why can Hecke operators be regarded as finite flat cohomological correspondence?

The first half of the question has been answered in the comments, so let me address the second half of the question.
We want to define Hecke operators on the complex $R\Gamma(X, \omega^k)$, because ...

4
votes

### Schur multiplier of a Chevalley group of type $D_5$

According to Theorem 5.3 of the first paper of Mike Stein that you mention, the universal central extension of the Chevalley group of type $D_5$ over the integers is given by the Steinberg generators ...

2
votes

Accepted

### Is this a valid method of extending convergence of the Maclaurin series for $\frac{x}{x+1}$ from $|x|<1$ to $\Re(x)>-1$?

In \eqref{5}, the inner sum is obviously $x(1-x)^{n-1}$. So, the limit in \eqref{5} (equal $\dfrac x{x+1}$ indeed) exists if and only if $|1-x|<2$.
In \eqref{6}, using the substitution $k=K-n-j$ in ...

6
votes

### What are the simplest sentences which might distinguish Zilber’s field from the complex numbers?

Kruckman's comment is a good answer. A counterexample to Schanuel's conjecture would be one such sentence. For example, a non-zero polynomial $p(x,y)$ with integer coefficients such that $p(\pi,e) = 0$...

9
votes

### Relation between different $E_8$ matrices

$M_1$, $M_4$ and $M_5$ are all the same quadratic form up to integer change of basis: To get from $M_1$ to $M_4$, conjugate by the diagonal matrix with digaonal entries $(1,-1,1,-1,1,-1,1,-1)$; to get ...

10
votes

Accepted

### On Zagier's missing continued fraction with multiple limits?

Set $Q=(1/2)L(\chi_{-3},2)$ (related to your Gieseking constant) and
$P=2\pi^2/81$. The limits are almost certainly (not proved),
\begin{align}
\lim_{m\to\infty}C_2(6m+0) &= -Q\\
\lim_{m\to\infty}...

17
votes

### A number theoretic conjecture by Chat GPT

The more natural conjecture, that every even number arises as the difference of two primes, was asked on math.SE years ago. That question remains open, and I have a hard time imagining that ...

9
votes

Accepted

### A number theoretic conjecture by Chat GPT

Ancient Greeks conjectured that there are infinitely many pairs of primes which differ by 2 (twin primes). A natural widely believed generalization is that 2 may be replaced by every even number. ...

2
votes

### On 12 cfracs: for Catalan's $K$, Gieseking's $\kappa$, and $\pi^2$, $\pi^3$, plus three for $\zeta(3)$ using Zagier's "six sporadic sequences"

Zagier answered your first question in his original paper. See the table in p. 11 here. He gives there evaluations of the continued fractions associated with his sequences A, C, D, E and F (note that ...

1
vote

### How to count to cusps of the modular curve $X_1(N)$, i.e., for the congruence subgroup $\Gamma_1(N)$

A good reference is Miyake, Modular forms (Springer, 1989), Section 4.2. There is an explicit bijection between the set of cusps of $X_1(N)(\mathbf{C})$ and the set
\begin{equation*}
\bigl\{(c,d) : c \...

2
votes

### Solutions to the Diophantine equation $a^xy+x=c$

As was observed by @JoshuaZ, for given $a, c\in\mathbb{N}$, the equation $(*)\; a^xy+x=c$ has only finitely many solutions. On the other hand, I show that for any $a$ and any $N\in\mathbb{N}$ one can ...

-1
votes

Accepted

### Does $n \mid \sigma(n^2)$, if $q^k n^2$ is an odd perfect number?

Let $p^k m^2$ be an odd perfect number with special prime $p$.
It follows that
$$\frac{\sigma(m^2)}{p^k}\cdot\frac{\sigma(p^k)}{2}=m^2.$$
Let $t_1 = \sigma(m^2)/p^k$, $t_2 = \sigma(p^k)/2$. It follows ...

4
votes

Accepted

### Automorphic classification of different types of abelian surfaces

Yes, knowing the endomorphism algebra of $A$ (conjecturally) translates to certain properties of an associated automorphic representation $\pi$.
First, you should look at the Galois type, which is ...

12
votes

Accepted

### On 12 cfracs: for Catalan's $K$, Gieseking's $\kappa$, and $\pi^2$, $\pi^3$, plus three for $\zeta(3)$ using Zagier's "six sporadic sequences"

We have $$C_2(-17,-6,-72)=-(5/8)L(\chi_{-3},2)$$ and
$$C_2(10,3,-9)=(1/2)L(\chi_{-3},2)$$ so both are proportional to what you
call Gieseking's constant but which is simply the value at 2 of the
L ...

3
votes

### Experiments with Voronoï summation

I think this is fine. Indeed, $S$ as a function of $B$ is of negligible size. This can also be checked as follows. Using Mellin transform and absolute convergence of the Dirichlet series of $\lambda$ ...

0
votes

### Computing explicit isogenies between elliptic curves over different kinds of fields

For 1.1 and 1.2: Vélu's formulae require the kernel of the isogeny, or the polynomial whose roots are the $x$-coordinates of the nontrivial kernel points. If you don't know the isogeny in advance, but ...

4
votes

Accepted

### Fibonacci and product polynomials

Question 2 follows from Theorem 6.1 of arXiv:2101.02131. (In this reference, I consider $\prod_{i=1}^n(1+x^{F_{i+1}})$ rather than $\prod_{i=1}^n(1+x^{F_i})$, but the proof still works.) The result ...

3
votes

### Jouanolou thesis on l-adic cohomology

You can see a project to make it available here
https://github.com/carmonamateo/Jouanolou
You can request a copy there.

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