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 ...
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6
votes
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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, ...
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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-...
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3
votes
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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 ...
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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 ...
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9
votes
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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 $y^2 = x^4 + 3x^2 + 3$ ...
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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}^{...
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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 \...
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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:
...
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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" ...
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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 ...
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8
votes
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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 ...
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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 ...
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7
votes
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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
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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 ...
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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 ...
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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. (...
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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 ...
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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 ...
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3
votes
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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 \...
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18
votes
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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
$$ A(7^u)^3 + 2 = y^2\quad\...
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1
vote
Accepted
Iterated exponential sums
This sum is equal to
$$
\frac{\left( \sum_{n \leq x} e(f(n)) \right)^2 - \sum_{n \leq x} e(2 f(n))}{2}
$$
In most situations, we do not expect there to be more than square root cancellation, and thus ...
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1
vote
On Zagier's missing continued fraction with multiple limits?
To complete the 12 cfracs in this post and the 4 in the next, all associated with 16 "sporadic sequences", then 13 of them have closed-forms, 1 has six limits (also with closed-forms but one ...
- 11.3k
4
votes
Sum of $\frac{1}{(\delta_1,\delta_2)}$ with congruence restrictions
Your sum is
\begin{align*}
L(Q)&=\sum _{d\leq Q}\frac {1}{d}\sum _{\delta _1,\delta _2\leq Q/d\atop {a|d\delta _1\atop {b|d\delta _2\atop {(\delta _1,\delta _2)=1}}}}1=\sum _{dh\leq Q}\frac {\mu (...
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0
votes
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On the continued fractions using Cooper's sequences $s_7,\, s_{10},\, s_{18}$ and the Zudilin-Cohen sequence
(This answers Question 2.)
Thanks to Cohen's 2022 paper, turns out there is a deg-$4$ and one can find polynomials $Q_k(n)$ for general deg-$k$ such that,
$$(n+1)^k s_{n+1} = Q_k (n)\, s_n - n^k s_{n-...
- 11.3k
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 ...
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7
votes
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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 $f \in \mathbb{Z}_p[[T]]$, ...
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71
votes
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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 ...
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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 ...
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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 $$\...
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15
votes
Does this conic have a rational point?
The answer is no. If there was, we could assume that $X,Y,Z$ are in $\mathbb Q[u,v]$ and are coprime (since that ring is a UFD). Setting $v=0$ we get $X(u,0)^2+uY(u,0)^2=0$ in $\mathbb Q[u]$, which (...
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5
votes
For a finite set A of positive reals, prove that the set A + A - A contains at least as many positive as negative elements
The following idea may work, the last step looks heuristic though.
Let $A=\{a_1, \ldots, a_n \}$ and let
$$C:= \{0 , \pm \alpha_1, \ldots \pm \alpha_k \}$$ with $\alpha_1, \ldots, \alpha_k >0$.
Let ...
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3
votes
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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 ...
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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 ...
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2
votes
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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 ...
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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$...
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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 ...
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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}...
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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 ...
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9
votes
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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. ...
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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 ...
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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 \...
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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 ...
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-1
votes
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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.$$
Let $t_1 = \sigma(m^2)/p^k$, $t_2 = \sigma(p^k)/2$. It follows ...
4
votes
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Automorphic classification of different types of abelian surfaces
Yes, knowing the endomorphism algebra of $A$ (conjecturally) translates to certain properties of an associated automorphic representation $\pi$.
First, you should look at the Galois type, which is ...
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12
votes
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On 12 cfracs: for Catalan's $K$, Gieseking's $\kappa$, and $\pi^2$, $\pi^3$, plus three for $\zeta(3)$ using Zagier's "six sporadic sequences"
We have $$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 ...
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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 $\lambda$ ...
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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 ...
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4
votes
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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 ...
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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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