All Questions
Tagged with nt.number-theory ag.algebraic-geometry
196 questions
0
votes
1
answer
746
views
A RKHS interpretation of the Rydberg formula for hydrogen and an application for physics?
I was thinking if it is possible to define an inner product between two small physical objects with a positive definite kernel and was led to look at the Rydberg formula:
The Rydberg formula for ...
- 3,064
150
votes
2
answers
22k
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 ...
- 13.6k
141
votes
0
answers
13k
views
Grothendieck-Teichmüller conjecture
(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmüller conjecture: the morphism
$$
G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})
$$
is an isomorphism.
Here $...
- 7,527
87
votes
12
answers
12k
views
Why do we make such big deal about the 'unsolvability' of the quintic?
The unsolvability of a general quintic equation in terms of the basic arithmetic operations and $n$th roots (i.e. the Abel–Ruffini theorem) is considered a major result in the mathematical canon. I ...
- 1,389
76
votes
2
answers
6k
views
Is it known that the ring of periods is not a field?
I have just learned here that we know numbers that are not periods; is it known meanwhile that the ring of periods is not a field? I know that it is conjectured that $1/\pi$ is not a period, but the ...
- 32.6k
70
votes
7
answers
28k
views
Have there been any updates on Mochizuki's proposed proof of the abc conjecture?
In August 2012, a proof of the abc conjecture was proposed by Shinichi Mochizuki. However, the proof was based on a "Inter-universal Teichmüller theory" which Mochizuki himself pioneered. It was known ...
64
votes
6
answers
52k
views
Consequences of Kirti Joshi's new preprint about p-adic Teichmüller theory on the validity of IUT and on the ABC conjecture
Today, somebody posted on the nLab a link to Kirti Joshi's preprint on the arXiv from last month: https://arxiv.org/abs/2210.11635
In that preprint, Kirti Joshi claims that
he agrees with Scholze and ...
- 2,624
40
votes
1
answer
10k
views
What actually is the idea behind the condensed mathematics?
Condensed mathematics is the (potential) unification of various mathematical subfields, including topology, geometry, and number theory. It asserts that analogs in the individual fields are instead ...
- 1,525
37
votes
2
answers
1k
views
varieties with points in number fields
Let $V$ be a projective variety, defined over $\mathbb{Q}$. Suppose that for every number field $K \neq \mathbb{Q}$, there is a $K$-point of $V$. Does it follow that $V$ has a $\mathbb{Q}$-point?
...
- 1,832
37
votes
7
answers
8k
views
Model theoretic applications to algebra and number theory(Iwasawa Theory)
One of my favorite results in algebraic geometry is a classical result of AX (see http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/) I'll recall ...
- 2,396
37
votes
3
answers
5k
views
Is there a nice proof of the fact that there are (p-1)/24 supersingular elliptic curves in characteristic p?
If $k$ is a characteristic $p$ field containing a subfield with $p^2$ elements (e.g., an algebraic closure of $\mathbb{F}_p$), then the number of isomorphism classes of supersingular elliptic curves ...
- 45.7k
34
votes
2
answers
3k
views
Shimura-Taniyama-Weil VS Grothendieck's dessins
When listening to the beautiful lectures by Gilles Schaeffer at
the SLC68, the following (perhaps crazy) question occurred to me:
did anyone attempt (succeed?) to combinatorially prove modularity of ...
34
votes
3
answers
3k
views
irreducibility of discriminant
This must be well-known to everyone but me, but here goes: take a general (monic) polynomial $p(x) = x^d + a_{d-1} x^{d-1} + \dotsc + a_0.$ The discriminant is a polynomial $D(a_0, \dotsc, a_{d-1}).$ ...
- 96.4k
33
votes
5
answers
8k
views
Why no abelian varieties over Z?
Motivation
I learned about this question from a wonderful article Rational points on curves by Henri Darmon. He gives a list of statements (some are theorems, some conjectures) of the form
the set $\{...
- 15.1k
32
votes
2
answers
2k
views
Etale cohomology can not be computed by Cech
It can be proven that if in a quasicompact scheme $X$ any finite subset is contained in an affine open subset then for any sheaf $\mathcal{F}$ on $X$ its Cech cohomology $\hat{H_{et}^{\bullet}}(X,\...
- 7,377
32
votes
1
answer
8k
views
$p$-adic Hodge Theory for rigid spaces, after P. Scholze
I was going over P. Scholze's paper on $p$-adic Hodge Theory for rigid analytic varieties.
This question is around the "Poincaré Lemma" in the paper.
Throughout, let $X$ be a proper smooth rigid ...
user113453
32
votes
1
answer
4k
views
How should a number theorist learn a modest amount of algebraic geometry?
A little bit vague, but I hope useful for the entire community.
I am, by training, an analytic number theorist. I have managed to learn some algebraic geometry, by reading parts of Silverman's ...
30
votes
4
answers
3k
views
Motivation for zeta function of an algebraic variety
If $p$ is a prime then the zeta function for an algebraic curve $V$ over $\mathbb{F}_p$ is defined to be
$$\zeta_{V,p}(s) := \exp\left(\sum_{m\geq 1} \frac{N_m}{m}(p^{-s})^m\right). $$
where $N_m$ is ...
- 901
29
votes
2
answers
4k
views
What is the algebraic closure of the field with one element?
If doing geometry over $\mathbb F_p$ means also using its algebraic closure, it must be interesting to talk about the algebraic closure of $\mathbb F_1$ - the field with one element.
I saw that the ...
- 4,593
28
votes
2
answers
5k
views
Status of (global) Langlands conjecture for $\mathrm{GL}_2$ over $\mathbb{Q}$
$\DeclareMathOperator\GL{GL}$Apologies if this question has already been dealt with on MO. I am wondering about the status of the global Langlands conjectures for $\GL_2$ over the rational numbers. ...
- 283
27
votes
2
answers
2k
views
Etale site is useful - examples of using the small fppf site?
Edit: After the answers and comments, I'm hoping for a little bit of elaboration (in the comment to the answer below.) Also, question 2 was discussed here:
Points in sites (etale, fppf, ... )
There, ...
- 3,555
27
votes
3
answers
4k
views
Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,...)
Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ...
On the other hand model theory, in particular after Hrushovski, found many ...
- 32.1k
26
votes
5
answers
3k
views
Existence of zero cycles of degree one vs existence of rational points
Let $k$ be a field (I'm mainly interested in the case where $k$ is a number field, however results for other fields would be interesting), and $X$ a smooth projective variety over $k$.
By a zero ...
- 21.3k
25
votes
1
answer
1k
views
How is Borger's approach to $\mathbb{F_{1}}$ related to previous approaches (e.g. Deitmar's)?
The "traditional" approach to the so-called "field with one element" $\mathbb{F}_{1}$ is by using monoids, or, to put it in another way, by forgetting the additive structure of ...
- 3,309
24
votes
3
answers
3k
views
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in\mathbb Z$ be unbounded as $n$ varies?
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in\mathbb Z$ be unbounded as $n$ varies?
$x,y$ are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of $n$ (...
- 454
23
votes
1
answer
2k
views
Does smooth and proper over $\mathbb Z$ imply rational?
Does smooth and proper over $\mathbb Z$ imply rational?
I think someone told me that this is a standard conjecture. Is it a widely held? held at all? Did someone in particular make this conjecture? ...
- 8,717
22
votes
1
answer
2k
views
Which elliptic curves over totally real fields are modular these days?
As the title says. In particular, every elliptic curve over $\mathbb{Q}$ is modular; but what is the current state of the art for general totally real number fields? I assume the answer is ...
- 13.1k
20
votes
5
answers
2k
views
Rational solutions of the Fermat equation $X^n+Y^n+Z^n=1$
As a generalisation to the equation of Fermat, one can ask for rational solutions of $X^n+Y^n+Z^n=1$ (or almost equivalently integer solutions of $X^n+Y^n+Z^n=T^n$).
Contrary to the case of Fermat, ...
- 7,722
20
votes
2
answers
4k
views
"Fermat's last theorem" and anabelian geometry?
Do I remember a remark in "Sketch of a program" or "Letter to Faltings" correctly, that acc. to Grothendieck anabelian geometry should not only enable finiteness proofs, but a proof of FLT too? If yes,...
- 10.8k
20
votes
5
answers
4k
views
Equivalent statements of the Riemann hypothesis in the Weil conjectures
In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with $q$ elements says that: the eigenvalues of ...
- 671
20
votes
1
answer
3k
views
Crystalline cohomology via the syntomic site
Hello,
Let $k$ be a field of characteristic $p > 0$, and let $Y$ be a $k$-scheme. Consider the
sites $Y_{syn}$ and $(Y/W_n)_{cris}$ (where $W_n$ are the Witt vectors of $k$ of length $n$), of $Y$ ...
- 2,842
20
votes
1
answer
2k
views
De Rham cohomology of formal groups
Let $G$ be some (dimension $1$, to simplify) formal group over a characteristic $0$ field $K$. The law of $G$ is denoted by $\oplus$. If $w(X) \in K[[X]] dX$ is a differential form, let $F_w(X)$ be ...
- 6,924
19
votes
1
answer
1k
views
Deligne's letter to Bhargava from March 2004
I am quite interested in moduli spaces for Rings and Ideals, a letter from Deligne to Bhargava is cited in Melanie Wood's thesis Moduli spaces for Rings and Ideals (pdf), studying the minimal free ...
- 461
19
votes
0
answers
4k
views
On 'A proof of ABC conjecture after Mochizuki' [closed]
It has been pointed out to me that Go Yamashita has a preprint on his website, A proof of ABC conjecture after Mochizuki, that is not in the RIMS Preprints online archive.
Is that work intended for ...
user75426
19
votes
1
answer
2k
views
What do formal group laws of height $\geq 3$ look like?
By the classification of formal groups in characteristic $p$, we know that isomorphism classes of connected smooth $1$-dimensional formal groups, equivalently group scheme structures on $\operatorname{...
- 148k
19
votes
1
answer
1k
views
Ehresmann's theorem over the $p$-adics
I am looking for a version of Ehresmann's theorem for analytic manifolds over the $p$-adic numbers $\mathbb{Q}_p$ or, more generally, local fields. I follow the conventions from Serre's book "Lie ...
- 21.3k
18
votes
1
answer
6k
views
Deligne's proof of Ramanujan's conjecture
I am trying to understand Deligne's proof of the Ramanujan conjecture and more generally how one associates geometric objects (ultimately, motives) to modular forms.
As the first step, which I ...
- 1,783
17
votes
1
answer
3k
views
Why is the section conjecture important?
As in the title, I want to know the reason for importance of the section conjecture. Of course, the statement of conjecture is important as itself, even I cannot fully grasp the soul of it. However, ...
- 601
17
votes
1
answer
2k
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Is there an analog of the Birch/Swinnerton-Dyer conjecture for abelian varieties in higher dimensions?
I am wondering if there is a multi-dimensional analog of the
Birch/Swinnerton-Dyer (BSD) conjecture.
The recent famous result inching toward resolution of that conjecture is:
Bhargava, Manjul, and ...
- 151k
17
votes
2
answers
1k
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 $...
- 11.3k
16
votes
2
answers
4k
views
Elliptic Curves over Rings?
So an elliptic curve $E$ over a field $K$ is a smooth projective nonsingular curve of genus $1$ together with a point $O \in E$.
I was reading Silverman's "Arithmetic of Elliptic Curves" and it ...
- 1,458
16
votes
2
answers
2k
views
Which languages could appear on Weil's Rosetta Stone?
André Weil's likening his research to the quest to decipher the Rosetta Stone (see this letter to his sister) continues to inspire contemporary mathematicians, such as Edward Frenkel in Gauge Theory ...
- 5,094
16
votes
3
answers
1k
views
Number of solutions to polynomial congruences
Suppose I have $R$ homogeneous polynomials $F_1, ..., F_R$ with integer coefficients. Let $V$ be the affine variety defined by these polynomials over $\mathbb{C}$. I was wondering if some bound that ...
- 3,625
16
votes
0
answers
546
views
What can be the dimension of a pointless smooth proper Z-scheme?
What is the smallest dimension $d$ such that there is a smooth proper morphism $X \to \operatorname{Spec} \mathbb Z$ of relative dimension $d$, with $X$ nonempty, without a section?
Of course, there ...
- 148k
16
votes
4
answers
1k
views
Geometric meaning of fiber of modular parameterization over a point of an elliptic curve?
Given an elliptic curve $E/\mathbb{Q}$ of conductor $N$, parameterization $\psi : X_0(N) \rightarrow E$, and a point $P \in E$, take the fiber $\psi^{-1}(P)$. Its points, being on $X_0(N)$, correspond ...
- 4,593
16
votes
1
answer
1k
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) $E_1[9] \cong E_2[9]$ ...
- 1,209
15
votes
2
answers
2k
views
sum of three cubes and parametric solutions
The first paragraph in the following link asserts that the equation $x^3+y^3+z^3=2$ has finite many parametric solutions over $\mathbb{Q}$, i.e., there are finite many polynomial triples $(x(t),y(t),z(...
- 3,337
15
votes
0
answers
591
views
For how many primes does an elliptic curve over a totally imaginary field have supersingular reduction?
An elliptic curve over a finite field, $k$, of characteristic p is called supersingular if it has no $p$-torsion over $k^{\mathrm{alg}}$, or equivalently, if $\mathrm{End}(E)$ is an order in a ...
- 605
15
votes
0
answers
2k
views
Inter-Universal Teichmuller Theory and the Field with One Element
The idea of the "field with one element", or $\mathbb{F}_{1}$, is supposed to allow us to do for number fields what we can do for function fields. Hence this idea often comes up regarding problems ...
- 3,309
14
votes
1
answer
1k
views
Geometry for Anderson's motives?
Anderson's $t$-motives satisfy most of what is expected of a reasonable category of mixed motives, except of course that everything is in positive characteristic. For instance, it is a linear category ...
- 4,015