All Questions
Tagged with nt.number-theory ag.algebraic-geometry
196 questions
296
votes
8
answers
143k
views
Philosophy behind Mochizuki's work on the ABC conjecture
Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy ...
43
votes
1
answer
4k
views
A mysterious connection between Ramanujan-type formulas for $1/\pi^k$ and hypergeometric motives
The question below is the follow-up of this question on MathOverflow.
Motivation: As is stated in the former question, those identities(formula (35)-(44)) of $1/\pi$ attributed to Ramanujan are ...
- 3,337
22
votes
6
answers
8k
views
A finitely generated $\mathbb{Z}$-algebra that is a field has to be finite
I was trying to understand completely the post of Terrence Tao on Ax-Grothendieck theorem. This is very cute. Using finite fields you prove that every injective polynomial map $\mathbb C^n\to \mathbb ...
- 14.3k
14
votes
3
answers
3k
views
Does isomorphic generic fibre imply isomorphic special fibre for smooth morphisms?
Let $X$ and $Y$ be regular integral Noetherian schemes. Assume that $X$ and $Y$ are smooth and proper over a base scheme $S=Spec R$, where $R$ is a discrete valuation ring.
If $X$ and $Y$ have ...
- 21.3k
101
votes
2
answers
11k
views
Riemann hypothesis via absolute geometry
Several leading mathematicians (e.g. Yuri Manin) have written or said publicly that there is a known outline of a likely natural proof of the Riemann hypothesis using absolute algebraic geometry over ...
- 5,232
43
votes
1
answer
19k
views
What is inter-universal geometry?
I wonder what Mochizuki's inter-universal geometry and his generalisation of anabelian geometry is, e.g. why the ABC-conjecture involves nested inclusions of sets as hinted in the slides, or why such ...
- 10.8k
42
votes
3
answers
5k
views
The Origin(s) of Modular and Moduli
In mathematics and in physics, people use the terms "modular..." and "moduli space" very often. I was puzzled by the etymology, the origins and the similarity/equivalence/differences for these usages/...
- 10.5k
26
votes
7
answers
6k
views
When is a product of elliptic curves isogenous to the Jacobian of a hyperelliptic curve?
David's question Families of genus 2 curves with positive rank jacobians reminded me of a question that once very much interested me: when is a product of elliptic curves isogenous to the jacobian of ...
- 4,593
24
votes
1
answer
3k
views
What is the precise relationship between Langlands and Tannakian formalism?
As anyone who's been reading the forums closely can see, I've been averaging a question a day about Tannakian formalism for the past few days. It's quite an interesting concept!
In any case, I wish ...
- 6,348
18
votes
1
answer
2k
views
Conjecture: The number of points modulo $p$ of certain elliptic curve is $p$ or $p+2$ for $p$ of form $p=27a^2+27a+7$
Numerical evidence suggests a conjecture that the number of
points of certain elliptic curve over $\mathbb{F}_p$ is
either $p$ or $p+2$ for $p$ of certain form.
Let $p$ be prime of the form $p=27a^2+...
- 25.4k
6
votes
1
answer
438
views
$y^3 = x^4 + x + 2$, and existence of rational points on rank 0 Picard curves
Do there exists rational numbers $x$ and $y$ such that
$$
y^3 = x^4 + x + 2 ?
$$
Context: There are a lot of publications about computing rational points on elliptic and hyperelliptic curves, and ...
- 7,182
6
votes
1
answer
472
views
Existence of elements in an Eichler order whose norm is minus one
Let $B$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $D$, and $\mathcal{O}_N$ be an Eichler order of level $N$. Is there an element $x\in \mathcal{O}_N$ such that its reduced ...
- 570
63
votes
1
answer
7k
views
Smooth proper scheme over Z
Does every smooth proper morphism $X \to \operatorname{Spec} \mathbf{Z}$ with $X$ nonempty have a section?
EDIT [Bjorn gave additional information in a comment below, which I am recopying here. -- ...
- 23.8k
55
votes
3
answers
5k
views
What are the higher homotopy groups of Spec Z ?
The homotopy groups of the étale topos of a scheme were defined by Artin and Mazur. Are these known for Spec Z? Certainly π1 is trivial because Spec Z has no unramified étale covers,...
- 8,004
55
votes
3
answers
11k
views
What is precisely still missing in Connes' approach to RH?
I have read Connes' survey article http://www.alainconnes.org/docs/rhfinal.pdf
and I am somewhat familiar with his classic paper on the trace formula: http://www.alainconnes.org/docs/selecta.ps
Very ...
- 1,338
48
votes
4
answers
4k
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 $\...
- 11.3k
48
votes
6
answers
5k
views
Smooth linear algebraic groups over the dual numbers
It is a standard and important fact that any smooth affine group scheme $G$ over a field $k$ is a closed $k$-subgroup of ${\rm{GL}}_n$ for some $n > 0$. (Smoothness can be relaxed to finite type, ...
36
votes
1
answer
4k
views
Special values of L-functions as periods
If $M$ is a pure motive over $\mathbb{Q}$, one cas define its $L$-function $L(M,s)$ which conjecturaly is a meromorphic function over $\mathbb{C}$ with finitely many poles.
For example, when $M=\...
- 26k
35
votes
4
answers
8k
views
What would a "moral" proof of the Weil Conjectures require?
At the very end of this 2006 interview (rm), Kontsevich says
"...many great theorems are originally proven but I think the proofs are not, kind of, "morally right." There should be better proofs......
- 1,764
33
votes
1
answer
1k
views
Coefficients of Weil Cohomology Theories
A Weil Cohomology theory is a functor from the category of smooth projective varieties (over some fixed field $k$) to graded $K$-algebras (for some fixed field $K$) satisfying various axioms. For ...
- 331
32
votes
9
answers
5k
views
Do there exist modern expositions of Klein's Icosahedron?
Reading Serre's letter to Gray
, I wonder if now modern expositions of the themes in Klein's book
exist. Do you know any?
- 10.8k
31
votes
2
answers
1k
views
The Sylvester-Gallai theorem over $p$-adic fields
The famous Sylvester-Gallai theorem states that for any finite set $X$ of points in the plane $\mathbf{R}^2$, not all on a line, there is a line passing through exactly two points of $X$.
What ...
- 20.8k
29
votes
3
answers
2k
views
$\zeta(n)$ as a mixed Tate motive
I am trying to understand why there exists, for each $n \geq 2$, a mixed Tate motive $M$ over $\mathbb{Q}$ such that
$M \in Ext^1_{MT(\mathbb{Q})}(\mathbb{Q}(0), \mathbb{Q}(n))$
and $\zeta(n)$, ...
- 291
27
votes
3
answers
3k
views
Where's the best place for an algebraic geometer to learn some algebraic number theory?
There are lots of introductions to number theory out there, but typically they are streamlined to assume as little prerequisite knowledge as possible. I'm looking for a text which does the opposite -- ...
- 63.9k
26
votes
3
answers
3k
views
Crux of Dwork's proof of rationality of the zeta function?
As the question title suggests, what is the crux of Dwork's proof of the rationality of the zeta function? What is the intuition behind the proof, what are the key steps that the proof boils down to?
user97565
26
votes
4
answers
1k
views
Variety acquiring rational point over any quadratic extension
Does there exist a variety $X$ over $\mathbb{Q}$ (or a number field) such that it has no rational points over $\mathbb{Q}$ but acquires points over any quadratic extension $\mathbb{Q}(\sqrt{d})$?
If ...
- 463
25
votes
2
answers
1k
views
Why it is difficult to define cohomology groups in Arakelov theory?
I have been puzzled by the following Faltings' remark in his paper Calculus on arithemetic surfaces (page 394) for a few months. He says:
If $D$ is a divisor on $X$, we would like to define a ...
- 6,249
22
votes
2
answers
8k
views
Geometric vs Arithmetic Frobenius
If an algebraic variety $X$ over a field characteristic p is given by equations $f_i(x_1,...,x_k) = 0$, we can consider the variety $X^{(p)}$ obtained by applying p-th powers to all the coefficients ...
- 1,783
20
votes
2
answers
2k
views
Integral points on varieties
I recently came across an interesting phenomenon which confused me slightly, concerning integral points on varieties.
For example, consider $X = \mathbb{A}_{\mathbb{Z}}^{n+1} \setminus \{0\}$, affine ...
- 21.3k
19
votes
3
answers
2k
views
Elkies' supersingularity theorem in higher dimension
The following is a theorem of Elkies:
Let $X$ be an elliptic curve over $\mathbb{Q}$. Then there are infinitely many primes $p$ such that the action of Frobenius on $H^1(\mathcal{O}, X)$ is zero.
...
- 156k
19
votes
2
answers
5k
views
New Geometric Methods in Number Theory and Automorphic Forms
The MSRI is organising a programme with the above title from Aug 11, 2014 to Dec 12, 2014. Here is a short description from their website :
The branches of number theory most
directly related to ...
- 18.7k
17
votes
0
answers
1k
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 ...
- 148k
16
votes
0
answers
878
views
L-Functions of Varieties, Zeta Functions of Their Models
Let $k$ denote a number field, with algebraic closure $\bar{k}$. Take a smooth, projective variety $X$ over $k$. If $\mathfrak{p}$ is a prime of $k$, and $l$ is a rational prime different to the ...
- 643
16
votes
0
answers
532
views
Are there smooth and proper schemes over $\mathbb Z$ whose cohomology is not of Tate type
Is there an example of smooth and proper scheme $X \to \mathrm{Spec}(\mathbb Z)$, and an integer $i$ such that $H^i(X, \mathbb Q)$ is not a Hodge structure of Tate type?
Alternatively: such that $H^...
- 171
14
votes
1
answer
2k
views
Some questions about the ring Z((x))
$\newcommand{\ZZ}{\mathbb{Z}}$
$\newcommand{\dim}{\text{dim }}$
Let me begin by apologizing for the length of this question, but I thought this might be interesting to some of you. This ring isn't ...
- 10.7k
13
votes
3
answers
2k
views
Estimates for Bezout coefficients
The answer to my question is probably well-known, but I was unable to find a reference.
The Bezout's identity states that for any positive non-zero integers $a_1, \ldots , a_n$ there exist integers $...
- 2,648
12
votes
2
answers
1k
views
An integrality question about expressing an integer as a product of numbers below $n$
Let $n\ge 2$ be a natural number. Suppose that $N$ is a natural number, composed only of primes below $n$, and that can be expressed as
$$
N= \prod_{j=1}^{n} j^{x_j}
$$
where $x_1$, $\ldots$, $x_n$...
- 43.7k
12
votes
1
answer
642
views
are the congruence subgroups $\Gamma(n)$ characteristic inside $\mathrm{SL}_2(\mathbb{Z})$?
For which $n$ is the "principal congruence subgroup" $\Gamma(n)\le \mathrm{SL}_2(\mathbb{Z})$, the subgroup consisting of matrices congruent to the identity modulo $n$, characteristic? I.e., for ...
- 7,527
11
votes
2
answers
679
views
Z/8Z elliptic curve with a missing generator
We are searching for the rank $6$ elliptic curves with the torsion subgroup $\mathbb{Z}/8\mathbb{Z}$ using the families similar to Allan MacLeod's as described in
A. J. MacLeod, A Simple Method for ...
- 913
10
votes
1
answer
2k
views
How does the conjectural Langlands group fit into the Tannakian point of view?
I've read that one way to formulate the Langlands program is the following:
Let $\mathcal{L}_ {\mathbb{Q}}$ be the conjectural Langlands group. Then the category of semi-simple (continuous) ...
9
votes
4
answers
3k
views
reduction of CM elliptic curves
Can someone indicate how to prove the following equivalences for a CM elliptic curve $E$:
(i) $p$ is inert in End($E$)
(ii) $E_p$ is supersingular
(iii) The trace of the Frobenius at $p$ is $0$ [...
user19475
9
votes
1
answer
1k
views
Modern Proof of the Theorem of the Base
I am looking for a modern proof of the so-called "Theorem of the Base"--that the Neron-Severi rank of a smooth projective variety is finite. One can prove this for varieties over $\mathbb{C}$ easily ...
- 23k
9
votes
2
answers
2k
views
Is an elementary symmetric polynomial an irreducible element in the polynomial ring?
Let $S=\mathbb{C}[x_1,x_2,\dots,x_n]$ be a polynomial ring. Let $e_a$ denotes the elementary symmetric polynomials of degree $a$ in $S$.
For $n=2$:
$e_1=x_1+x_2$;
$e_2=x_1x_2$.
For $n=3$:
$e_1=x_1+...
- 446
9
votes
3
answers
2k
views
Supersingular elliptic curves and their "functorial" structure over F_p^2
In a letter to Tate from 1987, Serre describes a beautiful Theorem relating mod p modular forms to quaternions ("Two letters on quaternions and modular forms (mod p)", Israel J. Math. 95 (1996), 281--...
- 2,406
8
votes
1
answer
308
views
Algebraic points of uniformly bounded degree on an algebraic variety
Let $k$ be a perfect field, and let $\bar k$ be a fixed algebraic closure of $k$.
Let $\overline{X}$ be a nonempty smooth algebraic variety over $\bar k$.
Does there exist a natural number $d=d(\...
- 14.1k
5
votes
1
answer
1k
views
Langlands conjectures in higher dimensions
Geometric class field theory (curves over a finite field) has been generalized
to higher dimensional varieties over a finite field (and other arithmetical fields). Some of the key names here are Lang, ...
- 3,867
2
votes
0
answers
96
views
Cryptography signature scheme based on hardness of finding points on varieties?
Related to this question Complexity of finding solutions of trapdoored polynomial.
I am trying to build signature scheme based on hardness
of finding points on varieties.
Let $K$ be field and $M=K[x_1,...
- 25.4k
2
votes
0
answers
87
views
Complexity of finding solutions of trapdoored polynomial?
Related to this question Cryptography signature scheme based on hardness of finding points on varieties.
Working over $K=\mathbb{Q}[x_1,...,x_n,y_1,...y_m]$.
By abuse of notation, for polynomial $f$, ...
- 25.4k
2
votes
1
answer
184
views
Centralizers of Cartan subgroups
Let $E$ be an elliptic curve with CM by an order $\mathcal O$ in an imaginary quadratic field $K$. Choose a basis for $E[N]$ to get an isomorphism $\operatorname{Aut}(E[N])\cong \operatorname{GL}_2(\...
- 2,375
0
votes
1
answer
744
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