Questions tagged [arithmetic-geometry]

Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.

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Rational points on a sphere in $\mathbb{R}^d$

Call a point of $\mathbb{R}^d$ rational if all its $d$ coordinates are rational numbers. Q1. Are the rational points dense on the unit sphere $S :\; x_1^2 +\cdots+ x_d^2 = 1$, i.e. does $S$ ...
Joseph O'Rourke's user avatar
145 votes
4 answers
66k views

What are "perfectoid spaces"?

This talk is about a theory of "perfectoid spaces", which "compares objects in characteristic p with objects in characteristic 0". What are those spaces, where can one read about them? Edit: A bit ...
Thomas Riepe's user avatar
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86 votes
9 answers
13k views

Why should I believe the Mordell Conjecture?

It was Faltings who first proved in 1983 the Mordell conjecture, that a curve of genus 2 or more over a number field has only finitely many rational points. I am interested to know why Mordell and ...
Barinder Banwait's user avatar
49 votes
8 answers
25k views

Roadmap for studying arithmetic geometry

I have read Hartshorne's Algebraic Geometry from chapter 1 to chapter 4, so I'd like to find some suggestions about the next step to study arithmetic geometry. I want to know how to use scheme ...
26 votes
2 answers
2k views

Are most curves over Q pointless?

Fresh out of the arXiv press is the remarkable result of Manjul Bhargava saying that most hyperelliptic curves over $\mathbf{Q}$ have no rational points. Don Zagier suggests the paraphrase : Most ...
Chandan Singh Dalawat's user avatar
16 votes
3 answers
1k views

Is Multilinear Hilbert's tenth problem version undecidable?

A multilinear polynomial $f\in\mathbb Z[x_1,\dots,x_t]$ has terms only of form $$b\prod_{i=1}^tx_i^{a_i}$$ where $a_i\in\{0,1\}$ and $b\in\mathbb Z$. Is there no general purpose algorithm for ...
Turbo's user avatar
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16 votes
2 answers
2k views

Good introductory references on moduli (stacks), for arithmetic objects

I've studied some fundation of algebraic geometry, such as Hartshorne's "Algebraic Geometry", Liu's "Algebraic Geometry and Arithmetic Curves", Silverman's "The Arithmetic of Elliptic Curves", and ...
k.j.'s user avatar
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42 votes
1 answer
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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 ...
Thomas Riepe's user avatar
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41 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 ...
Y. Zhao's user avatar
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20 votes
3 answers
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what is the maximum number of rational points of a curve of genus 2 over the rationals

Conjecturally, there exists an integer $n$ such that the number of rational points of a genus $2$ curve over $\mathbf{Q}$ is at most $n$. (This follows from the Bombieri-Lang conjecture.) We are ...
Dirk's user avatar
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14 votes
3 answers
3k views

Existence of fine moduli space for curves and elliptic curves

For the moduli problem of a curve of genus $g$ with $n$ marked points, how large an $n$ is needed to ensure the existence of a fine moduli space? For this question, terminology is that of Mumford's ...
Anweshi's user avatar
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9 votes
1 answer
1k views

Nonabelian $H^2$ and Galois descent

I would like to know whether the following metatheorem on nonabelian $H^2$ has been ever stated and/or proved. Let $k$ be a perfect field and $k^s$ its fixed separable closure. Let $X^s$ be a variety ...
Mikhail Borovoi's user avatar
54 votes
8 answers
8k views

Questions about analogy between Spec Z and 3-manifolds

I'm not sure if the questions make sense: Conc. primes as knots and Spec Z as 3-manifold - fits that to the Poincare conjecture? Topologists view 3-manifolds as Kirby-equivalence classes of framed ...
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 $\...
Pablo's user avatar
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41 votes
2 answers
9k views

Intuition behind the Eichler-Shimura relation?

The modular curve $X_0(N)$ has good reduction at all primes $p$ not dividing $N$. At such a prime, the Eichler-Shimura relation expresses the Hecke operator $T_p$ (as an element of the ring of ...
Qiaochu Yuan's user avatar
35 votes
2 answers
2k views

Durov approach to Arakelov geometry and $\mathbb{F}_1$

Durov's thesis on algebraic geometry over generalized rings looks extremely intriguing: it promises to unify scheme based and Arakelov geometry, even in singular cases, as well as including geometry ...
Andrea Ferretti's user avatar
32 votes
4 answers
4k views

Over which fields does the Mordell-Weil theorem hold?

According to a well-known theorem of Mordell, the group of rational points $E(\mathbf{Q})$ of an elliptic curve $E/\mathbf{Q}$ is finitely generated. Weil generalized this theorem to abelian varieties ...
François Brunault's user avatar
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 ...
François Brunault's user avatar
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 ...
Gazerun's user avatar
  • 463
24 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 ...
Bombyx mori's user avatar
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24 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?
user avatar
22 votes
2 answers
2k views

unboundedness of number of integral points on elliptic curves?

If $E/\mathbf{Q}$ is an elliptic curve and we put it into minimal Weierstrass form, we can count how many integral points it has. A theorem of Siegel tells us that this number $n(E)$ is finite, and ...
Kevin Buzzard's user avatar
18 votes
7 answers
3k views

SAT and Arithmetic Geometry

This is an agglomeration of several questions, linked by a single observation: SAT is equivalent to determining the existence of roots for a system of polynomial equations over $\mathbb{F}_2$ (note ...
Vanessa's user avatar
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16 votes
2 answers
2k views

Period rings for Galois representations

I have some questions concerning period rings for Galois representations. First, consider the case of $p$-adic representations of the absolute Galois group $G_K$, where $K$ denote a $p$-adic field. ...
A M's user avatar
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16 votes
0 answers
485 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^...
user114562's user avatar
13 votes
3 answers
1k views

Is the map on étale fundamental groups of a quasi-projective variety, upon base change between algebraically closed fields, an isomorphism?

$\DeclareMathOperator\Spec{Spec}$Let $k \subset L$ be two algebraically closed fields of characteristic $0$. Let $U \subset \mathbb P^n_k$ be a smooth quasi-projective variety and let $U_L$ denote the ...
Aaron Landesman's user avatar
13 votes
2 answers
907 views

Belyi's theorem for function fields

Belyi's theorem states that every smooth projective algebraic curve $C$ defined over $\bar{\mathbb{Q}}$ admits a map $C\to\mathbb{P}^1$ ramified only over $0,1,\infty$. Is there an analogue of this ...
Alex's user avatar
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11 votes
1 answer
1k views

What geometric properties do properties of ell-adic Galois representations imply?

This is the converse question to an earlier question. More precisely, Let $X/K$ be a curve(or variety) over a global field $K$. We consider the Galois representation obtained by the absolute Galois ...
Regenbogen's user avatar
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9 votes
3 answers
3k views

Some arithmetic terminology: "universal domain", "specialization", "Chow point"

As a non-connoisseur of arithmetic and arithmetic geometry, I would like to ask about some terminology, which meaning I haven't been able to find out on some books, nor on wikipedia, nor by google. ...
Qfwfq's user avatar
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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 $0$ [...
user avatar
8 votes
1 answer
302 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(\...
Mikhail Borovoi's user avatar
6 votes
1 answer
395 views

Number of points of parabolic Springer fibres

Let $P$ be a parabolic subgroup of $\mathrm{GL}_n$ and $u\in P$ a unipotent element. The parabolic Springer fibre associated to $(P,u)$ can be defined by $$ \mathcal{P}_u:=\{gP\in G/P \mathrel\vert g^{...
Dr. Evil's user avatar
  • 2,641
4 votes
1 answer
242 views

Does the modular form associated to cubic twist of a elliptic curve $E$ corresponds to some twist of $f_E$?

Let $E$ be an elliptic curve defined over $\Bbb Q$ and $f_E$ be the modular form associated with the elliptic curve $E$. Suppose the elliptic curve $E^D$ is a quadratic twist of $E$. I understand that ...
SUNIL PASUPULATI's user avatar
2 votes
2 answers
631 views

transcendence of canonical heights

Are there known examples of rational points on elliptic curves/abelian varieties over number fields with transcendental canonical height? Thanks.
SGP's user avatar
  • 3,847
2 votes
0 answers
220 views

On a class of loci in Chow varieties

Let $k$ be a field, $i:X\hookrightarrow \mathbf{P}(\mathscr{E})$ be a fixed projective embedding of a smooth projective $k$-variety $X$, whose dimension is pure and equals $d\ge 0$. For $0\le p\le d$,...
user avatar
148 votes
2 answers
21k 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 ...
Minhyong Kim's user avatar
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75 votes
1 answer
14k views

What is an étale theta function?

Let me start out by urging you to take seriously that whatever I write about the papers surrounding IUTT really are questions. If you would like to use it as a guide to the mathematics in any way, ...
Minhyong Kim's user avatar
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70 votes
7 answers
27k 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 ...
60 votes
1 answer
6k views

What were the main ideas and gaps in Yoichi Miyaoka's attempted proof (1988) of Fermat's Last Theorem?

Out of sheer curiosity I have been reading Stewert and Tall's "Algebraic Number Theory and Fermat's Last Theorem" (2001). As it contains various bits of history, I found out to my own shame that I was ...
M.G.'s user avatar
  • 6,683
50 votes
6 answers
6k views

Intuition for the last step in Serre's proof of the three-squares theorem

Serre's A Course in Arithmetic gives essentially the following proof of the three-squares theorem, which says that an integer $a$ is the sum of three squares if and only if it is not of the form $4^m (...
Qiaochu Yuan's user avatar
45 votes
3 answers
5k views

"Cute" applications of the étale fundamental group

When I was an undergrad student, the first application that was given to me of the construction of the fundamental group was the non-retraction lemma : there is no continuous map from the disk to the ...
Libli's user avatar
  • 7,210
44 votes
2 answers
3k views

Langlands in dimension 2: the Yoshida conjecture

Background: One prominent part of the Langlands program is the conjecture that all motives are automorphic. It is of interest to consider special cases that are more precise, if less sweeping. ...
Laie's user avatar
  • 1,694
40 votes
4 answers
3k views

Why are Green functions involved in intersection theory?

I've been learning Arakelov geometry on surfaces for a while. Formally I've understood how things work, but I'm still missing a big picture. Summary: Let $X$ be an arithmetic surface over $\...
Dubious's user avatar
  • 1,237
39 votes
1 answer
5k views

Clausen's modified Hodge Conjecture

In a recent talk at the University of Geneve, Dustin Clausen presented a "modified Hodge Conjecture". I found the abstract intriguing but couldn't find videos or notes available online. If I'...
user avatar
33 votes
1 answer
1k views

Is the group of integer points on a finite-type group scheme over Z finitely presented?

Let $G$ be a group scheme of finite type over $\mathbf{Z}$. Must $G(\mathbf{Z})$ be finitely presented? (The question is inspired by a not yet successful attempt to answer a question of Brian Conrad....
Bjorn Poonen's user avatar
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33 votes
3 answers
3k views

Arithmetic geometry examples

(This is inspired by Algebraic geometry examples.) I want to collect here (counter)examples in arithmetic geometry. Curves violating the Hasse principle: The Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$. ...
32 votes
1 answer
7k 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 ...
user avatar
28 votes
3 answers
8k views

Learning path for the proof of the Weil Conjectures

Assume you are an algebraic geometry advanced student who has mastered Hartshorne's book supplemented on the arithmetic side by the introduction of Lorenzini - "An Invitation to Arithmetic Geometry" ...
Javier Álvarez's user avatar
24 votes
0 answers
800 views

Smooth proper schemes over Z with points everywhere locally

This is a variation on Poonen's question, taking Buzzard's fabulous example into account. It was earlier a part of this other question. Question. Is there a smooth proper scheme $X\to\operatorname{...
Chandan Singh Dalawat's user avatar
22 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? ...
Ben Wieland's user avatar
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