# Questions tagged [arithmetic-geometry]

Diophantine equations, elliptic curves, Mordell conjecture, Arakelov theory, Iwasawa theory, Mochizuki theory.

**19**

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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 $S$ ...

**76**

votes

**9**answers

10k 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 ...

**25**

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 ...

**16**

votes

**3**answers

1k views

### 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 ...

**108**

votes

**4**answers

42k 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 ...

**36**

votes

**8**answers

15k 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 ...

**39**

votes

**1**answer

16k 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 ...

**37**

votes

**1**answer

3k 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 ...

**11**

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 ...

**10**

votes

**2**answers

616 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 ...

**52**

votes

**8**answers

6k 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 ...

**28**

votes

**2**answers

6k 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 ...

**21**

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 ...

**16**

votes

**6**answers

2k 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 ...

**9**

votes

**3**answers

2k 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.
...

**5**

votes

**4**answers

2k 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 $0$ [...

**23**

votes

**3**answers

2k 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?

**22**

votes

**2**answers

943 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 ...

**2**

votes

**2**answers

452 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.

**8**

votes

**1**answer

248 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(\...

**2**

votes

**0**answers

159 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 $0\le p\le d$,...

**127**

votes

**1**answer

14k 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 ...

**71**

votes

**1**answer

11k 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, ...

**44**

votes

**6**answers

5k 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 (...

**20**

votes

**3**answers

4k 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" ...

**53**

votes

**5**answers

16k 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 ...

**12**

votes

**1**answer

3k views

### Kapranov's analogies

I just wonder about Kapranov's "Analogies between Langlands Correspondence and topological QFT". I would like to read a more detailed exposition and how one turns that analogy into concrete ...

**17**

votes

**1**answer

3k views

### Crystalline cohomology of abelian varieties

I am trying to learn a little bit about crystalline cohomology (I am interested in applications to ordinariness). Whenever I try to read anything about it, I quickly encounter divided power ...

**19**

votes

**2**answers

3k 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,...

**40**

votes

**2**answers

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### 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. ...

**14**

votes

**3**answers

1k views

### Integer points (very naive question)

Well, I don't have any notion of arithmetic geometry, but I would like to understand what arithmetic geometers mean when they say "integer point of a variety/scheme $X$" (like e.g. in "integer points ...

**42**

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**

votes

**9**answers

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### Proofs of Mordell-Weil theorem

I would like to ask if there exist pedagogical expositions of the Mordell-Weil theorem (wikipedia). What parts of number theory (algebraic geometry) one should better learn first before starting to ...

**36**

votes

**4**answers

2k 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 $\...

**31**

votes

**1**answer

3k 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 ...

**19**

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 ...

**12**

votes

**3**answers

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### What is the intuition behind the definition of cuspidal representations?

Let $\mathbb{G}$ be a reductive group defined over a number field $K$, let $Z$ be its center, and let $\mathbb{A}:=\mathbb{A}_K$ be the ring of adeles of $K$. Reasonably, we care about the $\mathbb{G}(...

**12**

votes

**5**answers

4k views

### Examples and intuition for arithmetic schemes

How should a beginner learn about arithmetic schemes (interpret this as you wish, or as a regular scheme, proper and flat over Spec(Z))? What are the most important examples of such schemes? Good ...

**8**

votes

**5**answers

2k views

### The significance of modularity for all Galois representations

On pg. 1 of the slides of a talk, Henri Darmon wrote:
Question: What is an interesting Diophantine equation?
A “working definition”. A Diophantine equation is interesting
if it reveals or ...

**31**

votes

**2**answers

733 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 ...

**21**

votes

**1**answer

1k 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? ...

**19**

votes

**4**answers

915 views

### Everywhere locally isomorphic abelian varieties

Is there a standard example of two abelian varieties $A$, $B$ over some number field $k$ which are $k_v$-isomorphic for every place $v$ of $k$ but not $k$-isomorphic ?

**18**

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**3**answers

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### Understanding Faltings's Theorem

I am soon to become a graduate student and so I started a personal project; I want to understand Faltings's proof of the Mordell conjecture.
I want to get into arithmetic geometry (since I always ...

**14**

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. ...

**10**

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 ...

**12**

votes

**0**answers

336 views

### Artin L-function and Zeta function of twisted Dirac operator

If one thinks of a Frobenius as an element in the fundamental group of an arithmetic curve and of a Galois representation $\sigma$ as a flat connection on the curve, then the definition of the Artin L-...

**10**

votes

**0**answers

293 views

### Extension of Messing-Mazur-Oda to general groups

The following may be well-known (or obviously false), but I can't find a counterexample or a reference.
Suppose that $k$ is some perfect field (one can assume algebraically closed, if that makes you ...

**32**

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....

**22**

votes

**0**answers

717 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{...

**19**

votes

**0**answers

711 views

### Finiteness of etale cohomology for arithmetic schemes

By an arithmetic scheme I mean a finite type flat regular integral scheme over $\mathrm{Spec} \, \mathbb{Z}$.
Let $X$ be an arithmetic scheme. Then is $H_{et}^2(X,\mathbb{Z}/n\mathbb{Z})$ finite ...