# Questions tagged [arithmetic-geometry]

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

**2**

votes

**4**answers

513 views

### a question on function fields (extending my previous question)

Consider the extension Q(a,b) of the field of rationals, where a,b are algebraically independent transcendentals. To Q(a,b) adjoin the roots of the polynomials x^5+a^5=1 and y^5+b^5=1. The resulting ...

**10**

votes

**1**answer

627 views

### a question on function fields

Consider the transcendental extension Q(t) of the field of rationals.
To Q(t) adjoin the root of the polynomial x^5+t^5=1. The resulting
field Q(t)[x] is a radical extension of Q(t). Is it true that ...

**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 ...

**4**

votes

**1**answer

800 views

### An inverse problem: Number fields attached to elliptic curves over Q

If I understand FC's remark under the post "Very strong multiplicity one for Hecke eigenforms," in the course of Faltings's proof of the Tate conjecture, Faltings proves the following statement: let E/...

**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 (...

**5**

votes

**1**answer

458 views

### Finiteness of Obstruction to a Local-Global Principle

Say that a projective variety V over Q satisfies the local-global principle up to finite obstruction (#) if there are only finitely many isomorphism classes of projective varieties over Q that are not ...

**11**

votes

**4**answers

623 views

### Behaviour of Zeta-function under Finite Morphism

Let X ---> Y be a finite surjective morphism of smooth, projective, connected varieties over a finite field F_q. Can one describe the zeta function Z(X, t) in terms of the zeta-function Z(Y,t) of ...

**10**

votes

**2**answers

875 views

### A comprehensive overview of finite fields

I've read numerous introductions to finite fields, but I feel like my intuition about them is fairly lacking. Considering that finite fields are the the most "inert" objects in algebraic geometry, I ...

**5**

votes

**3**answers

561 views

### Solving “a, b, a+b have given divisors” problem

I've read an interesting article, math.NT/0409456 where you're just trying to solve a simple problem:
For a given (finite) set of primes S find all solutions to an equation ...

**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 ...

**4**

votes

**1**answer

486 views

### Existence of proper regular models for varieties over Q and other global fields

What is known about regular proper models for smooth projective varieties over Q? Results for other global fields would also be interesting, as well as general comments and suggested references for ...

**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,...

**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 ...

**11**

votes

**1**answer

2k views

### Reference for the `standard' Tate curve argument.

I'd like a reference (e.g. something published somewhere that I can cite in a paper) for the proof of the following:
Let $E$ be an elliptic curve over $\mathbb Q$ with minimal discriminant $\Delta$...