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Questions tagged [arithmetic-geometry]

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

2
votes
4answers
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
1answer
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
8answers
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
1answer
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
6answers
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
1answer
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
4answers
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
2answers
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
3answers
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
5answers
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
1answer
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
2answers
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
1answer
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
1answer
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 $\Delta$...