Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [arithmetic-geometry]

A subject that lies at the intersection of algebraic geometry and number theory dealing with varieties, the Mordell conjecture, Arakelov theory, and elliptic curves.

A subject that lies at the intersection of algebraic geometry and number theory dealing with varieties, the Mordell conjecture, Arakelov theory, and elliptic curves.

990 questions
111
votes
1 answer

Tate conjecture for Fermat varieties

I've been looking at Tate's Algebraic Cycles and Poles of Zeta Functions (hard to find online... Google books outline here) and have a question about his work on (conjecturing!) the Tate conjecture for the Fermat variety $X_m^r$, defined by the…
Alex
  • 4,178
45
votes
3 answers

The resemblance between Mordell's theorem and Dirichlet's unit theorem

The first one states that if $E/\mathbf Q$ is an elliptic curve, then $E(\mathbf Q)$ is a finitely generated abelian group. If $K/\mathbf Q$ is a number field, Dirichlet's theorem says (among other things) that the group of units $\mathcal…
Bruno Joyal
  • 55,975
43
votes
1 answer

What is Spec of the Adeles?

Let $K$ be a global field and $A_K$ the ring of adeles. What are the prime ideals of $A_K$? I have been told that a full proof of this is quite subtle, but have been unable to find a reference for this result.
Miles Lake
  • 968
  • 5
  • 12
38
votes
1 answer

Has SGA 4½ been typeset in TeX?

The title says it all. I've CW'd the question since I'm answering it, as this seemed like the best way to get the news out.
Daniel Miller
  • 2,395
38
votes
1 answer

Geometric intuition behind The Mordell Conjecture

The Mordell Conjecture/Faltings Theorem says roughly that if $K$ is an algebraic number field and $X$ is an algebraic curve defined over $K$ of genus $g >1$ then the set of $K$-rational points $X(K)$ is finite. Now in the cases when the genus is $g…
Adrián Barquero
  • 15,454
37
votes
5 answers

Why is it "easier" to work with function fields than with algebraic number fields?

I just bought a copy of Jürgen Neukirch's book Algebraic Number Theory. While browsing through it I found a section titled § 14. Function Fields in chapter I. In it the author describes some aspects of an analogy between function fields and…
Adrián Barquero
  • 15,454
31
votes
2 answers

Intuition for Class Numbers

So I've been thinking about the analytic class number formula lately, and class numbers in general and I'm trying to develop a good intuition for them. My basic question, which may be too general/difficult is: $\textbf{What forces the class number…
Dylan Yott
  • 7,189
31
votes
1 answer

Learning path to the proof of the Weil Conjectures and étale topology

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" and by Liu - "Algebraic Geometry and Arithmetic…
Javier Álvarez-Vizoso
  • 15,674
28
votes
1 answer

What is an intuitive meaning of genus?

I read from the Finnish version of the book "Fermat's last theorem, Unlocking the Secret of an Ancient Mathematical Problem", written by Amir D. Aczel, that genus describes how many handles there are on a given surface. But now I read the…
Jaakko Seppälä
  • 2,773
26
votes
1 answer

Why are period integrals naïve periods?

Apologies for the long question. I recall the definition of a (naïve) period according to Kontsevitch and Zagier [KS]: A (naïve) period is a complex number whose real and imaginary parts are absolutely convergent integrals of rational functions…
Bruno Joyal
  • 55,975
25
votes
1 answer

what is a "dévissage" argument?

In Serre's "Local Fields" (Chapter 2, section 2, proposition 3) he proves something about field extensions which he breaks into parts: first he dealt with the separable case, and then with the inseparable case. In between the two cases he writes "a…
edo arad
  • 1,045
23
votes
2 answers

Some basic examples of étale fundamental groups

I'm trying to get a better understanding of étale fundamental groups, and I think that the overall idea -- the big picture -- is beginning to become clear, but my computational ability seems to be essentially nonexistent. I think that I understand…
user101616
23
votes
1 answer

Prop. 3.2.15 in Liu: Geometrically reduced algebraic variety

I have a problem with proposition 3.2.15 of Algebraic geometry and arithmetic curves of Qing Liu: Let $X$ be an integral algebraic variety over $k$. If $X$ is geometrically reduced then $K(X)$ is a finite separable extension of a purely…
Gabriel Soranzo
  • 2,890
  • 16
  • 32
23
votes
2 answers

Translation for EGA/SGA

People often recommend Grothendieck's EGA (Elements de Geometrie Algebrique) and SGA (seminaire de geometrie algebrique) as a good reference for learning arithmetic geometry. However, as the title suggests, these books are in french. Does anyone…
Eugene
  • 7,802
22
votes
4 answers

Manageable project to learn some arithmetic geometry

OK, this may be an unusual question, since I'm asking you for advice, the way I would ask a supervisor were I a grad student. I am a thirty-something maths teacher with an unresolved attraction to arithmetic geometry, and would like to know more…
PseudoNeo
  • 10,358
1
2 3
…
65 66