Talk:Abelian variety

Map from fraction field to residue field

The third paragraph of the introduction says: "This induces a map from the fraction field to any such finite field." There is no such map (there are no homomorphisms between fields of different characteristic). Instead, one constructs a so-called Néron model for the abelian variety over the Dedekind domain (meaning that the generic point of the model is the original abelian variety over the number field). This model, being an abelian scheme over the Dedekind domain, can be reduced to an abelian variety over any residue field of the Dedekind domain. — Preceding unsigned comment added by 217.123.117.211 (talk) 07:38, 30 September 2022 (UTC)

Comment on article validity

This article is crap. For instance abelian varieties need not be over C, so they need not be complex tori, and I can certainly count Q-bar. Can someone more knowledgable with wikipedia put up the sign "In desperate need of expert review"?--unsigned

You would make a better impression if you adopted a more constructive tone, and also read the third sentence of the lead section. There is no need to put the most general sentence in the second sentence of an introduction. The 'algebraic definition' section down the page goes to the general field case; but the complex manifold case is important enough. Articles here do not strive initially for the greatest generality. Charles Matthews 19:22, 21 March 2006 (UTC)

Your remark still doesn't make the sentence true. Nah well, I'll just refer people to the literature.--unsigned

Well, you could refer them to Mumford's Abelian Varieties, if you want a ridiculously steep learning curve. There they will find a definition as complete variety with group law - on p.39. Chapter One is all about the complex manifold instance. Rest my case. Charles Matthews 12:14, 22 March 2006 (UTC)

expansion

I have expanded and re-organized this article. It is still far from perfect so feel free to improve on it. It is my first (serious) contbutto wikipedia so comments and critique are most welcome. --Lenthe 13:52, 15 Jun 2005 (UTC)

Fantastic work. I'll try to find time later on to do my grammar-nazi thing. - Gauge 21:20, 17 Jun 2005 (UTC)

User:Charles Matthews I'm fairly new round here - I have created a number of new pages to fill gaps in mathematics. I'm finding the list of mathematical topics, which I believe people follow through the 'related changes', somewhat clunky. I believe as a good citizen I should be adding to this the new pages, and also unlinked pages that I come across. Well, I have done some of that, but the page is very big at well over 100K.

Also, as in the case of 'Abelian variety' which was on the LOMT already and for which I made a page, it seems that the Related Changes doesn't flag that as a new page on its creation. This seems odd, in the case of a 'wanted' page.

This is a confusing piece. You could define words so that it is clear what is the same as what.

Question

I'm sort of a novice, but in the section on the algebraic defintion, isn't it true that all abelian varieties are projective (ie. over any field, not just complex) CraigDesjardins 14:45, 1 July 2006 (UTC)

That is correct, and is also what was intended in the article. I have changed the wording slightly, hoping to avoid confusion. --Lenthe 10:12, 7 August 2006 (UTC)

Abelian Schemes

This page should discuss the fact that there are no abelian schemes over $\mathbb {Z}$ (Fontaine, Il n’y a pas de vari´et´e ab´elienne sur Z, Invent. Math. 81 (1985), no. 3, 515–538.) and use this as motivation for semi-stable reduction. — Preceding unsigned comment added by Username6330 (talk • contribs) 08:30, 10 November 2017 (UTC)

Unclear paragraph

The section Important theorems is as follows:

"One important structure theorem of abelian varieties is Matsusaka's theorem. It states that over an algebraically closed field every abelian variety is the quotient of the Jacobian of some curve; that is, there is some surjection of abelian varieties where is a Jacobian. This theorem remains true if the ground field is infinite."

Since the initial statement of the theorem says nothing about the cardinality of the ground field, it is entirely unclear why the last sentence was necessary.

Unless it is attempting to say that the theorem remains true if the algebraically closed condition is removed and the condition that the ground field is infinite is added.

But if this is the case, it needs to be stated explicitly instead of forcing readers to guess what is meant.