Answer by Minhyong Kim for Why should I believe the Mordell Conjecture?
Since summer is here, I will bore people a bit with my own ahistorical viewof Mordell. The main point is that Serge Lang's strategy, mentioned by several people, isessentially correct. Nevertheless, we...
View ArticleAnswer by Felipe Voloch for Why should I believe the Mordell Conjecture?
Mordell's reasoning might have been the following. If a curve of genus zero has a (smooth) point, then it has infinitely many. If a curve of genus one has two smooth points, it has a good chance of...
View ArticleAnswer by Denis Chaperon de Lauzières for Why should I believe the Mordell...
I have a vague memory that Weil (or someone else) actually complained in print somewhere about people making conjectures unsupported by much evidence, and using Mordell's conjecture as an example, with...
View ArticleAnswer by S. Carnahan for Why should I believe the Mordell Conjecture?
Here's a quick and dirty version of George Lowther's calculation that I learned from Bjorn Poonen. It is presented in a bit more generality in the next-to-last slide of this talk, so it is in a sense...
View ArticleAnswer by Stergios for Why should I believe the Mordell Conjecture?
You can also have a look at the following paper of Curt McMullen : http://www.math.harvard.edu/~ctm/papers/home/text/papers/fermat/fermat.pdfIn his own words "This paper is an appreciation of some of...
View ArticleAnswer by Emerton for Why should I believe the Mordell Conjecture?
If the curve $X$ (over the number field $k$) has no $k$-points at all, then Mordell's conjecture is true for $X$.Otherwise, if $O$ is a given $k$-point on $X$, we can get a map (the Albanese map)$X \to...
View ArticleAnswer by Sam Lichtenstein for Why should I believe the Mordell Conjecture?
One reason to believe Mordell's conjecture is Chaubaty's proof of it (from 1940ish) in the case where the Mordell-Weil rank of the Jacobian is less than the genus of the curve. This basically builds on...
View ArticleAnswer by George Lowther for Why should I believe the Mordell Conjecture?
I'm not an expert here, but reading J.S.Milne's notes on elliptic curves (available from his website), I noticed the following statement.ASIDE 2.4 There is a heuristic explanation for Mordell’s...
View ArticleAnswer by Sándor Kovács for Why should I believe the Mordell Conjecture?
Disclaimer In the spirit of the question I am not trying to be very precise below. The key word is motivation. :)Reason #1 (simple, but lame)Fermat's Last Theorem says that a curve of that kind of a...
View ArticleWhy 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...
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