Editing Hasse–Weil zeta function (section)

==Birch and Swinnerton-Dyer conjecture==
The [[Birch and Swinnerton-Dyer conjecture]] states that the [[Rank of an abelian group|rank]] of the [[abelian group]] ''E''(''K'') of points of an elliptic curve ''E'' is the order of the zero of the Hasse–Weil ''L''-function ''L''(''E'',&nbsp;''s'') at ''s'' = 1, and that the first non-zero coefficient in the [[Taylor expansion]] of ''L''(''E'',&nbsp;''s'') at ''s'' = 1 is given by more refined arithmetic data attached to ''E'' over ''K''.<ref>{{Cite encyclopedia
| last=Wiles
| first=Andrew
| author-link=Andrew Wiles
| chapter=The Birch and Swinnerton-Dyer conjecture
| editor1-last=Carlson
| editor1-first=James
| editor2-last=Jaffe
| editor2-first=Arthur
| editor2-link=Arthur Jaffe
| editor3-last=Wiles
| editor3-first=Andrew
| editor3-link=Andrew Wiles
| title=The Millennium prize problems
| publisher=American Mathematical Society
| year=2006
| isbn=978-0-8218-3679-8
| chapter-url=http://www.claymath.org/sites/default/files/birchswin.pdf
| pages=31–44
| mr=2238272
| access-date=2022-04-13
| archive-date=2018-03-29
| archive-url=https://web.archive.org/web/20180329033023/http://www.claymath.org/sites/default/files/birchswin.pdf
| url-status=dead
}}</ref> The conjecture is one of the seven [[Millennium Prize Problems]] listed by the [[Clay Mathematics Institute]], which has offered a $1,000,000 prize for the first correct proof.<ref>[http://www.claymath.org/millennium-problems/birch-and-swinnerton-dyer-conjecture Birch and Swinnerton-Dyer Conjecture] at Clay Mathematics Institute</ref>