Local zeta function: Difference between revisions

Revision as of 02:10, 9 December 2020

Revision as of 02:14, 9 December 2020

Line 94:

:<math>Z(t) = \frac{P(t)}{(1 - t)(1 - qt)}\ ,</math>

with ''P''(''t'') a polynomial, of degree 2''g'', where ''g'' is the [[genus (mathematics)|genus]] of ''C''. Rewriting

:<math>P(t)=\prod^{2g}_{i=1}(1-\omega_i t)\ ,</math>

Line 104:

For example, for the elliptic curve case there are two roots, and it is easy to show the absolute values of the roots are ''q''^1/2. [[Hasse's theorem on elliptic curves|Hasse's theorem]] is that they have the same absolute value; and this has immediate consequences for the number of points.

[[André Weil]] proved this for the general case, around 1940 (''Comptes Rendus'' note, April 1940): he spent much time in the years after that writing up the [[algebraic geometry]] involved. This led him to the general [[Weil conjectures]],. [[Alexander Grothendieck]] developed the [[scheme (mathematics)|scheme]] theory for the sakepurpose of resolving itthese.
A andgeneration finally,later [[Pierre Deligne]] hadcompleted provedthe a generation laterproof.
(See [[étale cohomology]] for the basic formulae of the general theory.)

==General formulas for the zeta function==