Hurwitz genus formula
The following formula, due to Hurwitz, is extremely useful when trying to compute the genus of an algebraic curve. In this entry is a perfect field (i.e. every algebraic extension of is separable). Recall that a non-constant map of curves over is separable if the extension of function fields is a separable extension of fields.
Theorem (Hurwitz Genus Formula).
As an application of the Hurwitz genus formula, we show that an elliptic curve defined over a field of characteristic has genus . Notice that the fact that is an elliptic curve over implies that and are distinct elements of , otherwise would be a singular curve. We define a map:
and notice that , the “point at infinity” of , maps to , the point at infinity of . The degree of this map is : generically every point in has two preimages, namely and . Moreover, the genus of is and the map is ramified exactly at points, namely and the point at infinity. It is easily checked that the ramification index at each point is . Hence, the Hurwitz formula reads:
We conclude that , as claimed.
|Title||Hurwitz genus formula|
|Date of creation||2013-03-22 15:57:15|
|Last modified on||2013-03-22 15:57:15|
|Last modified by||alozano (2414)|