Editing Riemann–Roch theorem (section)

====Genus one====
[[File:Torus_cycles2.svg|right|thumb|A torus]]
The next case is a Riemann surface of genus <math>g=1</math>, such as a [[torus]] <math>\Complex/\Lambda</math>, where <math>\Lambda</math> is a two-dimensional [[lattice (group)|lattice]] (a group isomorphic to <math>\Z^2</math>). Its genus is one: its first singular homology group is freely generated by two loops, as shown in the illustration at the right. The standard complex coordinate <math>z</math> on <math>C</math> yields a one-form <math>\omega=dz</math> on <math>X</math> that is everywhere holomorphic, i.e., has no poles at all. Therefore, <math>K</math>, the divisor of <math>\omega</math> is zero.

On this surface, this sequence is

:1, 1, 2, 3, 4, 5 ... ;

and this characterises the case <math>g=1</math>. Indeed, for <math>D=0</math>, <math>\ell(K-D)=\ell(0)=1</math>, as was mentioned above. For <math>D=n\cdot P</math> with <math>n>0</math>, the degree of <math>K-D</math> is strictly negative, so that the correction term is 0. The sequence of dimensions can also be derived from the theory of [[elliptic function]]s.