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Extra resources for Compact Riemann surfaces
7 Let P0 be a Weierstrass point on R and z a local parameter at P0 , with z(P0 ) = 0. The order τ (P0 ) of the zero of ∆ at P0 ∆ = z τ (P0 ) O(1) (96) is called the weight of the Weierstrass point P0 . It turnes out that ∆ is well defined on R globally. 8 If to every local coordinate z : U ⊂ R → V ⊂ C there assigned a holomorphic function r(z) such that r = r(z)dz q , q∈Z (97) is invariant under holomorphic coordinate changes (49) one says that the holomorphic q-differential r is defined on R. In the same way as for the Abelian differentials one defines the divisor (r) of the qdifferentials.
G. 6 The differentials ΩR , ΩRQ with the singularities (63), (64) and all zero a-periods (67) are called the normalized Abelian differentials of the second and third kind. 12 Given a compact Riemann surface R with a canonical basis of cycles a1 , b1 , . . , ag , bg , points R, Q ∈ R, a local parameter z at R and N ∈ N there exist unique (N ) normalized Abelian differentials of the second ΩR and of the third ΩRQ kind. 4. The proof of the uniqueness is simple. 6. 7 Abelian differentials of the second and third kind can be normalized by a more symmetric then (67) condition.
In Fig. 20 the parts of the cycles lying on the ”lower” sheet of the covering are marked by dotted lines. b1 a1 Πg ag bg Figure 19: Canonical basis of cycles on the planar model Πg of compact Riemann surface. b1 b2 λ1 λ2 a1 λ3 λ4 a2 bg λ2g−1 λ2g λ2g+1 a3 Figure 20: Canonical basis of cycles of a hyperelliptic Riemann surface. λ2g+2 4 ABELIAN DIFFERENTIALS 4 32 Abelian differentials Our main goal is to construct functions on compact Riemann surfaces with prescribed analytical properties (for example, meromorphic functions with prescribed singularities).