By Robert D.M. Accola
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Extra info for Riemann Surfaces, Theta Functions, and Abelian Automorphisms Groups
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Lsj = 0 (mod n) and (ii) ~8jcj = 0 j=l J Then t h e r e Proof: I. hand side zero. form characteristic. has period are of the is complete. Lemma 5: there J(Wl) period Lemma 8, P a r t formula proof in is a X e R Suppose first so t h a t that Xj = Bj PO = O. for all By f o r m u l a in J(Wl) j. (4) o f P a r t I 46 ~BjUl(Xlj) ~ ~Bjvj-luI(~zo) or (XSjvj'1)~z0 ~ ZBjxIj Let a be a function on its divisor. definition Then • a = Bj W1 o. with the above invariant divisor as corresponds to some character for all X and by j.
In the limit ! diag where for ~A 1 Bq is W0 - A0 du o ~ 0 in the a 0. becomes ! (Bq, BO, BO, . . B matrix in the B1 limit. for A1 and Moreover, Thus i n t h e l i m i t ,Bo) B0 is the du0~ § 0 in the right B matrix 41 hand side and of % formula Thus (7) b e c o m e s a p o i n t cj has the indicated Now c o n s i d e r Since the left hand side (5) and t h e r i g h t formula (6) b o t h s i d e s (i) Continue (eq,0,0, ... ,0) ~ . s aE x ~ ffil X j C j . ~ j characteristic as i n F o r any • has period Since a characteristic is as i n an i s o m o r p h i s m , the the hypotheses o f Lemma 4.
A s, N1 § W0 (= W1/G) W 1 § WI/H § N 0. unramified, al,a2, be the is not G factorization is seen to be contradiction. a topological description of covers. b : W1 * W0 is a disc be a completely A0 c W0 properties. (let ramified A 1 = b'l(A0)) All the ramification of abelian with b cover. the occurs over D A0 and W1 homeomorphic Proof: If The c o v e r A1 - to PO = 0 group of curves group of sects Again let 50 . which corresponds corresponds to where a~j ~ O. on it N topy class 7~ Xoj under Let suffices on to W0 described by a r e p r e s e n t a t i o n ~ ...