, periodic: ?(? + k? 2 ) = ?(?)

, We claim that O 1 (?)?(?) is holonomic in the variable x(?). First, since it is (? 1 , ? 2 )-elliptic, O 1 (?) is algebraic in x(?). Second, ? (?) is a rational function of x(?), as ? (?) = ??(?). So O 1 (?)?(?) is holonomic in the variable x(?), proving the claim. We conclude using closure properties of holonomic functions. Proof of Corollary 43. If the group ? 1 , ? 2 of birational transformation of P 1 (C) 2 is finite, With the ? 1-periodicity, we find using Lemma 44 that ? is algebraic in the variable x(?)

, Let f (?) be a (? 1 , k? 2 )-elliptic function. Then f (?) is algebraic in x(?)

. Proof, The functions f (?) and ?(?) are (? 1 , k? 2 )-elliptic. Using a well-known property of elliptic functions

, Thus f (?) is algebraic in ?(?), and hence also in x(?) due to Proposition 18

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