 By Shafarevich I.R. (ed.)

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C) Let ψ : C2 → C3 be another nonconstant map of smooth curves. Then for all P ∈ C1 , eψ◦φ (P ) = eφ (P )eψ (φP ). P ROOF. 9] with Y = P1 and D = (0), or see [142, Proposition 2], [233, I Proposition 10], or [243, III §2, Theorem 1]. 8]. (c) Let tφP and tψφP be uniformizers at the indicated points. By definition, the functions eψ (φP ) and ψ ∗ tψφP tφP have the same order at φ(P ). Applying φ∗ and taking orders at P yields ψ ordP φ∗ tφP = ordP (ψφ)∗ tψφP , e (φP ) which is the desired result. 7.

Let {α1 , . . , αn } be a basis for L/K, and let {σ1 , . . , σn } = GL/K . For each 1 ≤ i ≤ n, consider the vector n (αi v)σj = TraceL/K (αi v). wi = j=1 invariant, so wi ∈ VK . A basic result from field theIt is clear that wi is GK/K ¯ σ ory [142, III, Proposition 9] says that the matrix αi j 1≤i,j≤n is nonsingular, so σj each v , and in particular v, is an L-linear combination of the wi ’s. ) Exercises 37 We conclude this section with a classic relationship connecting the genera of curves linked by a nonconstant map.

D) Find a basis for the holomorphic differentials on C. (Hint. Consider the set of differential forms {xi dx/y : i = 0, 1, 2, . }. 15. Let C/K be a smooth curve defined over a field of characteristic p > 0, and let t ∈ K(C). Prove that the following are equivalent: (i) K(C) is a finite separable extension of K(t). (ii) For all but finitely many pointsP ∈ C, the function t − t(P ) is a uniformizer at P . (iii) t ∈ / K(C)p . 16. Let C/K be a curve that is defined over K and let P ∈ C(K). , prove that there are uniformizers that are defined over K.