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I am aware of two forms of the Hurwitz formula. The first is more common, and deals only with the degrees. So if $f:X \rightarrow Y$ is a non-constant map of degree $n$ between two projective non-singular curves, with genera $g_X$ and $g_Y$, then $$ 2(g_X-1) = 2n(g_Y-1) + \deg(R), $$ where $R$ is the ramification divisor of $f$. The proof of this was given to me as an exercise when I started my PhD, and I am very happy with it.

However, in some other work that I was doing it appeared that one could strengthen this to say rather that if $K_X={\rm div}(f^*(dx))$ and $K_Y={\rm div}(dx)$ are canonical divisors of $X$ and $Y$, then $$ K_X = n\cdot K_Y + R. $$ I have found this alluded to in a number of places, and even stated in Algebraic Curves Over Finite Fields by Carlos Moreno. However, this was without proof, and every idea of a proof that I have seen is in sheaf theoretic language. I am slowly getting through sheaves and schemes, but I am currently trying to prove this in an elementary manner (fiddling around with orders of $dx$ etc.), in the wildly ramified case (the tamely ramified case is fine).

I would like to know if

  1. It is possible to prove this without using sheaves etc in the wildly ramified case
  2. If so, are there any references that would help with this.

edit: Also, is there a different name for the "more specific" Hurwitz formula?

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    There is an IMHO relatively accessible proof in Stichtenoth's *Algebraic Function Fields and Codes*. It is written without using differential forms (canonical divisor being defined in an adelic way as the divisor for whichs Riemann-Roch holds). Not surprisingly the proof gets a bit technical when dealing with wild ramification.2012-07-23
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    I thought the only proof in there was in terms of degrees, I must not have looked hard enough. Thanks for the recommendation, I will reread it more thoroughly.2012-07-23

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