# Elementary proof of the Hurwitz formula

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 $K_Y=f^*{\rm div}(dx)$ are canonical divisors of $X$ and $Y$, then $$K_X = n\cdot K_Y + R.$$ For ease of reference I will call this the "strong" Hurwitz formula.

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).

This has led to me the following questions:

1. Is it possible to prove this "strong" Hurwitz without using sheaves etc in the wildly ramified case?
2. If so, are there any references that would help with this?
3. Is there a different name for this "strong" Hurwitz formula.

I imagine/assume that an elementary proof would rely on computing the order of $dx$ at a point, but the best I am able to get from this is a lower bound on the order, not the precise value.

Previously asked at Stack Exchange (https://math.stackexchange.com/questions/174168/elementary-proof-of-the-hurwitz-formula/174189) but with no joy.

• You'll find a proof of what you call the strong Hurwitz formula, not using sheaves, in full generality, in e.g. in "Introduction to the Theory of Algebraic Functions of One Variable" by C. Chevalley. If you want a more recent reference, look up Stichtenoth's book "Algebraic Function Fields and Codes". – Felipe Voloch Jul 30 '12 at 11:42
• Thank you for the references, I was unaware of the first one and will check that. I did look in Stichtenoth's book, but I really can't find it in there for the wildly ramified case. It may be because it is an old edition though. Thank you. – Tait Jul 30 '12 at 11:49
• Your putative equation of divisors has one divisor on X and the other partly on Y and partly on X, so puzzles me. Maybe you mean div(f*(dx)) = f*(div(dx)) + R? then it seems just an elementary computation. But I am not an expert in "wild ramification". – roy smith Jul 30 '12 at 14:21
• Thanks for pointing out the error, I have corrected that. In the tamely ramified case it certainly is elementary, but it doesn't seem so elementary otherwise (to me at least), and no-one on stack exchange seemed able to provide a reference. – Tait Jul 30 '12 at 15:32
• Maybe your confusion stems from the following. The coefficient at a point $P$ of the ramification divisor is the order of vanishing at $P$ of $dx/dt$ for a local parameter $t$ at $P$ and the Hurwitz formula is almost trivial. If you expect this local multiplicity to be $e_P - 1$, where $e_P$ is the multiplicity of $P$ in $f^*(f(P))$, then this is simply false in the wild ramification case. – Felipe Voloch Jul 31 '12 at 12:41