Suppose $f,g:X \rightarrow Y$ are finite morphisms between connected smooth curves over $\mathbb{C}$, with $Y$ of genus at least $2$. 


If $f$ and $g$ induce the same morphism $H^\*(Y,\mathbb{C}) \rightarrow H^\*(X,\mathbb{C})$, does $f=g$?



Can distinct morphisms between curves induce the same morphism on singular cohomology?
Yes. Since $Y$ embeds into its Jacobian $B$, it is enough to prove the statement for pairs of maps to an abelian variety $f, g\colon X\to B$ sending a base point $x\in X$ to $0\in B$. Every such map factors uniquely through the Albanese variety $A$ of $X$, so we reduce further to the case of pairs of maps $f, g\colon A\to B$ between abelian varieties (sending $0$ to $0$). Every such map is necessarily a group homomorphism, and is uniquely determined by what it does on $\pi\_1 = H\_1$, or on $H^1(-, \mathbf{C})$.


