Morphisms of closed Riemann surfaces and Lefschetz trace formula

Tanabe, Masaharu

doi:10.1090/S0002-9939-09-10210-1

Morphisms of closed Riemann surfaces and Lefschetz trace formula
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Proc. Amer. Math. Soc. 138 (2010), 1295-1303 Request permission

Abstract:

We study the number of coincidences of two distinct morphisms $f_i :X\to Y (i=1,2)$ between closed Riemann surfaces of genera greater than zero. We give a necessary and sufficient condition for the existence of a coincidence in terms of the inner product defined on the free abelian group of homomorphisms between the Jacobian varieties J$(X)$ and J$(Y)$. We use the Hodge decomposition and the holomorphic Lefschetz number to study the number of coincidences in detail.

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