On cohomology groups attached to towers of algebraic curves
Motivated by a work of \textit{Y. Ihara} [Invent. Math. 86, 427-459 (1986; Zbl 0585.14020)], we develop a ``one-dimensional \(\ell\)-adic cohomology theory'' for ``almost pro-\(\ell\)'' towers of algebraic curves consisting of étale coverings of a fixed base curve. More precisely, we describe the projective limits of (three kinds of) \(\ell\)-adic étale cohomology groups of the members of a given tower in terms of continuous cochain cohomology groups and an analogue of the group of modular symbols. When the base curve is \(\mathbb{P}^ 1 - \{0,1, \infty\}\), we show that our cohomology theory is closely related to Ihara's theory. We also study the cohomology groups of towers of elliptic modular curves, especially in connection with the theory of \(\ell\)-adic representations attached to elliptic cusp forms.
scientific article; zbMATH DE number 469056