Faithful actions of the absolute Galois group on marked moduli spaces (40')
speaker: Fabrizio Catanese (Universität Bayreuth)abstract: I will report on joint work with Ingrid Bauer and the late professor Fritz Grunewald. I will present a canonical procedure associating to an algebraic number a a hyperelliptic curve \(C_a\) and a triangle curve \((D_a, G)\) obtained through the normal closure of the Belyi process. In this way we obtain a faithful action on the set of isomorphism classes of marked triangle curves. I will then illustrate the application to surfaces isogenous to a product, i.e., free quotients of a product \(C_1 \times C_2\) of curves of genus at least \(2\) by a finite group \(G\). Using polynomials with only two critical values, we can exhibit infinitely many examples of pairs of real surfaces which are Galois conjugate, but have nonisomorphic fundamental groups. I will then show that the absolute Galois group acts faithfully on the moduli space of \(G\)-marked surfaces, and will discuss the problem whether this action is faithful also when one forgets the marking (as was incorrectly asserted in the previous arxiv version of the paper).
timetable:
Thu 4 Oct, 10:30 - 11:10, Sala Conferenze Centro De Giorgi
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