Non-homeomorphic Galois conjugate Beauville structures on PSL(2, p)

作者：Gonzalez Diez G^*; Torres Teigell D

来源：Advances in Mathematics, 2012, 229(6): 3096-3122.

DOI：10.1016/j.aim.2012.02.014

• Beauville surfaces
• Galois action
• Fundamental group
• Triangle curves
• Triangle Fuchsian groups

Catanese%26apos;s rigidity results for surfaces isogenous to a product of curves indicate that Beauville surfaces should provide a fertile source of examples of Galois conjugate varieties that are not homeomorphic, a phenomenon discovered by J.P. Serre in the sixties. %26lt;br%26gt;In this paper, we construct Beauville surfaces S = (C-1 x C-2)/G with group G = PSL(2, p) for p %26gt;= 7, and curves C-1, C-2 such that the orbit of S under the action of the absolute Galois group Gal ((Q) over bar /Q) contains non-homeomorphic conjugate surfaces. When p = 7 the orbit consists exactly of two surfaces that have non-isomorphic fundamental groups, and the curves C-1, C-2 have genera 8 and 49, which is shown to be the minimum for which there is a pair of non-homeomorphic Galois conjugate Beauville surfaces. As p grows the orbits contain an arbitrarily large number of non-homeomorphic surfaces. %26lt;br%26gt;Along the way we prove a metric rigidity theorem for Beauville surfaces which provides an elementary proof of the part of Catanese%26apos;s theory needed to prove our results.

• 出版日期2012-4-1