Caterina Consani Matilde Marcolli (Eds.) Noncommutative Geometry and Number Theory Where Arithmetic meets Geometry and Physics A Publication of the Max-Planck-Institute for Mathematics, Bonn vieweg Contents Preface v The Hecke algebra of a reductive p-adic group: a view from noncommutative geometry ANNE-MARIE AUBERT, PAUL BAUM, ROGER PLYMEN 1 Hilbert modular forms and the Ramanujan conjecture DON BLASIUS 35 Farey fractions and two-dimensional tori FLORIN P. BOCA AND ALEXANDRU ZAHARESCU 57 Transgression of the Godbillon-Vey class and Rademacher functions ALAIN CONNES AND HENRI MOSCOVICI 79 Archimedean cohomology revisited CATERINA CONSANI AND MATILDE MARCOLLI 109 A twisted Burnside theorem for countable groups and Reidemeister numbers ALEXANDER FEL'SHTYN AND EVGENIJ TROITSKY 141 Introduction to Hopf cyclic cohomology MASOUD KHALKHALI AND BAHRAN RANGIPOUR 155 The non-abelian (or non-linear) method of Chabauty MINHYONG KIM 179 The residues of quantum field theory - numbers we should know DIRK KREIMER 187 Phase transitions with spontaneous symmetry breaking on Hecke C*-algebras from number fields MARCELO LACA AND MACHIEL VAN FRANKENHUIJSEN 205 On harmonic maps in noncommutative geometry GIOVANNI LANDI 217 Towards the fractional quantum Hall effect: a noncommutative geometry perspective MATILDE MARCOLLI AND VARGHESE MATHAI 235 Homological algebra for Schwartz algebras of reductive p-adic groups RALPH MEYER 263 viii CONTENTS A non-commutative geometry approach to the representation theory of reductive p-adic groups: Homology of Hecke algebras, a survey and some new results VICTOR NISTOR 301 Three examples of non-commutative boundaries of Shimura varieties FREDERIC PAUGAM 323 Holomorphic bundles on 2-dimensional noncommutative toric orbifolds ALEXANDER POLISHCHUK 341 A new short proof of the local index formula of Atiyah-Singer RAPHAEL PONGE 361.