2008/12/19 by Yves André, André, Yves · 1 citation
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Quantum Mechanics and Applications #advanced mathematical theories #math.AG #msc:14Fxx #msc:19Exx #msc:19Fxx #msc:32Gxx #msc:81Qxx
paper · pdf · doi:10.48550/arxiv.0812.3920
arxiv created 2008/12/19 · arxiv updated 2009/12/01
It oftens occurs that Taylor coefficients of (dimensionally regularized) Feynman amplitudes I with rational parameters, expanded at an integral dimension D= D0, are not only periods (Belkale, Brosnan, Bogner, Weinzierl) but actually multiple zeta values (Broadhurst, Kreimer). In order to determine, at least heuristically, whether this is the case in concrete instances, the philosophy of motives - more specifically, the theory of mixed Tate motives - suggests an arithmetic approach (Kontsevich): counting points of algebraic varieties related to I modulo sufficiently many primes p and checking that the number of points varies polynomially in p. On the other hand, Kapranov has introduced a new "zeta function", the role of which is precisely to "interpolate" between zeta functions of reductions modulo different primes p. In this survey, we outline this circle of ideas and some of their recent developments.