An introduction to motivic zeta functions of motives

Abstract

It oftens occurs that Taylor coefficients of (dimensionally regularized) Feynman amplitudes $I$ with rational parameters, expanded at an integral dimension $D=D_0$, 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.

References in corpus (8)

• Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops
• Counting Points on Hyperelliptic Curves using Monsky-Washnitzer Cohomology
• Resolution of singularities for multi-loop integrals
• The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups
• Galois theory, motives and transcendental numbers
• Ambiguity theory, old and new
• A motivic version of Pellikaan's two variable zeta function
• Motivic Weight Complexes for Arithmetic Varieties

Cited by in corpus (3)

• Feynman graph polynomials
• Unramified Euler sums and Hoffman $\star$ basis
• Arithmetic of Potts model hypersurfaces