0,2 Heterotic Gauge Couplings and their M-Theory Origin

We work out the relation between automorphic forms on SO(2+s,2,Z) and gauge one-loop corrections of heterotic K3 x T^2 string compactifications for the cases s=0,1. We find that one-loop gauge corrections of any orbifold limit of K3 with arbitrary choice of gauge bundles can always be expressed by their instanton numbers and generic automorphic forms. These functions classify also one--loop gauge thresholds of N=1 (0,2) heterotic compactifications based on toroidal orbifolds T^6/Z_N$. We compare these results with the gauge couplings of M-theory compactified on S^1/Z_2 x T^6/Z_N using Witten's Calabi--Yau strong coupling expansion.