Level Compatibility in the Passage from Modular Symbols to Cup Products.

摘要

For a positive integer M and an odd prime p , Sharifi defined a map piM from the first homology group of the modular curve X1( M) with Zp-coefficients to a second Galois cohomology group over Q(mu M) with restricted ramification and Z p(2)-coefficients that takes Manin symbols to certain cup products of cyclotomic M-units. Fukaya and Kato showed that if p|M and p ≥ 5, then pi (M/p) and piM are compatible via the map of homology induced by the quotient X1( Mp) → X1(M) and corestriction from Q(mu(M/p)) to Q(muM). We show that for a prime ℓ [special character omitted] M, ℓ ≠ p ≥ 5, the maps piMℓ and pi M are again compatible under a certain combination of the two standard degeneracy maps from level Mℓ to level M and corestriction.