Level Compatibility in the Passage from Modular Symbols to Cup Products

Abstract

For a positive integer 𝛭 and an odd prime p, Sharifi defined a map 𝜛M from the first homology group of the modular curve X₁(𝛭) with Zₚ-coefficients to a second Galois cohomology group over ℚ(µM) with restricted ramification and Zₚ(2)-coefficients that takes Manin symbols to certain cup products of cyclotomic 𝛭-units. Fukaya and Kato showed that if p|𝛭 and p ≥ 5, then 𝜛Mₚ and 𝜛M are compatible via the map of homology induced by the quotient X₁(𝛭p) -> X₁ (𝛭) and corestriction from ℚ(µMₚ) to ℚ(µM). We show that for a prime 𝓁∤𝛭,𝓁≠p ≥ 5, the maps 𝜛M𝓁 and 𝜛M are again compatible under a certain combination of the two standard degeneracy maps from level 𝛭𝓁 to level 𝛭 and corestriction.