Level compatibility in the passage from modular symbols to cup products

Abstract

For a positive integer M and an odd prime p there exists a map $\varpi _M$ from the first homology group of the modular curve $X_1(M)$ with ${\mathbb {Z}}_p$-coefficients to a second Galois cohomology group over ${\mathbb {Q}}(\mu _M)$ with restricted ramification and ${\mathbb {Z}}_p(2)$-coefficients. This map takes Manin symbols to certain cup products of cyclotomic M-units. It has previously been shown that if $p\mid M$ and $p\ge 5$, then $\varpi _{Mp}$ and $\varpi _M$ are compatible via the map of homology induced by the quotient $X_1(Mp)\rightarrow X_1(M)$ and corestriction from ${\mathbb {Q}}(\mu _{Mp})$ to ${\mathbb {Q}}(\mu _M)$. We show that for a prime $\ell \not \mid M$, $\ell \ne p\ge 5$, the maps $\varpi _{M\ell }$ and $\varpi _M$ are again compatible under a certain combination of the two standard degeneracy maps from level $M\ell$ to level M and corestriction.