Level compatibility in the passage from modular symbols to cup products


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.

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This paper originated from the author’s 2016 Ph.D. thesis. The author’s research was supported in part by the National Science Foundation under Grant No. DMS-1360583. The author thanks the reviewers for their insightful comments and suggestions. And a special thanks to Romyar Sharifi for all of the immensely helpful discussions, and feedback he gave to the author. Without his invaluable guidance this work would not have been possible.

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Correspondence to R. Scott Williams.

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Williams, R.S. Level compatibility in the passage from modular symbols to cup products. Res. number theory 7, 9 (2021). https://doi.org/10.1007/s40993-020-00234-w

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  • Iwasawa theory
  • Modular symbols
  • Cup products
  • Degeneracy maps

Mathematics Subject Classification

  • 11R23
  • 11F67