Modular symbols and the integrality of zeta elements

TL;DR

This paper proves the integrality of maps from Manin symbols to zeta elements in modular curves, removing previous restrictions and advancing Iwasawa theory applications.

Contribution

It introduces modifications of Manin symbols that generate homology in primitive eigenspaces and proves their integrality, improving prior results in Iwasawa theory.

Findings

Generated homology in primitive eigenspaces using modified symbols

Proved integrality of maps from Manin symbols to zeta elements

Removed the need to invert p in Iwasawa-theoretic conjectures

Abstract

We consider modifications of Manin symbols in first homology groups of modular curves with p-adic integer coefficients for an odd prime p. We show that these symbols generate homology in primitive eigenspaces for the action of diamond operators, with a certain condition on the eigenspace that can be removed on Eisenstein parts. We apply this to prove the integrality of maps taking compatible systems of Manin symbols to compatible systems of zeta elements. In the work of the first two authors on an Iwasawa-theoretic conjecture of the third author, these maps are constructed with certain bounded denominators. As a consequence, their main result on the conjecture was proven after inverting p, and the results of this paper allow one to remove this condition.