A 2-adic control theorem for modular curves

TL;DR

This paper investigates the structure of homology modules of modular curves associated with specific congruence subgroups and establishes a control theorem describing their behavior across a sequence of levels.

Contribution

It introduces a 2-adic control theorem for the ordinary parts of homology modules of modular curves, extending understanding of their algebraic structure.

Findings

Proves a control theorem for homology modules of modular curves

Analyzes ordinary parts of homology in relation to decreasing congruence subgroups

Provides new insights into the 2-adic properties of modular curve homology

Abstract

We study the behaviour of ordinary parts of the homology modules of modular curves, associated to a decreasing sequence of congruence subgroups ${\Gamma}_1(N2^r)$ for $r \geq 2$, and prove a control theorem for these homology modules.