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1 results

  • 6 - Congruence subgroups and modular curves

  • from Part I - Modular forms and their variants
  • Eric D'Hoker, University of California, Los Angeles, Justin Kaidi, Kyushu University
  • Book:
    Modular Forms and String Theory
    Print publication:
    12 December 2024, pp 121-136

  • Summary

    Congruence subgroups form a countable infinite class of discrete non-Abelian subgroups of SL(2,Z) and play a particularly prominent role in deriving the arithmetic properties of modular forms. In this chapter, we study various aspects of congruence subgroups, including their elliptic points, cusps, and topological properties of the associated modular curve. Jacobi theta-functions, theta-constants, and the Dedekind eta-function are used as examples of modular forms under congruence subgroups that are not modular forms under the full modular group SL(2,Z).