Modular Curves

Abstract

In this chapter, we translate the result in the latter part of the previous chapter into the language of schemes, and at the same time, we sketch a proof of the vertical control theorem in the p-ordinary case of elliptic modular forms. There are several different proofs of the vertical control theorem: (1) Through the moduli theory of elliptic curves; this is what we show (Ann. Sci. Ec. Norm. Sup. 19 (1986) and [GME] Chapter 3); (2) Through the study of topological cohomology groups and Jacobians of modular curves. This way has the advantage of simultaneously producing at the same time Galois representations into GL2(II), where II is a quotient of the universal p-ordinary Hecke algebra. The ring II could be large and may be free of finite type over W[[X]] (Inventiones 85 (1986)); (3) Through the theory of p-adic Eisenstein measures and p-adic Rankin convolution theory. This method was found by A. Wiles in [Wi] and is presented in the elliptic modular case in my book [LFE] in Chapter 7; (4) As an application of the identification of Hecke algebras and universal Galois deformation rings at many different weights (done by Wiles and Taylor). This method is presented in my book [MFG] 5.3.5. KeywordsModular FormElliptic CurveElliptic CurfGroup SchemeEisenstein SeriesThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.