Modular curves over Spec Z are covered in GEOMETRIC MODULAR FORMS AND ELLIPTIC CURVES by Haruzo Hida.
arXiv:1003.1935 The Langlands-Kottwitz approach for the modular curve from arXiv Front: math.AG by Peter Scholze We show how the Langlands-Kottwitz method can be used to determine the local factors of the Hasse-Weil zeta-function of the modular curve at places of bad reduction. On the way, we prove a conjecture of Haines and Kottwitz in this special case.
http://mathoverflow.net/questions/51147/what-objects-do-the-cusps-of-modular-curve-classify
arXiv:0909.0714 Higher order modular forms and mixed Hodge theory from arXiv Front: math.AG by Ramesh Sreekantan In this paper we introduce a certain space of higher order modular forms of weight 0 and show that it has a Hodge structure coming from the geometry of the fundamental group of a modular curve. This generalizes the usual structure on classical weight 2 forms coming from the cohomology of the modular curve. Further we construct some higher order Poincare series to get higher order higher weight forms and using them we define a space of higher weight, higher order forms which has a mixed Hodge structure as well.
arXiv:1209.0046 The Geometry of Hida Families and \Lambda-adic Hodge Theory from arXiv Front: math.NT by Bryden Cais We construct \Lambda-adic de Rham and crystalline analogues of Hida’s ordinary \Lambda-adic etale cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of \Q_p, we prove appropriate finiteness and control theorems in each case. We then employ integral p-adic Hodge theory to prove \Lambda-adic comparison isomorphisms between our cohomologies and Hida’s etale cohomology. As applications of our work, we provide a “cohomological” construction of the family of (\phi,\Gamma)-modules attached to Hida’s ordinary \Lambda-adic etale cohomology by Dee, and we give a new and purely geometric proof of Hida’s finitenes and control theorems. We are also able to prove refinements of theorems of Mazur-Wiles and of Ohta; in particular, we prove that there is a canonical isomorphism between the module of ordinary \Lambda-adic cuspforms and the part of the crystalline cohomology of the Igusa tower on which Frobenius acts invertibly.
arXiv:1205.5896 Approximate computations with modular curves fra arXiv Front: math.AG av Jean-Marc Couveignes, Bas Edixhoven This article gives an introduction for mathematicians interested in numerical computations in algebraic geometry and number theory to some recent progress in algorithmic number theory, emphasising the key role of approximate computations with modular curves and their Jacobians. These approximations are done in polynomial time in the dimension and the required number of significant digits. We explain the main ideas of how the approximations are done, illustrating them with examples, and we sketch some applications in number theory.
nLab page on Modular curve