Real components of modular curves
Number Theory
2011-08-17 v1 Algebraic Geometry
Abstract
We study the real components of modular curves. Our main result is an abstract group-theoretic description of the real components of a modular curve defined by a congruence subgroup of level N in terms of the corresponding subgroup of SL_2(Z/NZ). We apply this result to several families of modular curves (such as X_0(N), X_1(N), etc.) to obtain formulas for the number of real components. Somewhat surprisingly, the multiplicative order of 2 modulo N has a strong influence in many cases: for instance, if N is an odd prime then the real locus of X_1(N) is connected if and only if -1 and 2 generate (Z/NZ)^*.
Keywords
Cite
@article{arxiv.1108.3131,
title = {Real components of modular curves},
author = {Andrew Snowden},
journal= {arXiv preprint arXiv:1108.3131},
year = {2011}
}
Comments
33 pages, 4 tables