Real Components of Modular Curves
نویسنده
چکیده
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 SL2(Z/NZ). We apply this result to several families of modular curves (such as X0(N), X1(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 X1(N) is connected if and only if −1 and 2 generate (Z/NZ)×.
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