Editing Modular curve (section)

===Genus zero===
In general a '''modular function field''' is a [[Function field of an algebraic variety|function field]] of a modular curve (or, occasionally, of some other [[moduli space]] that turns out to be an [[irreducible variety]]).  [[Genus (mathematics)|Genus]] zero means such a function field has a single [[transcendental function]] as generator: for example the [[J-invariant|j-function]] generates the function field of ''X''(1) = PSL(2, '''Z''')\'''H'''*.  The traditional name for such a generator, which is unique up to a [[Möbius transformation]] and can be appropriately normalized, is a '''Hauptmodul''' ('''main''' or '''principal modular function''', plural '''Hauptmoduln''').

The spaces ''X''<sub>1</sub>(''n'') have genus zero for ''n''&nbsp;=&nbsp;1, ..., 10 and ''n'' = 12.  Since each of these curves is defined over '''Q''' and has a '''Q'''-rational point, it follows that there are infinitely many rational points on each such curve, and hence infinitely many elliptic curves defined over '''Q''' with ''n''-torsion for these values of ''n''.  The converse statement, that only these values of ''n'' can occur, is [[Mazur's torsion theorem]].