  A Teichm\"uller curve is an algebraic and isometric immersion of an algebraic
curve into the moduli space of Riemann surfaces. We give the first explicit
algebraic models of Teichm\"uller curves of positive genus. Our methods are
based on the study of certain Hilbert modular forms and the use of Ahlfors's
variational formula to identify eigenforms for real multiplication on genus two
Jacobians. We also present evidence that Teichm\"uller curves admit a rich
arithmetic geometry by exhibiting examples with small primes of bad reduction
and notable divisors supported at their cusps.
