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A Teichm\"uller curve is an algebraic and isometric immersion of an algebraiccurve into the moduli space of Riemann surfaces. We give the first explicitalgebraic models of Teichm\"uller curves of positive genus. Our methods arebased on the study of certain Hilbert modular forms and the use of Ahlfors'svariational formula to identify eigenforms for real multiplication on genus twoJacobians. We also present evidence that Teichm\"uller curves admit a richarithmetic geometry by exhibiting examples with small primes of bad reductionand notable divisors supported at their cusps.

Algebraic models and arithmetic geometry of Teichm\"uller curves in genus two