A Teichmüller curve is a finite-volume Riemann surface C = H/G together with an isometric immersion C --> M_g, where M_g is the moduli space of genus-g Riemann surfaces, equipped with the Teichmüller metric. The trace field F(C) is the number field obtained by adjoining to Q the traces of the elements of G (a subgroup of SL(2,R)). It is known that F(C) has degree at most g, and C is said to be algebraically primitive if F(C) has degree g.

McMullen and Calta have independently discovered an infinite family of algebraically primitive Teichmüller curves in M_2. It is an open question whether there are infinitely many algebraically primitive Teichmüller curves in M_g for any g>2. In this talk I will present some recent theorems and computer evidence indicating there should be only finitely many algebraically primitive Teichmüller curves in M_3. This is joint work with Martin Möller.