Differential Equations Associated with Nonarithmetic Fuchsian Groups

We describe globally nilpotent differential operators of rank 2 defined over a number field whose monodromy group is a nonarithmetic Fuchsian group. We show that these differential operators have an S-integral solution. These differential operators are naturally associated with Teichmüller curves in genus 2. They are counterexamples to conjectures by Chudnovsky–Chudnovsky and Dwork. We also determine the field of moduli of primitive Teichmüller curves in genus 2, and an explicit equation in some cases. Let L be a Fuchsian differential operator of order 2 defined over a number field K. In the literature, one finds several conjectures which connect that L “comes from geometry”, is globally nilpotent, or admits an integral solution. Here “coming from geometry” could mean, for example, that L is a direct factor of the Picard–Fuchs differential equation of a family of curves. The most famous of these is Grothendieck’s p-curvature conjecture which says that every globally nilpotent differential equation comes from geometry. Another conjecture says that if L admits an integral solution then L comes from geometry. We refer to § 5 for definitions, and to [1], [2], [6] and [15] for partial results and precise formulations of the conjectures. In this paper, rather than proving a version of these conjectures, we show the existence of integral solutions of a certain interesting class of differential equations which come from geometry. Here a solution u is integral if there exists a finite set of primes S such that the coefficients of u are in the ring OS ⊂ K of S-integral elements. For hypergeometric differential equations the existence of an integral solution is well understood. Differential equations with 4 singularities which admit an integral solution are very rare. Zagier ([6], [17], § 2.4) found in a huge computer search essentially only 6 of such differential operators with Q-coefficients. All of these are pullbacks of a hypergeometric differential operators, and are associated to families of elliptic curves. Other known examples of 2nd order differential equations with an integral solution are associated to elliptic K3-surfaces ([5]). This includes the differential equations that came up in the proof of the transcendence of ζ(2) and ζ(3). The differential equations with integral solution we consider in this paper have 5 singularities, and are not the pullback of a hypergeometric differential equation. Our differential equations also come from geometry, though they are of a different nature. Namely, they are the uniformizing differential equations of Teichmüller curves in genus 2. These curves were discovered by Calta and by McMullen and intensely studied from a complex-analytic point of view. This paper starts to explore arithmetic aspects of Teichmüller curves. For an introduction to Teichmüller curves, we refer to § 1. We restrict to Teichmüller curves parameterizing curves of genus 2. Let C be such a Teichmüller curve. 2000 Mathematics Subject Classification. Primary 14H25; Secondary 32G15, 12H25.