Differential equations associated with nonarithmetic Fuchsian groups

Abstract

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.2000 Mathematics Subject Classification. Primary 14H25; Secondary 32G15, 12H25. This paper contains results in two direction. Firstly, we prove new results on Teichmüller curves in genus 2. Secondly, we show that the uniformizing differential equation of these Teichmüller curves have interesting arithmetic properties.A new ingredient we use for studying Teichmüller curves in genus 2 is the construction of genus-2 fibrations as double coverings of ruled surfaces (following [10]). This allows us, for example, to compute the Lyapunov exponents of C ( § 2) for all Teichmüller curves C in genus two. This extends a result of Bainbridge ([3]).Teichmüller curves in genus 2 whose generating translation surface (X, ω) has a double zero are classified by McMullen ([21], [23]). These Teichmüller curves are characterized by two invariants: the discriminant D ∈ N and, if D ≡ 1 (mod 8), the spin invariant ε ∈ Z/2Z which is the signature of a quadratic form on a certain subspace of H 1 (X, Z/2Z) ( § 1). We denote by W ε D the Teichmüller curve with discriminant D and spin invariant ε. Surprisingly, the field of moduli of W ε D depends on whether D is a square or not, even though the spin invariant may be defined the same way in both cases. Theorem 3.3. If D ≡ 1 (mod 8) is not a square, the field of moduli of the Teichmüller curves W 0 D and W 1 D is Q( √ D). Otherwise, the field of moduli of W ε D is Q. Theorem 3.3 allows to significantly simplify Bainbridge's calculation of the orbifold Euler characteristic of W ε D ([3]). It is still an open problem to determine what the W ε

Algebraic curves uniformized by congruence subgroups of triangle groups

“…This is implied by a conjecture of Chudnovsky and Chudnovsky [10,Section 7]. The Chudnovskys' conjecture is false if the group is not cocompact-this is implicit in work of McMullen and made explicit in work of Bouw and Möller [7,8]-but may still be true in the compact case. See also work of Ricker [51].…”

“…(See also Wolfart [85, §6.5] and Wohlfahrt [83].) Genus 8, (2,3,8), PGL 2 (F 7 ) and (3,3,4), PSL 2 (F 7 ). The curve X 0 (2, 3, 8; 7) has genus 0, and the triangle group ∆(2, 3, 8) arises from the quaternion algebra over Q( √ 2) ramified at the prime above 2: the map is φ(t) = t 8 .…”

Modular embeddings of Teichmüller curves

“…In Part II we show conversely, in a specific example, how to obtain from these differential equations the Hilbert modular embedding ϕ. The example that we will consider in detail is D = 17, for which the differential equations needed were computed in [5]. In Section 6 we will sketch how these were obtained, referring to that paper for the full details.…”

“…3 and §5.4.) Starting from the flat geometry definition we briefly explain the derivation of the equation of the Teichmüller curve as family of hyperelliptic curves and computation of the Picard-Fuchs differential equations, following [5].…”

The equation of the Kenyon-Smillie (2, 3, 4)-Teichmüller curve

“…Notes and references. The strategy that we use to determine the equation of the universal family has already been successfully implemented before by Bouw and Möller [BM10a], who have worked out the equations of two of the Weierstraß curves in genus two. The main challenge is to find a suitable structural result about the canonical ring of quartic hypersurfaces analogous to the one used by Bouw and Möller for genus two curves.…”