We construct a Teichmüller curve uniformized by a Fuchsian triangle group commensurable to .m; n; 1/ for every m; n Ä 1. In most cases, for example when m ¤ n and m or n is odd, the uniformizing group is equal to the triangle group .m; n; 1/. Our construction includes the Teichmüller curves constructed by Veech and Ward as special cases. The construction essentially relies on properties of hypergeometric differential operators. For small m, we find billiard tables that generate these Teichmüller curves. We interpret some of the so-called Lyapunov exponents of the Kontsevich-Zorich cocycle as normalized degrees of a natural line bundle on a Teichmüller curve. We determine the Lyapunov exponents for the Teichmüller curves we construct.
Abstract. We give an explicit description of the stable reduction of superelliptic curves of the form y n = f (x) at primes p whose residue characteristic is prime to the exponent n. We then use this description to compute the local L-factor and the exponent of conductor at p of the curve.
Let G = Dp be the dihedral group of order 2p, where p is an odd prime. Let k an algebraically closed field of characteristic p. We show that any action of G on the ring k[[y]] can be lifted to an action on R [[y]], where R is some complete discrete valuation ring with residue field k and fraction field of characteristic 0.
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
We study indigenous bundles in characteristic p > 0 with nilpotent p-curvature, and show that they corresponds to simpler objects which we call deformation data. We consider the existence problem of a certain class of deformation data, which arise from the reduction of Belyi maps from characteristic zero to characteristic p.
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