A compact Riemann surface of genus g > 1 has different uniform dessins d'enfants of the same type if and only if its surface group S is contained in different conjugate Fuchsian triangle groups Δ and αΔα −1 . Tools and results in the study of these conjugates depend on whether Δ is an arithmetic triangle group or not. In the case when Δ is not arithmetic the possible conjugators are rare and easy to classify. In the arithmetic case, i.e. for Shimura curves, the problem is much more complicated, but the arithmetic of quaternion algebras controls the growth of the number of uniform dessins of given type with respect to the genus. This number grows at most as O(g 1/3 ) and this bound is sharp. As a tool, localization of the quaternion algebras and the representation of p-adic maximal orders as vertices of Serre-Bruhat-Tits trees turn out to be crucial. In low genera, the results shed a surprising new light on the uniformization of some classical curves like Klein's quartic and other Macbeath-Hurwitz curves.

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A Beauville surface is a rigid surface of general type arising as a quotient of a product of curves C1, C2 of genera g1, g2 ≥ 2 by the free action of a finite group G. In this paper we study those Beauville surfaces for which G is abelian (so that G ∼ = Z 2 n with gcd(n, 6) = 1 by a result of Catanese). For each such n we are able to describe all such surfaces, give a formula for the number of their isomorphism classes and identify their possible automorphism groups. This explicit description also allows us to observe that such surfaces are all defined over Q.

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We calculate the Masur–Veech volume of the gothic locus scriptG in the stratum H(23) of genus 4. Our method is based on the use of the formulae for the Euler characteristics of gothic Teichmüller curves to determine the number of lattice points of given area. We also use this method to recalculate the Masur–Veech volumes of the Prym loci scriptP3⊂Hfalse(4false) and scriptP4⊂Hfalse(6false) in genus 3 and 4.