Shimura curves with many uniform dessins

Abstract

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.

Geometric point-circle pentagonal geometries from Moore graphs

“…Note that Γ(7, 2, 5) is a non-arithmetic triangle group [27], and that it is maximal with respect to inclusion [25]. By Theorem 1 in [11], two groups G and G , contained 48 10 39 21 30 17 15 36 9 7 5 8 41 40 23 36 37 16 26 27 22 20 25 15 18 31 5 37 2 4 29 25 10 39 18 28 8 35 36 23 Figure 3: A drawing in a non-orientable surface of genus 57 (identification along the border according to the labelling of the points). The edges colored magenta give the pentagonal geometry of order (7,7).…”

Fields of definition of uniform dessins on quasiplatonic surfaces

“…For the first claim about the role of ∆ 0 (p j ) as norm 1 subgroup of the intersection M 0 ∩ M j of two maximal orders M 0 and M j one may consult [10] to see that these orders correspond to two vertices which are at distance j from each other in the Bruhat-Tits tree.…”

“…In [10] we studied under which conditions a surface S contains different uniform Belyi functions of a given type (r, s, t). This is equivalent to determine when the uniformising group K of S is contained in different triangle groups of that signature.…”

“…In this case, inclusions between triangle groups always induce new uniform dessins, so one may ask the more interesting question of when and how many different uniform dessins of the same signature may exist on one curve. The answer is given in [10]: an unexpected large number of different uniform dessins of the same signature exists if and only if Γ is contained in a congruence subgroup of an arithmetic triangle group ∆ , for details see Section 3.…”

“…The present paper is in some sense a continuation of [10]. We determine fields of moduli and some fields of definition of the curves involved and of these exotic uniform dessins -at least in the case of Shimura curves whose surface groups are principal congruence subgroups Γ = ∆(p n ) of prime power level in an arithmetic triangle group ∆ .…”