Generalized Humbert curves

Abstract

In this note we consider a certain class of closed Riemann surfaces which are a natural generalization of the so called classical Humbert curves. They are given by closed Riemann surfaces S admitting H ∼ = Z k 2 as a group of conformal automorphisms so that S/H is an orbifold of signature (0, k + 1; 2, . . . , 2). The classical ones are given by k = 4. Mainly, we describe some of its generalities and provide Fuchsian, algebraic and Schottky descriptions.

CONJUGACY CLASSES OF AUTOMORPHISMS p-GROUPS

“…If p = 2 and n = 4, under the above conditions, we have (see [1]) that S is a closed Riemann surface of genus 5 for which Aut(S) = H. If we set…”

Non-hyperelliptic Riemann surfaces with real field of moduli but not definable over the reals

“…In [2] it was noted that H < Aut(S λ1,λ2,λ3 ) is the unique subgroup satisfying that H ∼ = Z 5 2 and S λ1,λ2,λ3 /H is an orbifold of signature (0, 6; 2, 2, 2, 2, 2, 2). We may identify S λ1,λ2,λ3 /H with the Riemann sphere with conical points of order 2 at the values ∞, 0, 1, λ 1 , λ 2 and λ 3 .…”

“…Let us consider the extended Möbius transformation η(z) = λ 1 z The transformation η defines an anticonformal involution of the orbifold C λ1,λ2 /H. As C λ1,λ2 is the homology cover of C λ1,λ2 /H (see [2]), it is uniformized by the derived subgroup of a Fuchsian group uniformizing the orbifold C λ1,λ2 /H. It follows that the transformation η lifts to an anticonformal automorphism of C λ1,λ2 (see also [6]).…”

Edmonds maps on Fricke-Macbeath curve

“…If S is the Fricke-Macbeath curve, then there is a regular branched cover Q : S → C having deck group G ∼ = Z ; a group of conformal automorphisms of S. Then there exists a set of generators of H, say a 1 ,..., a 6 , so that the only elements of H acting with fixed points are these and a 7 = a 1 a 2 a 3 a 4 a 5 a 6 . In [4,5] it was noted that S corresponds to the generalized Fermat curve of type (2, 6) (also called the homology cover of S/H)…”