Klein Quartic - Quaternion Algebra Construction

Quaternion Algebra Construction

The compact Klein quartic can be constructed as the quotient of the hyperbolic plane by the action of a suitable Fuchsian group Γ(I) which is the principal congruence subgroup associated with the ideal in the ring of integers of the field where . Note the identity

exhibiting as a prime factor of 7 in the ring of integers.

The group Γ(I) is a subgroup of the (2,3,7) hyperbolic triangle group. Namely, Γ(I) is a subgroup of the group of norm 1 elements in the quaternion algebra generated as an associative algebra by the generators i,j and relations . One chooses a suitable Hurwitz quaternion order in the quaternion algebra, Γ(I) is then the group of norm 1 elements in . The least absolute value of a trace of a hyperbolic element in Γ(I) is, corresponding the value 3.936 for the systole of the Klein quartic, one of the highest in this genus.