Genus
The covering X(N) → X(1) is Galois, with Galois group SL(2,N)/{1,-1}, which is equal to PSL(2,N) if N is prime. Applying the Riemann–Hurwitz formula and Gauss–Bonnet theorem, one can calculate the genus of X(N). For a prime level p ≥ 5,
where χ = 2 − 2g is the Euler characteristic, |G| = (p+1)p(p−1)/2 is the order of the group PSL(2,p), and D = π − π/2 − π/3 − π/p is the angular defect of the spherical (2,3,p) triangle. This results in a formula
Thus X(5) has genus 0, X(7) has genus 3, and X(11) has genus 26. For p equal to 2 or 3, one must additionally take into account the ramification, that is, the presence of order p elements in PSL(2,Z), and the fact that PSL(2,2) has order 6, rather than 3. There is a more complicated formula for the genus of the modular curve X(N) of any level N that involves divisors of N.
Read more about this topic: Modular Curve
Famous quotes containing the word genus:
“Methinks it would be some advantage to philosophy if men were named merely in the gross, as they are known. It would be necessary only to know the genus and perhaps the race or variety, to know the individual. We are not prepared to believe that every private soldier in a Roman army had a name of his own,—because we have not supposed that he had a character of his own.”
—Henry David Thoreau (1817–1862)