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Title: Optimal Special Polygons for the Congruence Subgroups Γ0(p) and Γ0(pq)

Abstract: For a primep, we compute the minimum of m(P )over all possible special fundamental polygons (in the sense of Kulkarni)Pfor0(p), where m(P )denotes the largest denominator in the cusp set of P. This minimum valuem(Gamma(0)(p))is expressed in termsof the solution set to a certain finite system of quadratic Diophantine equations and inequalities, and can be explicitly computed with time complexity O(p(2)).From this computation, we obtain freely independent generators of Gamma(0)(p) that have 0 orpintheir(2,1)components, answering a question of Kulkarni. By an analogous argument, we establish that Gamma(0)(N)admits freely independent generators whose Frobeniusnorms satisfy O(N) for N = p or N = pq, where p and q are odd primes satisfying 0 <= |root p - root q| < root 2.

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