By Larry Joel Goldstein

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These points represent the cuspidal and the elliptic orbits of Γ. ˜ ber g is the genus of Γ. ˜ mod . 2 Examples. 1 on p. 2 is not canonical. 8 for other examples. 3 Generators and relations. The advantage of canonical fundamental domains is the resulting uniform description of all coﬁnite discrete groups. ¯ is We see in Lehner, [32], Ch. 2, p. 230, or Petersson, [42], §3, that Γ generated by πj (1 ≤ j ≤ p), εj (1 ≤ j ≤ q), and γj , ηj (1 ≤ j ≤ g), with explicitly ˜ is generated by given relations.

2 we have seen the holomorphic Eisenstein series Gk . As we do not impose a growth condition here, the modular invariant J, and also eJ , fall under this deﬁnition. Usually a function like J is called an automorphic function, and eJ is not mentioned at all. 3 for an example of a non-holomorphic form of weight 0 with eigenvalue 0 = 12 · 0(1 − 12 · 0). 7 Maass’s deﬁnition. If we take an automorphic form F of type {Γ, α, β, v} in the sense of Maass, [35], p. 3. The α+β weight is α−β, the multiplier system is v, and the eigenvalue equals α+β 2 (1− 2 ).

The exponent 2g − 2 + p + q in the last relation requires more work. Petersson discusses it in terms of multiplier systems, see [42], p. 64, and [43], p. 192, Satz 8. 1) = 2π ⎝2g − 2 + p + q − y2 v F j=1 j which can be proved by considering divisors of automorphic forms on the compact ˜ Riemann surface related to Γ\H. , Shimura, [53], Thm. 20, p. 42. 9 a choice of canonical generators for the modular group: ζ, n(1), k(π/2), and k(π/2)n(−1). 2 shows that n(1) and k(π/2) already ˜ mod . This is a general phenomenon: Γ ˜ can be generated with less gengenerate Γ erators, but with canonical generators one gets a uniform description.