On the Non-existence of Exceptional Automorphisms on Shimura Curves

We study the group of automorphisms of Shimura curves X0(D,N) attached to an Eichler order of square-free level N in an indefinite rational quaternion algebra of discriminant D > 1. We prove that, when the genus g of the curve is greater than or equal to 2, Aut(X0(D,N)) is a 2-elementary abelian group which contains the group of Atkin-Lehner involutions W0(D,N) as a subgroup of index 1 or 2. It is conjectured that Aut(X0(D,N)) = W0(D,N) except for finitely many values of (D,N) and we provide criteria that allow us to show that this is indeed often the case. Our methods are based on the theory of complex multiplication of Shimura curves and the Cerednik-Drinfeld theory on their rigid analytic uniformization at primes p | D. 1. The automorphism group of Shimura curves 1.1. Congruence subgroups of PSL2(R) and automorphisms. Let Γ be a congruence subgroup of PSL2(R). As explained in [16, §4], Γ is a congruence subgroup of PSL2(R) if there exists • A quaternion algebra B/F over a totally real number field F of degree d ≥ 1, • An embedding φ : B ↪→ M2(R)×D× (d−1) ... ×D, • An integral two-sided ideal I of a maximal order O of B, such that Γ contains φ({α ∈ O : α ∈ 1 + I}). Here, we let D denote Hamilton’s skew-field over R and n : B −→ F stand for the reduced norm. We write O = {α ∈ O : n(α) = 1}. We refer the reader to [27] for generalities on quaternion algebras. Examples of congruence subgroups of PSL2(R) with F = Q will be described in detail below. Let XΓ denote the compactification of the Riemann surface Γ\H. Let N = NormPSL2(R)(Γ) denote the normalizer of Γ in PSL2(R). The group BΓ = N/Γ is a finite subgroup of Aut(XΓ). 1991 Mathematics Subject Classification. 11G18, 14G35.