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Bulletin of the London Mathematical Society Advance Access originally published online on April 26, 2008
Bulletin of the London Mathematical Society 2008 40(3):363-374; doi:10.1112/blms/bdn012
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© 2008 London Mathematical Society

On the non-existence of exceptional automorphisms on Shimura curves

Aristides Kontogeorgis

Department of Mathematics
University of the Ægean
83200 Karlovassi
Samos
Greece
kontogar@aegean.gr
http://eloris.samos.aegean.gr

Victor Rotger

Escola Universitària Politècnica de Vilanova i la Geltrú
Av. Víctor Balaguer s/n
E-08800 Vilanova i la Geltrú
Spain

Received 19 March 2007. Revision received 7 November 2007.

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.

2000 Mathematics Subject Classification 11G18, 14G35.


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