On the non-existence of exceptional automorphisms on Shimura curves

Details On the non-existence of exceptional automorphisms on Shimura curves On the non-existence of exceptional automorphisms on Shimura curves

2008-04-27

arxiv.org/abs/0804.4276v2

Mathematics

Number Theory

Title correction. 15 pages. This is an expanded version of the second half of version 1 of math/0604099

Scientific paper

10.1112/blms/bdn012

We study the group of automorphisms of Shimura curves $X_0(D, N)$ 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, $\Aut (X_0(D, N))$ is a 2-elementary abelian group which contains the group of Atkin-Lehner involutions $W_0(D, N)$ as a subgroup of index 1 or 2. It is conjectured that $\Aut (X_0(D, N)) = W_0(D, N)$ 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 $p\mid D$.

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