Hecke module structure of quaternions. (English) Zbl 1040.11044

Miyake, Katsuya (ed.), Class field theory – its centenary and prospect. Proceedings of the 7th MSJ International Research Institute of the Mathematical Society of Japan, Tokyo, Japan, June 3–12, 1998. Tokyo: Mathematical Society of Japan (ISBN 4-931469-11-6/hbk). Adv. Stud. Pure Math. 30, 177-195 (2001).
Generalizing a construction of Hecke modules on supersingular points due to Mestre and Oesterlé, the author gives a construction of Hecke modules on quaternion ideals. Let \(E/k\) be a supersingular elliptic curve over the algebraic closure \(k\) of the finite prime field of characteristic \(p\), and let \(C\) be a cyclic subgroup of order \(m\). The pair \({\mathbf E}=(E,C)\) is called an enhanced elliptic curve. Then \(R :=\text{End} _{k}({\mathbf E})\) is an Eichler order (i.e. an intersection of two maximal orders) in the quaternion algebra ramified at \(p\) and infinity. The functor \(\hom_{k}({\mathbf E},-)\) provides an equivalence between the category of enhanced supersingular elliptic curves over \(k\) and the category of locally free rank one \(R\)-modules. Therefore, instead of starting with elliptic curves, the construction of Hecke modules can be based on an Eichler order \(R\) in a definite quaternion algebra. The author then proceeds to an equivalent construction in terms of Shimura curves and studies Hecke modules on Shimura curves by means of Eichler orders in quaternion algebras.
For the entire collection see [Zbl 0968.00031].


11G18 Arithmetic aspects of modular and Shimura varieties