Sur quelques propriétés des algèbres de Hecke quaternioniques

Terracini, Lea

Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 3, pp. 677-700.

Abstract

Let $B$ be an indefinite quaternion algebra over $\mathbf{Q}$, of discriminant divisible by a prime $p$. We introduce the space of quaternionic automorphic forms of level $p^{s}$ and the algebra of Heche operators acting on it. By making use of the Jacquet-Langlands correspondence we show that this algebra is a quotient of a classical Hecke algebra (without the $T_{p}$ operator). We deduce that the quaternionic Hecke algebra is free of finite rank over $\mathbf{Z}$ and that it is compatible with base change.

Riassunto

Sia $B$ un'algebra di quaternioni indefinita su $\mathbf{Q}$ di discriminante divisibile per un primo $p$. Introduciamo lo spazio delle forme automorfe quaternioniche di livello $p^{s}$ e l'algebra degli operatori di Hecke che vi agisce. Utilizzando la corrispondenza di Jacquet-Langlands mostriamo che quest'algebra è un quoziente di un'algebra di Hecke classica (privata dell'operatore $T_{p}$). Ne deduciamo proprietà di finitezza e di compatibilità per cambiamento di base per l'algebra di Hecke quaternionica.

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