The semi-simple zeta function of quaternionic Shimura varieties

Abstract

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Mod $p$ points on Shimura varieties of abelian type

“…[Re,§B3]). We remark that since conjugation by µ h (−1) is a Cartan involution of G, and w h is central, I ξ∞ = G * R , the (inner) form of G R with compact adjoint group.…”

Hilbert modular forms and the Ramanujan conjecture

“…To prove RC at the place v, it is enough to choose any totally real quadratic extension K of F . Then, defining B over K as above, one proves RC, by the method here, at each place of K for the base change π K of π from F to K. But it is easy to see that RC holds for π K at a place w of K iff it holds for π at the place of F under w. Thus, RC may be proved for all Hilbert modular forms which satisfy the congruence condition at infinity by a uniform method which reduces the problem to the calculation of [Re1] and Shahidi's estimate.…”

“…By the l-adic Cebotarev Theorem ( [Se]), it suffices to show that the semisimplification of the Galois action on H r (Sh B,W , Q l )(π ′ f,W )|L is a multiple of R l (JL(π ′ )). But, up to notation and the base change to L, this is given by Theorem 11.6 of [Re1], and by the main theorem of [BrLa] in the non-compact case. See Section 5.3 for some explicit review of the zeta function.…”

On the zeta function of quaternionic Shimura varieties

“…This is a stronger version of Theorem 11.6 in [9]. In fact, let C ⊂ D × (A f ) be a sufficiently small open compact subgroup containing a maximal compact subgroup of D × (Q p ), and consider the local factor…”

“…This paper is a supplement to the monograph [9]. We consider the Shimura variety Sh D associated with the multiplicative group D × of a quaternion division algebra D over a totally real number field F .…”