The semi-simple zeta function of quaternionic Shimura varieties
This valuable monograph is most adequately described by its preface and parts of the author's introduction. The aim of this monograph is the study of the Shimura variety \(\text{Sh}_{D,C}\) attached to the multiplicative group of a quaternion division algebra \(D\) over a totally real number field \(F\) and a sufficiently small open compact subgroup \(C\) of \(D^\times (\mathbb{A}_f)\). For every rational prime \(p\) which is unramified in \(F\) and such that \(C= C_p\times C^p\) where \(C_p\) is a maximal compact subgroup of \(D^\times (\mathbb{Q}_p)\) and \(C^p\) is a subgroup of \(D^\times (\mathbb{A}_f^p)\), the semisimple local zeta-function \(Z_p^{ss} (\text{Sh}_{D,C},s)\) is expressed as a product of integral powers of semisimple local \(L\)-functions of automorphic representations of \(D^\times (\mathbb{A})\). In an appendix a mistake occuring in the constructions of Langlands and Rapoport needed to state their conjecture on the reduction of a Shimura variety in the most general case is corrected, and the conjecture is restated in a slightly stronger form. It is proved for \(\text{Sh}_{D,C}\) up to a question concerning the reduction of Abelian varieties of CM-type, which is discussed in another appendix. If, however, either the quaternion algebra \(D\) is totally indefinite or the condition for the presence of good reduction which is part of the conjecture is satisfied, the proof for \(\text{Sh}_{D,C}\) is complete. In the latter of these two cases we obtain the expected expression for the local factor of the Hasse-Weil zeta-function in terms of local factors of automorphic \(L\)-functions. We now give a short survey of the contents of the present monograph. Its first part comprising \S\S 1-7 is devoted to the construction and detailed study of a reduction of the Shimura variety attached to \(D^\times\) and \(C\) at a certain place \(\wp\). Apart from the totally indefinite case, this variety is not of PEL type. In \S 1 we recall the main idea of its construction [cf. \textit{P. Deligne}, Travaux de Shimura. Séminaire Bourbaki 1970/71, Exposé No. 389, Lect. Notes Math. 244, 123-165 (1971; Zbl 0225.14007)], namely the fact that there are embeddings as open and closed subschemes into Galois twists of various Shimura varieties which are coarse moduli schemes parameterizing Abelian varieties with some additional structure. Moreover, although these embeddings are defined only over certain finite extensions of the original reflex field, for a given place \(\wp\), the data \(G\) and \(h\) defining such a Shimura variety can be chosen so that this extension has an embedding into the local field \(E\). In \S 2 we extend the moduli problem to the category of \(O_E\)-schemes and show that it admits a coarse moduli scheme \(\widetilde{\text{Sh}}_{G,C}\) which is proper over \(O_E\). The main point is to provide a suitable replacement of the trace condition of Deligne (loc. cit.). In \S 3 we study the completion of the local ring of this moduli scheme with respect to the étale topology at a geometric point in \(\widetilde{\text{Sh}}_{G,C} (\mathbb{F})\), and we show that the geometric special fibre is a union of smooth projective varieties which intersect transversally. The corresponding results of Zink in a special case have served as a guide to find the correct assertions. In \S 4 we classify the set of \(\mathbb{F}\)-valued points of the moduli problem up to isogeny. At this, unless \(D\) is totally indefinite or \(\widetilde{\text{Sh}}_{G,C}\) is smooth, there appear obstructions for the existence of those isogeny types which cannot be obtained by reducing special points of the general fibre. These obstructions which complicate the situation are discussed in a somewhat more general setting in Appendix A. Roughly speaking the problem is the following. Consider a complex Abelian variety \(A_0\) of CM-type which is defined over a number field \(L\subset \mathbb{C}\) and has good reduction \(A\) over the algebraic closure of the residue class field of a prime of \(L\) dividing \(p\). Then \(H= H_1(A_0,\mathbb{Q})\) admits a canonical embedding into the rational Tate module \(\widehat{V}^p(A)\). If an endomorphism of \(A\) does not come from an endomorphism of \(A_0\), it does not respect \(H\). The question arises whether there are other ``compatibilities'' between \(H\) and the action of \(\text{End}^0A\) on \(\widehat{V}^p(A)\). This question is made precise under certain restrictive assumptions in \S A1. The discussion in \S A2 and \S A3 shows that an affirmative answer would ensure the existence of ``all possible isogeny classes'' in \(\widetilde{\text{Sh}}_{G,C} (\mathbb{F})\). In \S 5 we describe the set of points within a fixed isogeny class together with the action of the Frobenius automorphism over \(\kappa(\wp)\) and with the information needed in \S 3 to determine the local structure at a given point. Using the embedding considered in \S 1 the results obtained for the Shimura variety attached to \(G\) can be translated into corresponding statements concerning the Shimura variety for \(D^\times\). This is done in \S 6. In particular, we obtain a projective flat \(O_E\)-scheme \(\widetilde{\text{Sh}}_{D,C}\) which satisfies the condition (PgD) of \textit{M. Rapoport} and \textit{T. Zink} [Invent. Math. 68, 21-101 (1982; Zbl 0498.14010)]. Hence we can use the results of loc. cit. which turn the description of the local structure obtained in \S 3 into a description of the sheaves of vanishing cycles. At this point the main result of the first part is achieved. It is summarized in Theorem 6.6. In the remainder of \S 6 we compare our description of the set of \(\mathbb{F}\)-valued points of \(\widetilde{\text{Sh}}_{D,C}\) with the general conjecture of Langlands and Rapoport which is reviewed in Appendix B. Due to a mistake in the constructions in [\textit{R. P. Langlands} and \textit{M. Rapoport}, J. Reine Angew. Math. 378, 113-220 (1987; Zbl 0615.14014)] preceding the statement of the conjecture, we have to give a new definition for a quasi-motivic groupoid replacing the definition of the quasi-motivic gerb (loc. cit.). This error has independently been found by \textit{M. Pfau} [The reduction of connected Shimura varieties at primes of good reduction. Dissertation, University of Michigan (1993)] who proposes a different solution to the problem. Note that we cannot work with the pseudo-motivic gerb (for which the mistake does not occur) since our Shimura variety does not have rational weight unless \(D\) is totally indefinite. Moreover, we formulate the conjecture more precisely and under slightly weaker assumptions, so that it applies to our situation. It turns out that \(\widetilde{\text{Sh}}_{D,C}\) has the properties predicted by this conjecture if and only if all of the obstructions for the existence of isogeny types mentioned above vanish. The first part is completed in \S 7 where the assertions of Theorem 6.6 are translated into a description of the set of points with values in any finite extension \(\kappa|\kappa(\wp)\) together with the function provided by the alternating sum of the traces of the Frobenius automorphism over \(\kappa\) acting on the stalks of the sheaves of vanishing cycles. In the second part, which consists of \S\S 8-11, we establish the expression for the semisimple local zeta-function in terms of semisimple automorphic \(L\)-functions asserted above. In \S 8 we use the description obtained in \S 7 to derive a formula for the semisimple traces which appear as coefficients in a power series expansion of the logarithm of the semisimple local zeta-function relating them to certain orbital and twisted orbital integrals. The procedure to obtain this formula is the same as that of \textit{M. Rapoport} [Math. Ann. 279, 673-697 (1988; Zbl 0617.14017)] which in turn is based on the method used by \textit{R. E. Kottwitz} [Math. Ann. 269, 287-300 (1984; Zbl 0533.14009)] in the case of a hyperspecial subgroup \(C_p\). In \S 9 we compare the twisted orbital integrals occurring in this formula with the orbital integrals of certain explicitly defined functions in the Hecke algebra \({\mathcal H} (D^\times (\mathbb{Q}_p)//C_p)\). For this we use again ideas of Rapoport (loc. cit.) (although some of the details in \S 5 of loc. cit. are incorrect). In particular, we obtain a vanishing result for these twisted orbital integrals which implies that the intersection of \(\widetilde{\text{Sh}}_{D,C} (\mathbb{F})\) with an isogeny class of the moduli problem considered in \S 4 does not contribute to the semisimple traces if it does not contain the reduction of a special point of the general fibre. In the special case treated in [Rapoport, loc. cit.] this corresponds to the fact that only the supersingular isogeny class contributes to the semisimple traces. An analogous fact was conjectured in [\textit{M. Rapoport}, On the bad reduction of Shimura varieties. Automorphic forms, Shimura varieties and \(L\)-functions. Vol. II, Perspectives in Mathematics 11, 253-321 (1990; Zbl 0716.14010) (Conjecture 5.7 and the note on p. 318)]. This observation eliminates the appearance of the obstructions mentioned above in the formula obtained in \S 8. Applying the Selberg trace formula and the comparison accomplished in \S 9 to the expression obtained so far, we can identify the semisimple traces with traces of certain functions in \(C_c^\infty (D^\times (\mathbb{A})/ F^\times (\mathbb{R}))\) acting on the space of automorphic forms on \(D^\times F^\times (\mathbb{R})\setminus D^\times (\mathbb{A})\). This is done in \S 10. Finally, in \S 11 we introduce the semisimple automorphic \(L\)-functions needed to deduce the main result which is summarized in Theorem 11.6.
scientific article; zbMATH DE number 1015090