Weissauer, Rainer Endoscopy for \(\mathrm{GSp}(4)\) and the cohomology of Siegel modular threefolds. (English) Zbl 1273.11089 Lecture Notes in Mathematics 1968. Berlin: Springer (ISBN 978-3-540-89305-9/pbk; 978-3-540-89306-6/ebook). xvii, 368 p. (2009). From the preface: “This volume grew out of a series of preprints which were written and circulated between 1993 and 1994. Around the same time, related work was done independently by G. Harder [“Eisensteinkohomologie und die Konstruktion gemischter Motive”, Berlin: Springer (1993; Zbl 0795.11024)] and G. Laumon [Compos. Math. 105, 267–359 (1997; Zbl 0877.11037)]. In writing this text based on a revised version of these preprints that were widely distributed in summer 1995, I finally did not pursue the original plan to completely reorganize the original preprints. After the long delay, one of the reasons was that an overview of the results is now available [Astérisque 302, 67–150 (2005; Zbl 1097.11027)]. Instead I tried to improve the presentation modestly, in particular by adding cross-references wherever I felt this was necessary. In addition, Chaps. 11 and 12 and Sects. 5.1, 5.4, and 5.5 were added; these were written in 1998.I will give a more detailed overview of the content of the different chapters below (pp. v–xii). Before that I should mention that the two main results are the proof of Ramanujan’s conjecture for Siegel modular forms of genus 2 for forms which are not cuspidal representations associated with parabolic subgroups (CAP representations), and the study of the endoscopic lift for the group \(\mathrm{GSp}(4)\). Both topics are formulated and proved in the first five chapters assuming the stabilization of the trace formula. All the remaining technical results, which are necessary to obtain the stabilized trace formula, are presented in the remaining chapters.”The author’s preface is highly recommended to read as it gives a very concise overview of the methods and results of this monograph and their relations to previous results. Its size, however, is too large to reproduce here, so there are only given the chapter headlines of the table of contents plus some parts of the chapter abstracts.1) An Application of the Hard Lefschetz Theorem. (Review of Eichler integrals; Automorphic representations; CAP and toric representations; The \(l\)-adic sheaves; Lefschetz maps; Weak Ramanujan; Residues of \(L\)-series). In the later parts of this chapter, we use the hard Lefschetz theorem to show that an irreducible cuspidal automorphic representation \(\pi=\pi_\infty \pi_{\text{fin}}\) of the group \(\mathrm{GSp}(4,\mathbb A)\), whose Archimedean component \(\pi_\infty\) belongs to the discrete series, gives a contribution to a cohomology group of some canonical associated locally constant sheaves on Siegel modular threefolds only if the cohomology degree is 3, provided \(\pi\) is not a cuspidal representation associated with parabolic subgroup (CAP representation). In other words, under the action of the adele group \(\mathrm{GSp}(4,\mathbb A_{\text{fin}})\), all irreducible constituents occur in the middle degree, if one discards so-called CAP representations. Since CAP representations are well understood for the group \(\mathrm{GSp}\), this result is important for the analysis of the supertrace of Hecke operators acting on the cohomology of Siegel modular threefolds, and hence for the proof of the generalized Ramanujan conjecture for holomorphic Siegel modular forms of genus 2 and weight 3 or more.2) CAP Localization. (Standard parabolic subgroups; The adelic reductive Borel-Serre compactification; Fixed points; Lefschetz numbers; Computation of an orbital integral; Elliptic traces; The Satake transform; Automorphic representations; The discrete series case).In this chapter, we express the \(\pi\)-isotypic Lefschetz numbers of Hecke operators acting on the cohomology of symmetric spaces \(S_K(G)\) attached to reductive groups \(G\) in terms of so-called elliptic traces \(T_\ell\), provided the underlying representation \(\pi\) is not a cuspidal representation associated with a parabolic subgroup (CAP representation) of \(G(\mathbb A)\). In the following two chapters we derive from these formulas all the essential information required.3) The Ramanujan Conjecture for Genus 2 Siegel Modular Forms. (Results obtained by Kottwitz, Milne, Pink, and Shpiz, Stabilization; Destabilization; The topological trace formula; \((G,M)\)-regularity removed; CAP localization revisited; The fundamental lemma).In this chapter we apply the results obtained in Chaps. 1 and 2 and work of Kottwitz in the special case of the symplectic group of similitudes GSp(4). The main result obtained in this chapter is the proof of the Ramanujan conjecture for \(\mathrm{GSp}(4)\) for cohomological automorphic forms which are not cuspidal representations associated with parabolic subgroups (CAP). We will assume certain statements regarding the trace formula, the proof of which occupies Chaps. 6–10.4) Character Identities and Galois Representations Related to the Group \(\mathrm{GSp}(4)\). (Galois Representations; The trace formula applied; Rationality; Orthogonality relations; Exponents and tempered representations; The classification of local representations; The Siegel parabolic \(P\) (results obtained by Shahidi); The Klingen parabolic \(Q\) (results obtained by Waldspurger); The Borel group \(B\) (results obtained by Tadić and Rodier); The Borel group \(B\) (the nonregular reducible principal series); Results obtained by Kudla and Rallis; List of irreducible unitary discrete series representations of \(G\); Asymptotics; The character lift; Whittaker models; Theta lifts; The local theta lift; The local L-packets; The endoscopic character lift; The anisotropic endoscopic theta lift; The global situation; The anisotropic theta lift (local theory); The Jacquet-Langlands lift; The intertwining map \(b\); The \(Q\)-Jacquet module of the anisotropic Weil representation \(\Theta-(\sigma)\); Whittaker models; The Siegel parabolic; The anisotropic theta lift \(\theta-(\sigma)\) (global theory)).In this chapter we study the \(\ell\)-adic representations attached to cohomological Siegel modular forms. To understand their associated \(\ell\)-adic representations defined by the Galois action on the etale cohomology, one needs to understand the endoscopic lifts and vice versa, since for all automorphic representations in this lift the associated \(\ell\)-adic representations turn out to be smaller than what would be expected a priori.The simple version of the Lefschetz trace formula used in Chap. 3 suffers from the particular restriction that for one specific prime – the chosen Frobenius prime \(p\) – the corresponding local Hecke operators at pp always have to be deleted from consideration. As a consequence, this particular chosen Frobenius prime \(p\) together with the Archimedean place play a distinguished role. This situation in fact looks similar to the situation in Arthur’s simple trace formula. Of course this is not a mere coincidence, but is definitely forced by the failure of the strong multiplicity 1 theorem for automorphic forms on \(\mathrm{GSp}(4)\). To bypass the technical difficulties that result from this, the \(\ell\)-adic representations of the absolute Galois group on the cohomology turn out to be very helpful. With their help we analyze the endoscopic lift. This allows us to understand the failure of the strong multiplicity 1 theorem caused by it.5) Local and Global Endoscopy for \(\mathrm{GSp}(4)\). (The Local Endoscopic Lift; The Global Situation; The Multiplicity Formula; Local and Global Trace Identities; Appendix on Arthur’s Trace Formula).In this chapter we refine the global description of the endoscopic lift obtained in Corollary 4.2. The main result obtained in this chapter is Theorem 5.2, which is a special case of a global multiplicity formula conjectured by Arthur. We consider the symplectic group \(\mathrm{GSp}(4)\) over an arbitrary totally real number field \(F\). In the special case \(F=\mathbb Q\) and for irreducible automorphic representations \(\pi\) with \(\pi_\infty\) in the discrete series the proof of this theorem is given in Sects. 5.2 and 5.3. In Sect. 5.1 we explain how the local endoscopic character lift is constructed for arbitrary local base fields of characteristic zero. In Sects. 5.4 and 5.5 we explain how the Arthur-Selberg trace formula can be used to extend the results obtained in Sects. 5.2 and 5.3 to the case of arbitrary totally real number fields and arbitrary irreducible automorphic not necessarily cohomological representations \(\pi\). The proof of Theorem 5.2 is based on two ingredients: the principle of exchange and the key formula (stated in Sect. 5.3). The latter can be directly deduced from weak versions of the trace formula (see Chap. 4, Corollary 4.2, or Sect. 5.5). It deals with simultaneous changes of global representations at two places. The principle of exchange, on the other hand, deals with an exchange of the representation at one place exclusively, and its proof boils down to a special case of the Hasse-Brauer-Noether theorem (Lemma 5.4). This is based on an explicit theta lift. At this point we use [S. Kudla, S. Rallis and D. Soudry, Invent. Math. 107, 483–541 (1992; Zbl 0776.11028)], which unfortunately forces us to make the restrictive assumption that \(F\) is a totally real number field. Since we apply results obtained in that paper in a rather specific case, it is very likely that this restriction on \(F\) can be removed.6) A Special Case of the Fundamental Lemma. I. 7) A Special Case of the Fundamental Lemma II. 8) The Langlands-Shelstad Transfer Factor. (In this chapter we give an explicit formula for the Archimedean transfer factor of R. P. Langlands and D. Shelstad [Math. Ann. 278, 219–271 (1987; Zbl 0644.22005)]).9) Fundamental Lemma (Twisted Case). 10) Reduction to Unit Elements; 11) Appendix on Galois Cohomology. 12) Appendix on Double Cosets. (We discuss a double coset decomposition for the symplectic group \(\mathrm{GSp}(2n, F)\), which in the case \(n=2\) was found by M. Schröder [Thesis University of Mannheim 1993]. Reviewer: Olaf Ninnemann (Berlin) Cited in 4 ReviewsCited in 45 Documents MSC: 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11F75 Cohomology of arithmetic groups 11-02 Research exposition (monographs, survey articles) pertaining to number theory 22-02 Research exposition (monographs, survey articles) pertaining to topological groups 11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11G18 Arithmetic aspects of modular and Shimura varieties 14G35 Modular and Shimura varieties 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings Citations:Zbl 0795.11024; Zbl 0877.11037; Zbl 1097.11027; Zbl 0776.11028; Zbl 0644.22005 PDFBibTeX XMLCite \textit{R. Weissauer}, Endoscopy for \(\mathrm{GSp}(4)\) and the cohomology of Siegel modular threefolds. Berlin: Springer (2009; Zbl 1273.11089) Full Text: DOI