\(l\)-adic representations associated to modular forms over imaginary quadratic fields. I: Lifting to \(GSp_ 4(\mathbb{Q})\)
This paper is part of a series which establishes an important new link between automorphic representations and arithmetic. Namely this paper, together with the subsequent papers of \textit{R. Taylor} [On the \(l\)-adic cohomology of Siegel threefolds, Invent. Math. 114, 289-310 (1993; Zbl 0810.11034); \(l\)- adic representations associated to modular forms over imaginary quadratic fields, II; Invent. Math. 116, 619-643 (1994; Zbl 0823.11020)]shows how one may associate a \(\lambda\)-adic representation to certain modular forms over an imaginary quadratic field. By way of background, recall that if \(f\) is a holomorphic elliptic modular newform then one can associate to \(f\) a compatible system of \(\lambda\)-adic representations, obtained from the cohomology of certain sheaves on modular curves. Similar statements are true for Hilbert modular forms. However, if \(g\) is in the space of an automorphic representation over an imaginary quadratic field, no such arithmetic object had previously been found. Indeed, the corresponding quotient of hyperbolic three-space is not an algebraic variety, and no link to arithmetic is apparent. The authors establish such a link by using the theta correspondence to associate to \(g\) a holomorphic Siegel modular form \(G\) in the same near- equivalence class. This paper is concerned with the details of this correspondence, and the subsequent papers of Taylor with applying the methods of algebraic geometry to \(G\) to attach to it a compatible system of \(\lambda\)-adic representations. In the paper at hand, the authors first observe that \(GL_ 2\) over an imaginary quadratic field \(K\) is closely related to a four-variable group of orthogonal similitudes \(GO\) over \(\mathbb{Q}\). (This similitude group is disconnected, and the disconnectedness plays an important role.) There is a theta lifting of automorphic representations on \(GO\) to automorphic representations on \(GSp_ 4\). After establishing some generalities on theta liftings for similitude groups, they calculate explicitly the local unramified theta lift in the case at hand. These results are then stitched together to give global results, and in particular to show that holomorphic Siegel modular forms are produced. The applications of Taylor's analysis of the \(l\)-adic cohomology of Siegel three-folds are then sketched. The paper concludes with a detailed discussion of the underlying Langlands' philosophy.
\(l\)-adic cohomology of Siegel three-folds
scientific article; zbMATH DE number 447108
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