On central critical values of the degree four \(L\)-functions for \(\mathrm{GSp}(4)\): the fundamental lemma. III. (English) Zbl 1303.11059
Mem. Am. Math. Soc. 1057, v, 134 p. (2013).
This article is the continuation of the authors’ long term project of establishing S. Böcherer’s conjecture [Math. Gottingensis, Schriftenr. Sonderforschungsbereichs Geom. Anal. 68, 36 S. (1986; Zbl 0593.10025)]. The conjecture relates the central critical value of the twisted spinor \(L\)-function of an automorphic representation of \(\mathrm{GSp}_4\) with a period integral of forms in the representation. It is in the same spirit of the other well known conjectures relating period integrals and \(L\)-values (Gan-Gross-Prasad conjecture, Ichino-Ikeda conjecture, etc.) and is also motivated by the fundamental work of J. L. Waldspurger [J. Math. Pures Appl. (9) 60, 375–484 (1981; Zbl 0431.10015); Compos. Math. 54, 173–242 (1985; Zbl 0567.10021)].
The authors use the relative trace formula approach to establish the conjecture. Previously, the authors developed three (conjectured) relative trace formulas, and established the fundamental lemma for the unit Hecke function. (The fundamental lemma is an equality of two local relative orbital integrals.) In this article the authors make another important progress on two of the formulas and establishe the fundamental lemma for the general Hecke functions. This boils down to a difficult problem of explicit comparison of orbital integrals. To simplify the problem the authors make an ingenious use of the theory of Macdonald polynomials. They obtain a relation between Whittaker models and Bessel models, and used it to reduce the issue to the proof of identities of orbital integrals of some base elements in the Hecke algebra. The rest is tour de force.
The authors use the relative trace formula approach to establish the conjecture. Previously, the authors developed three (conjectured) relative trace formulas, and established the fundamental lemma for the unit Hecke function. (The fundamental lemma is an equality of two local relative orbital integrals.) In this article the authors make another important progress on two of the formulas and establishe the fundamental lemma for the general Hecke functions. This boils down to a difficult problem of explicit comparison of orbital integrals. To simplify the problem the authors make an ingenious use of the theory of Macdonald polynomials. They obtain a relation between Whittaker models and Bessel models, and used it to reduce the issue to the proof of identities of orbital integrals of some base elements in the Hecke algebra. The rest is tour de force.
Reviewer: Zhengyu Mao (Newark)
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F46 | Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms |
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