Pullbacks of Eisenstein series from GU(3,3) and critical L-values for GSp(4) × GL(2)

Abhishek Saha

Vol. 246 (2010), No. 2, 435–486

DOI: 10.2140/pjm.2010.246.435

Abstract

Let F be a genus two Siegel newform and g a classical newform, both of squarefree levels and of equal weight ℓ. We prove a pullback formula for certain Eisenstein series—thus generalizing a construction of Shimura—and use this to derive an explicit integral representation for the degree eight L-function L(s,F × g). This integral representation involves the pullback of a simple Siegel-type Eisenstein series on the unitary group GU(3,3). As an application, we prove a reciprocity law—predicted by Deligne’s conjecture—for the critical special values L(m,F × g) where m ∈ ℤ with 2 ≤ m ≤ ℓ∕2 − 1.

Keywords

modular form, automorphic form, automorphic representation, L-function, special values, Siegel modular form, Eisenstein series, GSp(4), Deligne’s conjecture, pullback formula

Mathematical Subject Classification 2000

Primary: 11F46, 11F67, 11F70

Milestones

Received: 20 May 2009

Revised: 2 October 2009

Accepted: 29 October 2009

Published: 1 June 2010