Jacquet–Langlands–Shimizu correspondence for theta lifts to GSp(2) and its inner forms I: An explicit functorial correspondence

Abstract

As was first essentially pointed out by Tomoyoshi Ibukiyama, Hecke eigenforms on the indefinite symplectic group GSp(1,1) or the definite symplectic group GSp^*(2) over ℚ right invariant by a (global) maximal open compact subgroup are conjectured to have the same spinor L-functions as those of paramodular new forms of some specified level on the symplectic group GSp(2) (or GSp(4)). This can be viewed as a generalization of the Jacquet–Langlands–Shimizu correspondence to the case of GSp(2) and its inner forms GSp(1,1) and GSp^*(2).

In this paper we provide evidence of the conjecture on this explicit functorial correspondence with theta lifts: a theta lift from GL(2) × B^× to GSp(1,1) or GSp^*(2) and a theta lift from GL(2) × GL(2) (or GO(2,2)) to GSp(2). Here B denotes a definite quaternion algebra over ℚ. Our explicit functorial correspondence given by these theta lifts are proved to be compatible with archimedean and non-archimedean local Jacquet–Langlands correspondences. Regarding the non-archimedean local theory we need some explicit functorial correspondence for spherical representations of the inner form and non-supercuspidal representations of GSp(2), which is studied in the appendix by Ralf Schmidt.