GLNT: p-adic companion forms for Yoshida lifts

Abstract: Coleman showed that the (k-1)st power of the theta operator q d/dq defines a map from overconvergent forms of weight 2-k and slope 0 to weight k and slope k-1. Moreover, the critical p-stabilization of a classical CM form is the image of a p-adic CM form, strengthening the fact that its Galois representation splits locally at p. In the GSp4 setting, the Galois representation of a Yoshida lift splits locally into two 2-by-2 blocks at p. In joint work in progress with Bharathwaj Palvannan, we aim to prove an analogous strengthening. The relevant theta operator arises from the last differential of the dual BGG complex. We computed its explicit effect on q-expansions for weight (k, 3), and expect that the effect for general weights to be a power of this. Using the explicit Fourier coefficients of Yoshida lifts by Hsieh--Namikawa, we show that Yoshida lifts lie in the image of this theta operator, up to choices of the p-stabilization.