On \(p\)-adic \(L\)-functions for \(\operatorname{GL}(2)\times\operatorname{GL}(3)\) via pullbacks of Saito--Kurokawa lifts

We build a one-variable \(p\)-adic \(L\)-function attached to two Hida families of ordinary \(p\)-stabilised newforms \(\mathbf{f}\), \(\mathbf{g}\), interpolating the algebraic part of the central values of the complex \(L\)-series \(L(f \otimes \textrm{Ad}(g), s)\) when \(f\) and \(g\) range over the classical specialisations of \(\mathbf{f}\), \(\mathbf{g}\) on a suitable line of the weight space. The construction rests on two major results: an explicit formula for the relevant complex central \(L\)-values, and the existence of non-trivial \(\Lambda\)-adic Shintani liftings and Saito--Kurokawa liftings studied in a previous work by the authors. We also illustrate that, under an appropriate sign assumption, this \(p\)-adic \(L\)-function arises as a factor of a triple product \(p\)-adic \(L\)-function attached to \(\mathbf{f}\), \(\mathbf{g}\), and \(\mathbf{g}\).