p-adic L-functions via local–global interpolation: the case of

Abstract

Let F be a totally real field, and let $E/F$ be a CM quadratic extension. We construct a p-adic L-function attached to Hida families for the group $\mathrm {GL}_{2/F}\times \mathrm {Res}_{E/F}\mathrm {GL}_{1}$ . It is characterized by an exact interpolation property for critical Rankin–Selberg L-values, at classical points corresponding to representations $\pi \boxtimes \chi $ with the weights of $\chi $ smaller than the weights of $\pi $ . Our p-adic L-function agrees with previous results of Hida when $E/F$ splits above p or $F=\mathbf {Q}$ , and it is new otherwise. Exploring a method that should bear further fruits, we build it as a ratio of families of global and local Waldspurger zeta integrals, the latter constructed using the local Langlands correspondence in families. In an appendix of possibly independent recreational interest, we give a reality-TV-inspired proof of an identity concerning double factorials.

THE p-ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES, II: NONSPLIT PRIMES

“…It should be L p,α (σ E ) ∈ O(Y ) b (with the interpolation property being correct for the choice of additive character ψ p as in Theorem A). For a correct discussion of the ring of rationality of L p,α (σ E ), within the context of a generalized construction, see [12,Corollary 4.5.4].…”