$p$-adic L-functions via local-global interpolation: the case of ${\mathrm{GL}}_2\times \mathrm{GU}(1)$

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 characterised by an exact interpolation property for critical Rankin-Selberg $L$-values, at classical points corresponding to representations $\pi ⊠\chi$ 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.

New title highlighting the methods. Final version, to appear in Canad. J. Math. 40 pages

References in corpus (3)

• A proof of Perrin-Riou's Heegner point main conjecture
• The universal $p$-adic Gross-Zagier formula
• The $p$-adic Gross-Zagier formula on Shimura curves, II: nonsplit primes

Cited by in corpus (2)

• The universal $p$-adic Gross-Zagier formula
• Theta cycles and the Beilinson--Bloch--Kato conjectures