Geodesic
p-adic L-functions via local–global interpolation: the case of $\mathrm {GL}_{2}\times \mathrm {GU}(\text{1})$
Canadian journal of mathematics, Tome 75 (2023) no. 3, pp. 965-1017

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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. 
Disegni, Daniel. p-adic L-functions via local–global interpolation: the case of $\mathrm {GL}_{2}\times \mathrm {GU}(\text{1})$. Canadian journal of mathematics, Tome 75 (2023) no. 3, pp. 965-1017. doi: 10.4153/S0008414X22000256
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     author = {Disegni, Daniel},
     title = {p-adic {L-functions} via local{\textendash}global interpolation: the case of $\mathrm {GL}_{2}\times \mathrm {GU}(\text{1})$},
     journal = {Canadian journal of mathematics},
     pages = {965--1017},
     year = {2023},
     volume = {75},
     number = {3},
     doi = {10.4153/S0008414X22000256},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X22000256/}
}
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