p-adic L-functions for GL_2 x GU(1)

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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 $GL_{2/F} \times GL_{1/E}$. It interpolates critical Rankin--Selberg $L$-values at all classical points corresponding to representations $\pi\boxtimes \chi$ with the weights of $\chi$ smaller than the weights of $\pi$. This confirms a conjecture of the author, making related results on the $p$-adic Beilinson conjecture and its Iwasawa-theoretic variants unconditional. Our $p$-adic $L$-function agrees with a previous result of Hida when $E/F$ splits above $p$, and it is new otherwise. 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.
Original languageEnglish
StatePublished - 2021


  • math.NT
  • 11F67, 11F33


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