The existence conjecture for $p$-adic Asai $L$-functions

Let $F$ be the CM field, let $p$ be a prime, let $\boldsymbol{\rm As}^+_{\mathcal M}(\pi)$ denote the positive Asai motive attached to $\pi$, and let $\mathcal O_\pi$ be the coefficient ring. Write $F(\mu_{p^\infty})$ for the extension obtained by adjoining all $p$-power roots of unity. For a Hecke character $\phi:F^\times\backslash F^\times_{\mathbf A}\to\mathbf C^\times$, let $\widehat\phi$ denote its associated character of $\operatorname{Gal}(F(\mu_{p^\infty})/F)$, and let $\mathcal E_p$ and $\mathcal E_\infty$ be the local interpolation factors. Let $\Omega(\boldsymbol{\rm As}^+_{\mathcal M}(\pi))$ be the period appearing in the critical-value formula.

The existence conjecture for $p$-adic Asai $L$-functions. There exists an element

$$\mathscr L_p(\boldsymbol{\rm As}^+_{\mathcal M}(\pi))\in\mathcal O_\pi[[\operatorname{Gal}(F(\mu_{p^\infty})/F)]]$$

such that, for every Hecke character $\phi$ satisfying

$$\phi_\infty(x)=x^{n-\alpha}\quad\text{for some }0\leq\alpha\leq n,$$

with $\phi$ of $p$-power conductor and $\boldsymbol{\rm As}^+_{\mathcal M}(\pi)(\phi)$ critical at $s=0$, one has

$$\widehat\phi(\mathscr L_p(\boldsymbol{\rm As}^+_{\mathcal M}(\pi)))=\mathcal E_\infty(\boldsymbol{\rm As}^+_{\mathcal M}(\pi)(\phi))\mathcal E_p(\boldsymbol{\rm As}^+_{\mathcal M}(\pi)(\phi))\frac{L(0,\boldsymbol{\rm As}^+_{\mathcal M}(\pi)(\phi))}{\Omega(\boldsymbol{\rm As}^+_{\mathcal M}(\pi))}.$$

This conjecture predicts a $p$-adic interpolation of critical Asai $L$-values; the supplied text does not establish the asserted existence in general.