Modular and $p$-adic methods in the theory of zeta functions

For a prime $p$ and a positive integer ${m}$, zeta function $L_{F}(s)$ is considered, attached to an Hermitian modular form $F=\displaystyle\sum_{H}A(H) q^H$ on the Hermitian upper half plane $\mathcal H_{m}$ of degree $m$, where $H$ extends over all semi-integral positive definite hermitian matrices of degree $m$, i.e. $H\in \Lambda_m({\mathcal O})$ over the integers ${\mathcal O}$ of an imaginary quadratic field $K$, where $q^{H}=\exp(2\pi i\,{\rm Tr}(HZ))$. Analytic $p$ -adic continuation of their zeta functions is constructed via $p$-adic measures, bounded or growing. Previously this problem was solved for the Siegel modular forms.
Main result is stated in terms of the Hodge polygon $P_{H}(t): [0,d]\to {\mathbb R}$ and the Newton polygon $P_{N}(t)=P_{N,\,p}(t): [0,d]\to {\mathbb R}$ of the zeta function $L_{F}(s)$, with $d=4m$.
Main theorem gives a $p$ -adic analytic interpolation of the $L$ values in the form of certain integrals with respect to Mazur-type measures.
These $p$-adic measures are constructed from the Fourier coefficients of hermitian modular forms, and from certain eigenvalues of Hecke operators on the unitary group. The integrity of such measures is proven under the condition of the equality at the central point $t=d/2$ of $P_{H}(t)$ and $P_{N}(t)$. In the case of positivity of the difference $h=P_{N}(d/2)-P_{H}(d/2)>0$ a weaker result is valid on the existence of $h$-admissible (growing) measures of Amice-Vélu-type which produced an unbounded $p$ -adic analytic interpolation of the $L$-values of growth $\log_{p}^h(\cdot)$, using the Mellin transform of the constructed measures.