I think this is an easy question, but I need some time to introduce it. I need to apply Yumiko Hironaka's computations on local densities of hermitian forms (see [1](http://www.sciencedirect.com/science/article/pii/S0022314X98922375)).
I would have liked to create the new tag [local-densities], but I don't have enough reputation. 


Let $\mathcal O$ be a ring of integers of an imaginary quadratic number field, like $\mathbb Z(\sqrt{-1})$. 
Let $H\in \mathrm{GL}(n,\mathcal O)$ be a Hermitian matrix and $\ell$ an integer number.
For $p$ an integer prime number, I need to compute the $p$-local density given by
$$
\delta\_p(H,\ell) = \lim\_{j\to\infty} \frac{A\_j(H,\ell)}{p^{j(2n-1)}}
$$
where 
$$
A\_j(H,\ell) = \# \{x\in ({{\mathcal O}/p^j{\mathcal O}})^n : x^\*Hx\equiv \ell\pmod {p^j} \}.
$$


Now I introduce the setting on Hironaka's paper: let $k$ be a nonarchimedian local field of characteristic $0$, $\mathcal O\_k$ the ring of integers in $k$ (note that $\mathcal O$ and $\mathcal O\_k$ are different rings), $\*$ an involution on $k$ and denote $k\_0$ the fixed field by $\*$. Assume that $k$ is unramified over $k\_0$. Let $q$ be the residue class field of $k\_0$, $\pi\in k\_0$ be a prime element of $k$ and $\mathfrak{p}=\pi\mathcal O\_k$.
She gives a formula for 
$$
\mu\_p(H,\ell) = \lim\_{j\to\infty} \frac{N\_j(H,\ell)}{q^{j(2n-1)}}
$$
where 
$$
N\_j(H,\ell)= \# \{x \in ({{\mathcal O\_k}/\mathfrak{p}})^n:x^\*Hx\equiv \ell\pmod {\mathfrak{p}^j} \}.
$$


Let $D\_{\mathcal O}(<0)$ be the discriminant of $\mathcal O$. 


If $(\frac{D\_{\mathcal O}}{p})=-1$, we have that $p\mathcal O$ is a prime ideal in $\mathcal O$, and we can apply Hironaka's formula to the nonarquimedian local field $k=\mathbb Q[\sqrt{D\_\mathcal O}]\otimes\_\mathbb Q \mathbb Q\_p$, where the involution $\*$ is given by $\alpha\otimes x\mapsto \bar\alpha\otimes x$, and $k\_0=\mathbb Q\_p$. 
This can be done because $\mathcal O\_k/\mathfrak{p} \simeq \mathcal O/p\mathcal O$, since $\mathfrak{p}=p\mathcal O\_k$.


Now, if $(\frac{D\_{\mathcal O}}{p})=+1$, we have that $p\mathcal O$ decomposes as a product of two different prime ideals (are conjugated). Also, $\mathbb Q[\sqrt{D\_\mathcal O}]\otimes\_\mathbb Q \mathbb Q\_p$ is isomorphic to $\mathbb Q\_p\times \mathbb Q\_p$. 
Finally, my question is: 


How do I apply Hironaka's formula to this case? Who is the local field $k$?
Who is the conjungation on $k$ and the fixed field $k\_0$? 
Who is the prime element $\pi$?


I'm sorry for this long question. I hope that somebody helps me with an understandable answer since I usually don't work in this area. Thanks.-.


[1](http://www.sciencedirect.com/science/article/pii/S0022314X98922375) Y. Hironaka. "Local zeta functions on hermitian forms and its application to local densities". Journal of Number Theory 71, 40--64 (1998). [Link](http://www.sciencedirect.com/science/article/pii/S0022314X98922375). 



Local densities of hermitian forms
Several months later, I have the answer to my question. I would like to share here.


Such as Maurice Mischler (Local densities of hermitian forms, in Contemporary Mathematics vol 272 (2000), 201--208) mentioned in his abstract, Hironaka compute the local density only in the inert prime case ($p\mathcal O$ is a prime ideal).


However, we can compute $\delta\_p(H,\ell)$ as follows: let $1,\omega$ a $\mathbb Z$-basis of $\mathcal O$. Writing $z\_j=x\_{2j-1}+x\_{2j}\omega$, we have that $H[z]$ induces a quadratic forms $Q\_H\in\mathrm{GL}(2n,\mathbb Z)$. Then
$$
\delta\_p^\mathbb C(H,\ell) = \delta\_p^\mathbb R(Q\_H,\ell).
$$
Now, we can compute the right-hand side by applying Yang's paper (An explicit formula for local densities of quadratic forms, J. Number Theory, vol 72 (1998), 309--356).


