# Local densities of hermitian forms

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). 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 Y. Hironaka. "Local zeta functions on hermitian forms and its application to local densities". Journal of Number Theory 71, 40--64 (1998). Link.

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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).