On the local density formula and the Gross–Keating invariant with an Appendix ‘The local density of a binary quadratic form’ by T. Ikeda and H. Katsurada

Abstract

Ikeda and Katsurada have developed the theory of the Gross–Keating invariant of a quadratic form in their recent papers Ikeda and Katsurada (Am J Math 140:1521–1565, 2018, Explicit formula of the Siegel series of a quadratic form over a non-archimedean local field, 2017. arXiv:1602.06617). In particular, they prove that the local factors of the Fourier coefficients of the Siegel–Eisenstein series are completely determined by the Gross–Keating invariants with extra datums, called the extended GK datums, in Ikeda and Katsurada (Explicit formula of the Siegel series of a quadratic form over a non-archimedean local field, 2017. arXiv:1602.06617). On the other hand, such a local factor is a special case of the local density for a pair of two quadratic forms. Thus we propose a general question if the local density can be expressed in terms of a certain series of extended GK datums. In this paper, we prove that the answer to this question is affirmative, for the local density of a single quadratic form defined over an unramified finite extension of ${\mathbb {Z}}_2$ and over a finite extension of ${\mathbb {Z}}_p$ with p odd. In the appendix, Ikeda and Katsurada compute the local density formula of a single binary quadratic form defined over any finite extension of ${\mathbb {Z}}_2$.

Appendix A: The local density of a binary quadratic form

Tamotsu IKEDA

Graduate school of mathematics, Kyoto University,

Kitashirakawa, Kyoto, 606-8502, Japan

ikeda@math.kyoto-u.ac.jp

Hidenori KATSURADA

Muroran Institute of Technology

27-1 Mizumoto, Muroran, 050-8585, Japan

hidenori@mmm.muroran-it.ac.jp

In this appendix, we calculate the local density of a binary form over a dyadic field F which may not be an unramified extension of ${{\mathbb {Q}}}_2$. We also calculate the Gross–Keating invariant ${\mathrm {GK}}(L\perp -L)$ and the truncated EGK invariant ${\mathrm {EGK}}(L\cap \varpi ^i L^\sharp )^{\le 1}$ for a binary quadratic lattice L. The local density formula is given in Proposition A.6. We also show that the local density is not determined by ${\mathrm {GK}}(L\perp -L)$ and ${\mathrm {EGK}}(L\cap \varpi ^i L^\sharp )^{\le 1}$, if we drop the assumption that $F/{{\mathbb {Q}}}_2$ is unramified (See Example A.1).

Let ${{\mathfrak {o}}}$ be the ring of integers of F, ${{\mathfrak {p}}}$ the maximal ideal of ${{\mathfrak {o}}}$, $\varpi$ a prime element of F, and q the order of the residue field. The ramification index of $F/{{\mathbb {Q}}}_2$ is denoted by e.

Let (L, Q) and $(L_1, Q_1)$ be quadratic lattices of rank n over ${{\mathfrak {o}}}$. We say that (L, Q) and $(L_1, Q_1)$ are weakly equivalent if there exist an isomorphism $\iota :L\rightarrow L_1$ and a unit $u\in {{\mathfrak {o}}}^\times$ such that $u Q_1(\iota (x))=Q(x)$ for any $x\in L$. Similarly, we say that $B, B_1\in {{\mathcal {H}}}_n({{\mathfrak {o}}})$ are weakly equivalent if there exist a unimodular matrix $U\in {\mathrm {GL}}_n({{\mathfrak {o}}})$ and a unit $u \in {{\mathfrak {o}}}^\times$ such that $u B_1=B[U]$. If B and $B_1$ are weakly equivalent, then ${\mathrm {GK}}(B)={\mathrm {GK}}(B_1)$. Recall that a half-integral symmetric matrix $B\in {{\mathcal {H}}}_n({{\mathfrak {o}}})$ is primitive if $\varpi ^{-1}B\notin {{\mathcal {H}}}_n({{\mathfrak {o}}})$. Put ${\mathrm {GK}}(B)=(a_1, a_2, \ldots , a_n)$. Then B is primitive if and only if $a_1=0$.

Let E/F be a semi-simple quadratic algebra. This means that E is a quadratic extension of F or $E=F\times F$. The non-trivial automorphism of E/F is denoted by $x\mapsto {\bar{x}}$. Note that if $E=F\times F$, we have $\overline{(x_1, x_2)}=(x_2, x_1)$. Let ${{\mathfrak {o}}}_E$ be the maximal order of E. In the case $E=F\times F$, ${{\mathfrak {o}}}_E={{\mathfrak {o}}}\times {{\mathfrak {o}}}$. The discriminant ideal of E/F is denoted by ${{\mathfrak {D}}}_E$. When $E=F\times F$, we understand ${{\mathfrak {D}}}_E={{\mathfrak {o}}}$. Put $d={\mathrm {ord}}({{\mathfrak {D}}}_E)$ and

$$\begin{aligned} \xi = {\left\{ \begin{array}{ll} 1 &{} \text { if } E=F\times F, \\ -1 &{} \text { if } E{/}F \text { is unramified quadratic extension, } \\ 0 &{} \text { if } E{/}F \text { is ramified quadratic extension. } \end{array}\right. } \end{aligned}$$

We say that E/F is unramified, if $d=0$. Note that $d\in \{2g\,|\, g\in {{\mathbb {Z}}}, 0\le g \le e\}\cup \{2e+1\}$. The order ${{\mathfrak {o}}}_{E, f}$ of conductor f for E/F is defined by ${{\mathfrak {o}}}_{E, f}={{\mathfrak {o}}}+{{\mathfrak {p}}}^f{{\mathfrak {o}}}_E$. Any open ${{\mathfrak {o}}}$-subring of ${{\mathfrak {o}}}_E$ is of the form ${{\mathfrak {o}}}_{E, f}$ for some non-negative integer f.

Proposition A.1

([10], Proposition 2.1) Let $B\in {{\mathcal {H}}}^{{\mathrm {nd}}}_2({{\mathfrak {o}}})$ be a primitive half-integral symmetric matrix of size 2 and (L, Q) its associated quadratic lattice. Put $E=F(\sqrt{D_B})/F$. When $D_B\in F^{\times 2}$, we understand $E=F\times F$. Put $f=({\mathrm {ord}}(D_B)-{\mathrm {ord}}({\mathfrak {D}}_E))/2$. Then f is a non-negative integer and (L, Q) is weakly equivalent to $({{\mathfrak {o}}}_{E, f}, {{\mathcal {N}}})$, where ${{\mathcal {N}}}$ is the norm form for E/F.

Proposition A.2

([10], Proposition 2.2) The Gross–Keating invariant of the binary quadratic form $(L, Q)=({{\mathfrak {o}}}_{E,f}, {{\mathcal {N}}})$ is given by

$$\begin{aligned} {\left\{ \begin{array}{ll} (0, 2f) &{} \text { if } \,\, E/F \text { is unramified,} \\ (0, 2f+1) &{} \text { if } \,\, E/F \text { is ramified.} \end{array}\right. } \end{aligned}$$

The following lemma is well-known.

Lemma A.3

We have

$$\begin{aligned}{}[{{\mathfrak {o}}}^\times :{{\mathfrak {o}}}^{\times 2}(1+{{\mathfrak {p}}}^f)]= {\left\{ \begin{array}{ll} q^{\left[ \frac{f}{2}\right] } &{} \text { if } \,\, 0<f\le 2e, \\ 2q^e &{} \text { if } \,\, f> 2e. \end{array}\right. } \end{aligned}$$

Choose $\omega \in {{\mathfrak {o}}}_E$ such that ${{\mathfrak {o}}}_E={{\mathfrak {o}}}+{{\mathfrak {o}}}\omega$. If E/F is unramified, then ${\mathrm {ord}}_E(\omega )=0$. It E/F is ramified, then we may assume $\omega$ is a prime element of E. Put $h={\mathrm {ord}}_E(\omega )$.

We fix an ${{\mathfrak {o}}}$-module isomorphism ${{\mathfrak {o}}}^2\simeq {{\mathfrak {o}}}_{E, f}$ by $(x, y)\mapsto x+\varpi ^f \omega y$. By this isomorphism, we identify ${{\mathfrak {o}}}^2$ and ${{\mathfrak {o}}}_{E,f}$. We consider a quadratic form Q(x, y) by

$$\begin{aligned} Q(x, y)={{\mathcal {N}}}(x+\varpi ^f \omega y)=x^2+\varpi ^f {\mathrm {tr}}(\omega ) xy +\varpi ^{2f}{{\mathcal {N}}}(\omega )y^2. \end{aligned}$$

An ${{\mathfrak {o}}}$-endomorphism of ${{\mathfrak {o}}}_{E,f}$ is expressed as

$$\begin{aligned} U_{\alpha , \beta }(x+ \varpi ^f \omega y) = \alpha x + \beta \varpi ^f \omega y \end{aligned}$$

for some $\alpha \in {{\mathfrak {o}}}_{E, f}$ and $\beta \in (\varpi ^f\omega )^{-1}{{\mathfrak {o}}}_{E, f}$. Note that $U_{\alpha , \alpha }\circ U_{\beta , \gamma }=U_{\alpha \beta , \alpha \gamma }$. Note also that

$$\begin{aligned} Q(U_{\alpha , \beta }(x,y))= {{\mathcal {N}}}(\alpha )x^2+\varpi ^f {\mathrm {tr}}(\bar{\alpha }\omega \beta ) xy +\varpi ^{2f}{{\mathcal {N}}}(\omega \beta )y^2. \end{aligned}$$

We shall determine when $Q\circ U_{\alpha , \beta }\equiv Q$ mod ${{\mathfrak {p}}}^N$, where N is a sufficiently large integer. Put

$$\begin{aligned} V_N=\{\alpha \in {{\mathfrak {o}}}_{E,f}\,|\, {{\mathcal {N}}}(\alpha )\equiv 1 \text { mod } {{\mathfrak {p}}}^N\}. \end{aligned}$$

Then $V_N\subset {{\mathfrak {o}}}_{E,f}^\times$. Clearly, if $Q\circ U_{\alpha , \beta }\equiv Q$ mod ${{\mathfrak {p}}}^N$, then $\alpha \in V_N$. Replacing $U_{\alpha , \beta }$ by $U_{\alpha , \alpha }^{-1}\circ U_{\alpha , \beta }$, we may assume $\alpha =1$. Then $\beta$ belongs to the set

Thus we have

$$\begin{aligned} \{U_{\alpha , \beta }\,|\,Q\circ U_{\alpha , \beta }\equiv Q \text { mod } {{\mathfrak {p}}}^N\} =\{U_{\alpha , \alpha }\circ U_{1, \beta }\,|\, \alpha \in V_N, \ \beta \in W_N\}, \end{aligned}$$

As we have assumed that N is sufficiently large, we have $W_N\subset {{\mathfrak {o}}}_{E,f}^\times$. Then the local density for $({{\mathfrak {o}}}_{E,f}, Q)$ is

$$\begin{aligned} \frac{1}{2} q^{3N} \frac{{\mathrm {Vol}}(V_N)}{{\mathrm {Vol}}({{\mathfrak {o}}}_{E,f})} \frac{{\mathrm {Vol}}(W_N)}{{\mathrm {Vol}}((\varpi ^f \omega )^{-1}{{\mathfrak {o}}}_{E,f})}. \end{aligned}$$

Lemma A.4

$$\begin{aligned} \frac{{\mathrm {Vol}}(W_N)}{{\mathrm {Vol}}((\varpi ^f \omega )^{-1}{{\mathfrak {o}}}_{E,f})} =2 q^{-2N+2f+d}. \end{aligned}$$

Proof

For $\beta \in W_N$, we have

$$\begin{aligned} (\bar{\beta }-1)(\omega \beta -\bar{\omega })&\equiv \omega {{\mathcal {N}}}(\beta )-{\mathrm {tr}}(\omega \beta )+\bar{\omega }\\&\equiv \omega -{\mathrm {tr}}(\omega )+\bar{\omega }\\&\equiv 0 \quad \text { mod } \varpi ^{N-2f}{{\mathfrak {o}}}_E. \end{aligned}$$

It follows that

$$\begin{aligned} W_N\subset (1+\varpi ^{N-2f-d}{{\mathfrak {o}}}_E)\cup \left( \frac{\bar{\omega }}{\omega }+\varpi ^{N-2f-d}{{\mathfrak {o}}}_E\right) . \end{aligned}$$

Put $W'_N=W_N\cap (1+\varpi ^{N-2f-d}{{\mathfrak {o}}}_E)$. Then we have

$$\begin{aligned} W_N=W'_N\cup \frac{\bar{\omega }}{\omega }\overline{W'_N}. \end{aligned}$$

Note that ${\mathrm {ord}}_E(1-\frac{\bar{\omega }}{\omega })={\mathrm {ord}}_E(\omega ^{-1}(\omega -\bar{\omega }))=d-h$, and so we have $W'_N\cap \frac{\bar{\omega }}{\omega }\overline{W'_N}=\emptyset$, since N is sufficiently large. Hence we have ${\mathrm {Vol}}(W_N)=2{\mathrm {Vol}}(W'_N)$. Note that

Observe that if $\gamma =x+\bar{\omega }y$, $x, y\in {{\mathfrak {o}}}$, then

$$\begin{aligned} \begin{pmatrix} {\mathrm {tr}}(\gamma ) \\ {\mathrm {tr}}(\omega \gamma ) \end{pmatrix} = \begin{pmatrix} 2 &{} {\mathrm {tr}}(\omega ) \\ {\mathrm {tr}}(\omega ) &{} 2{{\mathcal {N}}}(\omega ) \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}. \end{aligned}$$

Since ${\mathrm {ord}}\left( \det \begin{pmatrix} 2 &{} {\mathrm {tr}}(\omega ) \\ {\mathrm {tr}}(\omega ) &{} 2{{\mathcal {N}}}(\omega ) \end{pmatrix}\right) ={\mathrm {ord}}(\omega -\bar{\omega })^2=d$, we have

$$\begin{aligned} \frac{{\mathrm {Vol}}(W'_N)}{{\mathrm {Vol}}(1+\varpi ^{N-2f-d}{{\mathfrak {o}}}_E)} = q^{-f-d+h}. \end{aligned}$$

Note also that

$$\begin{aligned} {\mathrm {Vol}}((\varpi ^f \omega )^{-1}{{\mathfrak {o}}}_{E,f})=q^{2f+h}{\mathrm {Vol}}({{\mathfrak {o}}}_{E,f})= q^{f+h}{\mathrm {Vol}}({{\mathfrak {o}}}_{E}). \end{aligned}$$

Hence we have

$$\begin{aligned} \frac{{\mathrm {Vol}}(W_N)}{{\mathrm {Vol}}((\varpi ^f \omega )^{-1}{{\mathfrak {o}}}_{E,f})}&=2 \frac{{\mathrm {Vol}}(W'_N)}{{\mathrm {Vol}}(1+\varpi ^{N-2f-d}{{\mathfrak {o}}}_E)} \frac{{\mathrm {Vol}}(1+\varpi ^{N-2f-d}{{\mathfrak {o}}}_E)}{{\mathrm {Vol}}((\varpi ^f \omega )^{-1}{{\mathfrak {o}}}_{E,f})} \\&= 2 q^{-f-d+h}\cdot q^{-2N+3f+2d-h} \\&=2 q^{-2N+2f+d}. \end{aligned}$$

$\square$

Lemma A.5

• (1) If E/F is unramified, then

  $$\begin{aligned} \frac{{\mathrm {Vol}}(V_N)}{{\mathrm {Vol}}({{\mathfrak {o}}}_{E,f})} = {\left\{ \begin{array}{ll} q^{-N}(1-\xi q^{-1}) &{} \text { if } \,\, f=0, \\ q^{-N+[f/2]} &{} \text { if } \,\, 0 < f \le 2e, \\ 2q^{-N+e} &{} \text { if } \,\, f>2e. \end{array}\right. } \end{aligned}$$

• (2) If E/F is ramified, then

  $$\begin{aligned} \frac{{\mathrm {Vol}}(V_N)}{{\mathrm {Vol}}({{\mathfrak {o}}}_{E,f})} = {\left\{ \begin{array}{ll} 2q^{-N+f} &{} \text { if } \,\, 0\le f < \left[ \frac{d+1}{2}\right] , \\ q^{-N+[\frac{f}{2}+\frac{d}{4}]} &{} \text { if } \,\, \left[ \frac{d+1}{2}\right] \le f \le 2e-\left[ \frac{d}{2}\right] , \\ {} 2q^{-N+e} &{} \text { if } \,\, f> 2e-\left[ \frac{d}{2}\right] . \end{array}\right. } \end{aligned}$$

Proof

We normalize the Haar measure of E and F by ${\mathrm {Vol}}({{\mathfrak {o}}}_E)={\mathrm {Vol}}({{\mathfrak {o}}})=1$. Since N is sufficiently large, ${{\mathcal {N}}}({{\mathfrak {o}}}_{E,f}^\times )\supset 1+{{\mathfrak {p}}}^N$. Then we have

$$\begin{aligned} {[}{{\mathfrak {o}}}_{E,f}^\times :V_N]=[{{\mathcal {N}}}({{\mathfrak {o}}}_{E,f}^\times ): 1+{{\mathfrak {p}}}^N]. \end{aligned}$$

We have

$$\begin{aligned} \frac{{\mathrm {Vol}}(V_N)}{{\mathrm {Vol}}({{\mathfrak {o}}}_{E,f})}&= \frac{{\mathrm {Vol}}({{\mathfrak {o}}}_{E,f}^\times )}{{\mathrm {Vol}}({{\mathfrak {o}}}_{E,f})} \frac{{\mathrm {Vol}}(1+{{\mathfrak {p}}}^N)}{{\mathrm {Vol}}({{\mathcal {N}}}({{\mathfrak {o}}}^\times _{E,f}))} \\&= q^{-N+f} \frac{{\mathrm {Vol}}({{\mathfrak {o}}}_{E,f}^\times )}{{\mathrm {Vol}}({{\mathcal {N}}}({{\mathfrak {o}}}^\times _{E,f}))}. \end{aligned}$$

It is easily seen that

$$\begin{aligned} {\mathrm {Vol}}({{\mathfrak {o}}}_{E,f}^\times ) = {\left\{ \begin{array}{ll} (1-q^{-1})(1-\xi q^{-1}) &{} \text { if } f=0, \\ q^{-f} (1-q^{-1}) &{} \text { if } f>0. \end{array}\right. } \end{aligned}$$

Thus it is enough to calculate ${\mathrm {Vol}}({{\mathcal {N}}}({{\mathfrak {o}}}_{E,f}^\times ))$. If $f=0$, then

$$\begin{aligned}{}[{{\mathfrak {o}}}^\times :{{\mathcal {N}}}({{\mathfrak {o}}}_E^\times )]= {\left\{ \begin{array}{ll} 1 &{} \text { if } E/F \text { is unramified, } \\ 2 &{} \text { if } E/F \text { is ramified.} \end{array}\right. } \end{aligned}$$

This settles the case $f=0$. Suppose that $f>0$. Then ${{\mathfrak {o}}}_{E,f}^\times ={{\mathfrak {o}}}^\times (1+\varpi ^f {{\mathfrak {o}}}_E)$ and so

$$\begin{aligned} {{\mathcal {N}}}({{\mathfrak {o}}}_{E,f}^\times )={{\mathfrak {o}}}^{\times 2} \cdot {{\mathcal {N}}}(1+\varpi ^f {{\mathfrak {o}}}_E). \end{aligned}$$

If E/F is unramified, then ${{\mathcal {N}}}(1+\varpi ^f{{\mathfrak {o}}}_E)=1+{{\mathfrak {p}}}^f$. By Lemma A.3, we have

$$\begin{aligned} {\mathrm {Vol}} ({{\mathcal {N}}}({{\mathfrak {o}}}_{E,f}^\times )) = {\left\{ \begin{array}{ll} (1-q^{-1}) q^{-\left[ \frac{f}{2}\right] } &{}\text { if } 0< f\le 2e, \\ \frac{1}{2} (1-q^{-1}) q^{-e} &{}\text { if } f> 2e, \end{array}\right. } \end{aligned}$$

and so

$$\begin{aligned} \frac{{\mathrm {Vol}}(V_N)}{{\mathrm {Vol}}({{\mathfrak {o}}}_{E,f})} = {\left\{ \begin{array}{ll} q^{-N+[f/2]} &{} \text { if } 0 < f \le 2e, \\ 2q^{-N+e} &{} \text { if } f>2e. \end{array}\right. } \end{aligned}$$

Now suppose F/F is ramified. By Serre [13], p.85, Corollary 3, we have

$$\begin{aligned} {{\mathcal {N}}}(1+\varpi ^f {{\mathfrak {o}}}_E)=1+{{\mathfrak {p}}}^{f+\left[ \frac{d}{2}\right] } \end{aligned}$$

for $f\ge \left[ \frac{d+1}{2}\right]$. It follows that

$$\begin{aligned} {{\mathcal {N}}}({{\mathfrak {o}}}_{E,f}^\times )={{\mathfrak {o}}}^{\times 2}(1+{{\mathfrak {p}}}^{f+\left[ \frac{d}{2}\right] }) \end{aligned}$$

for $f\ge \left[ \frac{d+1}{2}\right]$. By Lemma A.3, we have

$$\begin{aligned} {\mathrm {Vol}}({{\mathcal {N}}}({{\mathfrak {o}}}_{E,f}^\times )) = {\left\{ \begin{array}{ll} (1-q^{-1})q^{-[\frac{f}{2}+\frac{d}{4}]} &{} \text { if } \left[ \frac{d+1}{2}\right] \le f \le 2e-\left[ \frac{d}{2}\right] , \\ \frac{1}{2} (1-q^{-1})q^{-e} &{} \text { if } f> 2e-\left[ \frac{d}{2}\right] , \end{array}\right. } \end{aligned}$$

and so

$$\begin{aligned} \frac{{\mathrm {Vol}}(V_N)}{{\mathrm {Vol}}({{\mathfrak {o}}}_{E,f})} = {\left\{ \begin{array}{ll} q^{-N+[\frac{f}{2}+\frac{d}{4}]} &{} \text { if } \left[ \frac{d+1}{2}\right] \le f \le 2e-\left[ \frac{d}{2}\right] , \\ 2q^{-N+e} &{} \text { if } f> 2e-\left[ \frac{d}{2}\right] . \end{array}\right. } \end{aligned}$$

Finally, suppose that $0<f<\left[ \frac{d+1}{2}\right]$. In this case, by Shimura [14], Lemma 21.13 (v), we have

$$\begin{aligned} {{\mathcal {N}}}(1+\varpi ^f {{\mathfrak {o}}}_E)=(1+{{\mathfrak {p}}}^{2f})\cap {{\mathcal {N}}}({{\mathfrak {o}}}_E^\times ). \end{aligned}$$

Since $1+{{\mathfrak {p}}}^{d-1}\not \subset {{\mathcal {N}}}({{\mathfrak {o}}}_E^\times )$, we have ${{\mathcal {N}}}(1+\varpi ^f {{\mathfrak {o}}}_E)\subsetneqq 1+{{\mathfrak {p}}}^{2f}$. Hence

$$\begin{aligned} {\mathrm {Vol}}({{\mathcal {N}}}(1+\varpi ^f{{\mathfrak {o}}}_E))=\frac{1}{2} {\mathrm {Vol}}(1+{{\mathfrak {p}}}^{2f}) =\frac{1}{2} q^{-2f}. \end{aligned}$$

On the other hand, we have

$$\begin{aligned} {{\mathcal {N}}}(1+\varpi ^f {{\mathfrak {o}}}_E)\cap {{\mathfrak {o}}}^{\times 2}=(1+{{\mathfrak {p}}}^{2f})\cap {{\mathcal {N}}}({{\mathfrak {o}}}_E^\times )\cap {{\mathfrak {o}}}^{\times 2}=(1+{{\mathfrak {p}}}^{2f})\cap {{\mathfrak {o}}}^{\times 2}. \end{aligned}$$

Hence

$$\begin{aligned} {[}{{\mathfrak {o}}}^{\times 2}:{{\mathcal {N}}}(1+\varpi ^f {{\mathfrak {o}}}_E)\cap {{\mathfrak {o}}}^{\times 2}]&= [{{\mathfrak {o}}}^{\times 2}:(1+{{\mathfrak {p}}}^{2f})\cap {{\mathfrak {o}}}^{\times 2}] \\&= [{{\mathfrak {o}}}^{\times 2}(1+{{\mathfrak {p}}}^{2f}): 1+{{\mathfrak {p}}}^{2f}] \\&= \frac{[{{\mathfrak {o}}}^\times : 1+{{\mathfrak {p}}}^{2f}]}{[{{\mathfrak {o}}}^\times :{{\mathfrak {o}}}^{\times 2}(1+{{\mathfrak {p}}}^{2f})]} \\&= q^f(1-q^{-1}). \end{aligned}$$

It follows that

$$\begin{aligned} {\mathrm {Vol}}({{\mathcal {N}}}({{\mathfrak {o}}}_{E,f}^\times ))&= {\mathrm {Vol}}({{\mathcal {N}}}(1+\varpi ^f{{\mathfrak {o}}}_E)) [{{\mathfrak {o}}}^{\times 2}\cdot {{\mathcal {N}}}(1+\varpi ^f {{\mathfrak {o}}}_E):{{\mathcal {N}}}(1+\varpi ^f {{\mathfrak {o}}}_E)] \\&=\frac{1}{2} q^{-2f}[{{\mathfrak {o}}}^{\times 2}:{{\mathcal {N}}}(1+\varpi ^f {{\mathfrak {o}}}_E)\cap {{\mathfrak {o}}}^{\times 2}] \\&=\frac{1}{2} q^{-f}(1-q^{-1}). \end{aligned}$$

Hence we have

$$\begin{aligned} \frac{{\mathrm {Vol}}(V_N)}{{\mathrm {Vol}}({{\mathfrak {o}}}_{E,f})} = 2q^{-N+f} \end{aligned}$$

in this case. This proves the lemma. $\square$

By Lemmas A.4 and A.5, we obtain the following formula.

Proposition A.6

• (1) Assume that E is unramified. Then the local density of $(L, Q)=({{\mathfrak {o}}}_{E,f}, {{\mathcal {N}}})$ is given by

  $$\begin{aligned} \beta (L)= {\left\{ \begin{array}{ll} 1-\xi q^{-1} &{} \text { if } f=0, \\ q^{[f/2]+2f} &{} \text { if } 0 < f \le 2e, \\ 2q^{e+2f} &{} \text { if } f>2e. \end{array}\right. } \end{aligned}$$

• (2) Assume that E is ramified and that $d=2g\le 2e$. Then the local density of $(L, Q)=({{\mathfrak {o}}}_{E,f}, {{\mathcal {N}}})$ is given by

  $$\begin{aligned} \beta (L)= {\left\{ \begin{array}{ll} 2q^{3f+2g} &{} \text { if } 0\le f < g, \\ q^{[\frac{f}{2}+\frac{g}{2}]+2f+2g} &{} \text { if } g \le f \le 2e-g, \\ 2q^{2f+e+2g} &{} \text { if } f> 2e-g. \end{array}\right. } \end{aligned}$$

• (3) Assume that E is ramified and that $d=2e+1$. Then the local density of $(L, Q)=({{\mathfrak {o}}}_{E,f}, {{\mathcal {N}}})$ is given by

  $$\begin{aligned} \beta (L)= {\left\{ \begin{array}{ll} 2q^{3f+2e+1} &{} \text { if } 0\le f < e+1, \\ 2q^{2f+3e+1} &{} \text { if } f\ge e+1. \end{array}\right. } \end{aligned}$$

Next, we calculate ${\mathrm {GK}}(L\oplus -L)$.

Proposition A.7

Suppose that $(L, Q)=({{\mathfrak {o}}}_{E,f}, {{\mathcal {N}}})$.

• (1) Assume that E is unramified. Then $L\oplus -L$ is equivalent to a reduced form of GK type $({\underline{a}}, \sigma )$, where

  $$\begin{aligned} ({\underline{a}}, \sigma )= {\left\{ \begin{array}{ll} ((0,0,0,0), \, (12)(34)) &{} \text { if } f=0, \\ ((0,f,f,2f),\, (14)(23)) &{} \text { if } 0< f < 2e, \\ ((0, 2e, 2f-2e, 2f),\, (12)(34)) &{} \text { if } f\ge 2e. \end{array}\right. } \end{aligned}$$

• (2) Assume that E is ramified and that $d\le 2e$. Put $g=d/2$. Then $L\oplus -L$ is equivalent to a reduced form of GK type $({\underline{a}}, \sigma )$, where

  $$\begin{aligned} ({\underline{a}}, \sigma )= {\left\{ \begin{array}{ll} ((0, 2f+1, 2g-1, 2g+2f),\, (14)(23)) &{} \text { if } 0\le f \le g-1, \\ ((0, g+f, g+f, 2g+2f),\, (14)(23)) &{} \text { if } g\le f < 2e-g, \\ ((0, 2e, 2g+2f-2e, 2g+2f),\, (12)(34)) &{} \text { if } f \ge 2e-g. \end{array}\right. } \end{aligned}$$

• (3) Assume that E is ramified and that $d=2e+1$. Then $L\oplus -L$ is equivalent to a reduced form of GK type $({\underline{a}}, \sigma )$, where

  $$\begin{aligned} ({\underline{a}}, \sigma )= {\left\{ \begin{array}{ll} ((0, 2f+1, 2e, 2e+2f+1),\, (13)(24)) &{} \text { if } 0\le f < e, \\ ((0, 2e, 2f+1, 2e+2f+1),\, (12)(34)) &{} \text { if } f \ge e. \end{array}\right. } \end{aligned}$$

Proof

Let $B\in {{\mathcal {H}}}_2({{\mathfrak {o}}})$ be a half-integral symmetric matrix associated to (L, Q). First we consider the case E/F is unramified. If $f=0$, then we have $B\perp -B\sim H\oplus H$, where H is the hyperbolic plane $\begin{pmatrix} 0 &{} 1/2 \\ 1/2 &{} 0 \end{pmatrix}$. In fact, it is easy to see that $B\perp -B$ expresses H, and so $B\perp -B\sim H\perp K$ for some $K\in {{\mathcal {H}}}_2({{\mathfrak {o}}})$. Since $-\det (2K)\in {{\mathfrak {o}}}^{\times 2}$, we have $K\sim H$. This settles the case $f=0$ of (1). Next, we consider the case $0<f$. Let $\{1, \omega \}$ be a basis for ${{\mathfrak {o}}}_E$ as an ${{\mathfrak {o}}}$-module. Then, since F is dyadic, we have ${\mathrm {tr}}(\omega )\in {{\mathfrak {o}}}^\times$. By multiplying $\omega$ by some unit, we may assume ${\mathrm {tr}}(\omega )=1$. By using this basis, the half-integral symmetric matrix associated to $({{\mathfrak {o}}}_E, {{\mathcal {N}}})$ is of the form $\begin{pmatrix} 1 &{} 1/2 \\ 1/2 &{} u\end{pmatrix}$ for some $u\in {{\mathfrak {o}}}$. Since $\{1, \varpi ^f\omega \}$ is a basis of ${{\mathfrak {o}}}_{E, f}$ over ${{\mathfrak {o}}}$, we may assume $B=\begin{pmatrix} 1 &{} \varpi ^f/2 \\ \varpi ^f/2 &{} u \varpi ^{2f} \end{pmatrix}$. If $0<f<2e$, we have

Here, $X\xrightarrow {A} Y$ means $Y=X[A]$. Since the last matrix is a reduced form of GK type $((0,f,f,2f),\, (14)(23))$, we have proved the case $0<f<2e$ of (1). Next, suppose $f\ge 2e$. Then we have

$$\begin{aligned} B=\begin{pmatrix} 1 &{} \varpi ^f/2 \\ \varpi ^f/2 &{} u \varpi ^{2f} \end{pmatrix} \xrightarrow {\left( {\begin{matrix} 1 &{} -\varpi ^f/2 \\ 0 &{} 1\end{matrix}}\right) } \begin{pmatrix} 1 &{} 0 \\ 0 &{} v\varpi ^{2f-2e} \end{pmatrix}. \end{aligned}$$

Here, $v=\varpi ^{2e}(4u-1)/4\in {{\mathfrak {o}}}^\times$. Then we have

It is easy to check that the lase matrix is a reduced form of GK type $(0, 2e, 2f-2e, 2f),\, (12)(34))$. Thus we have proved the last case of (1).

Suppose that E/F is ramified and $d=2g\le 2e$. In this case, E is generated by an element $\varpi _E=\varpi ^g (-1+\sqrt{\varepsilon })/2$, such that ${\mathrm {ord}}(\varepsilon -1)=2e-2g+1$. Then $\{1, \varpi ^f\varpi _E\}$ is a basis of ${{\mathfrak {o}}}_{E, f}$. By using this basis, $B=\begin{pmatrix} 1 &{} \varpi ^{g+f}/2 \\ \varpi ^{g+f}/2 &{} u\varpi ^{2f+1} \end{pmatrix}$. Here, $u=\varpi ^{2g-1}(1-\varepsilon )/4\in {{\mathfrak {o}}}^\times$. If $f<2e-g$, we have

The last matrix is a reduced form of GK type $((0, 2f+1, 2g-1, 2g+2f),\, (14)(23))$ if $0\le f \le g-1$, and a reduced form of GK type $((0, g+f, g+f, 2g+2f),\, (14)(23))$, if $g\le f < 2e-g$. This proves the first and the second case of (2). Suppose that $f\ge 2e-g$. Then we have

$$\begin{aligned} B=\begin{pmatrix} 1 &{} \varpi ^{g+f}/2 \\ \varpi ^{g+f}/2 &{} u\varpi ^{2f+1} \end{pmatrix} \xrightarrow {\left( {\begin{matrix} 1 &{} -\varpi ^{g+f}/2 \\ 0 &{} 1\end{matrix}}\right) } \begin{pmatrix} 1 &{} 0 \\ 0 &{} v \varpi ^{2g+2f-2e} \end{pmatrix}. \end{aligned}$$

Here, $v=-\varpi ^{2e}\varepsilon /4\in {{\mathfrak {o}}}^\times$. In this case, by a similar calculation as before, we have

This matrix is a reduced form of GK type $((0, 2e, 2g+2f-2e, 2g+2f),\, (12)(34))$, and this settles the last case of (2).

Finally, suppose that E/F is ramified and $d=2e+1$. In this case, the quadratic extension E/F is generated by $\varpi _E=\sqrt{-\varpi u}$ for some unit $u\in {{\mathfrak {o}}}^\times$. Then $\{1, \varpi ^f\varpi _E\}$ is a ${{\mathfrak {o}}}$-basis of ${{\mathfrak {o}}}_{E, f}$. By using this basis, we may assume $B=\begin{pmatrix} 1 &{} 0 \\ 0 &{} u \varpi ^{2f+1} \end{pmatrix}$. Then, by a similar calculation as before, we have

If $0\le f < e$, then the first matrix is a reduced form of GK type $((0, 2f+1, 2e, 2e+2f+1),\, (13)(24))$. If $f \ge e$, then the second matrix is a reduced form of GK type $((0, 2e, 2f+1, 2e+2f+1),\, (12)(34))$. Hence we have proved the proposition. $\square$

By Theorem 4.1 (Corollary 5.1 of [10]), we obtain the following proposition.

Proposition A.8

Suppose that $(L, Q)=({{\mathfrak {o}}}_{E,f}, {{\mathcal {N}}})$.

• (1) Assume that E is unramified. Then we have

  $$\begin{aligned} {\mathrm {GK}}(L\oplus -L)= {\left\{ \begin{array}{ll} (0,0,0,0) &{} \text { if } \,\, f=0, \\ (0,f,f,2f) &{} \text { if } \,\, 0< f < 2e, \\ (0, 2e, 2f-2e, 2f) &{} \text { if } \,\, f\ge 2e. \end{array}\right. } \end{aligned}$$

• (2) Assume that E is ramified and that $d=2g\le 2e$. Then we have

  $$\begin{aligned} {\mathrm {GK}}(L\oplus -L)= {\left\{ \begin{array}{ll} (0, 2f+1, 2g-1, 2g+2f) &{} \text { if }\,\, 0\le f \le g-1, \\ (0, g+f, g+f, 2g+2f) &{} \text { if } \,\, g\le f < 2e-g, \\ (0, 2e, 2g+2f-2e, 2g+2f) &{} \text { if } \,\, f \ge 2e-g. \end{array}\right. } \end{aligned}$$

• (3) Assume that E is ramified and that $d=2e+1$. Then we have

  $$\begin{aligned} {\mathrm {GK}}(L\oplus -L)= {\left\{ \begin{array}{ll} (0, 2f+1, 2e, 2e+2f+1) &{} \text { if } \,\, 0\le f < e, \\ (0, 2e, 2f+1, 2e+2f+1) &{} \text { if } \,\, f \ge e. \end{array}\right. } \end{aligned}$$

We shall give a Jordan splitting for $L=({{\mathfrak {o}}}_{E,f}, {{\mathcal {N}}})$. Let $L=\bigoplus L_i$ be a Jordan splitting such that $L_i$ is i-modular. Put ${\mathrm {Jor}}(L)=\{i\in {{\mathbb {Z}}}\,|\, L_i \text { is nonzero.}\}$.

Lemma A.9

• (1) Suppose that E/F is unramified. If $f<e$, then ${\mathrm {Jor}}(L)=\{f-e\}$ and L is an indecomposable $(f-e)$-modular lattice. If $f\ge e$, then ${\mathrm {Jor}}(L)=\{0, 2f-2e\}$ and $L\sim (1)\perp (u\varpi ^{2f-2e})$, with $u\in {{\mathfrak {o}}}^\times$.

• (2) Suppose that E/F is ramified and $d=2g\le 2e$. If $f<e-g$, then ${\mathrm {Jor}}(L)=\{f+g-e\}$ and L is an indecomposable $(f+g-e)$-modular lattice. If $f\ge e-g$, then ${\mathrm {Jor}}(L)=\{0, 2f+2g-2e\}$ and $L\sim (1)\perp (u\varpi ^{2f+2g-2e})$, with $u\in {{\mathfrak {o}}}^\times$.

• (3) Suppose that E/F is ramified and $d=2e+1$. In this case, ${\mathrm {Jor}}(L)=\{0, 2f+1\}$ and $L\sim (1)\perp (u\varpi ^{2f+1})$, with $u\in {{\mathfrak {o}}}^\times$.

Proof

Suppose that E/F is unramified. As we have seen in the proof of Proposition A.7, L is expressed by $B=\begin{pmatrix} 1 &{} \varpi ^f/2 \\ \varpi ^f/2 &{} u \varpi ^{2f} \end{pmatrix}$ for some $u\in {{\mathfrak {o}}}^\times$. If $f<e$, then B is indecomposable by Lemma 2.1 of [10]. In this case, it is easy to see $\varpi ^{e-f}B$ is unimodular. If $f\ge e$, then we have $B\sim (1) \perp ((-1+u\varpi ^{2e}) \varpi ^{2f-2e})$. This proves (1). The other cases can be proved similarly. $\square$

We shall calculate the Gross–Keating invariant ${\mathrm {GK}}(L\cap \varpi ^i L^\sharp )$ for $(L\cap \varpi ^i L^\sharp , \varpi ^{-i}Q)$ for each $i\in {\mathrm {Jor}}(L)$. Recall that the Gross–Keating invariant $(a_1, a_2)$ of a binary form $(L^{\prime }, Q')$ is determined by

$$\begin{aligned} a_1&={\mathrm {ord}}({{\mathbf {n}}}(L^{\prime })), \\ a_1+a_2&= {\left\{ \begin{array}{ll} {\mathrm {ord}}(4\det Q') &{} \text { if } {\mathrm {ord}}({{\mathfrak {D}}}_{Q'})=0, \\ {\mathrm {ord}}(4\det Q')-{\mathrm {ord}}({{\mathfrak {D}}}_{Q'})+1 &{} \text { if } {\mathrm {ord}}({{\mathfrak {D}}}_{Q'})>0. \end{array}\right. } \end{aligned}$$

Here, ${{\mathfrak {D}}}_{Q'}$ is the discriminant of $F(\sqrt{-\det Q'})/F$. These formula follows form Proposition A.2, since a binary quadratic form is isomorphic to some $({{\mathfrak {o}}}_{E,f}, {{\mathcal {N}}})$ up to multiplication by a unit ([10], Proposition 2.1). In terms of $B=\begin{pmatrix} b_{11} &{} b_{12} \\ b_{12} &{} b_{22}\end{pmatrix}\in {{\mathcal {H}}}_2({{\mathfrak {o}}})$, the Gross–Keating invariant $(a_1, a_2)$ of B is given by

$$\begin{aligned} a_1&=\min \{{\mathrm {ord}}(b_{11}), {\mathrm {ord}}(2b_{12}), {\mathrm {ord}}(b_{22})\}, \\ a_1+a_2&= {\left\{ \begin{array}{ll} {\mathrm {ord}}(4\det B) &{} \text { if } {\mathrm {ord}}({{\mathfrak {D}}}_{B})=0, \\ {\mathrm {ord}}(4\det B)-{\mathrm {ord}}({{\mathfrak {D}}}_{B})+1 &{} \text { if } {\mathrm {ord}}({{\mathfrak {D}}}_{B})>0. \end{array}\right. } \end{aligned}$$

Note also that ${\mathrm {GK}}(\varpi ^i B)=(a_1+i, a_2+i)$.

Proposition A.10

Suppose that $(L, Q)=({{\mathfrak {o}}}_{E,f}, {{\mathcal {N}}})$ and $i\in {\mathrm {Jor}}(L)$.

• (1) Assume that E is unramified. Then we have

  $$\begin{aligned} {\mathrm {GK}}(L\cap \varpi ^i L^\sharp )= {\left\{ \begin{array}{ll} (e-f, e+f) &{} \text { if } f< e, \\ (0, 2f) &{} \text { if } f\ge e. \end{array}\right. } \end{aligned}$$

• (2) Assume that E is ramified and that $d=2g\le 2e$. Then, we have

  $$\begin{aligned} {\mathrm {GK}}(L\cap \varpi ^i L^\sharp )= {\left\{ \begin{array}{ll} (e-g-f, e-g+f+1) &{} \text { if } f<e-g, \\ (0, 2f+1) &{} \text { if } f\ge e-g, \end{array}\right. } \end{aligned}$$

• (3) Assume that E is ramified and that $d=2e+1$. Then we have

  $$\begin{aligned} {\mathrm {GK}}(L\cap \varpi ^i L^\sharp )= (0, 2f+1). \end{aligned}$$

Proof

Suppose that L is i-modular. In this case, $L\cap \varpi ^i L^\sharp =L$. Then ${\mathrm {GL}}(L\cap \varpi ^i L^\sharp )=(a_1-i, a_2-i)$, where $(a_1, a_2)={\mathrm {GL}}(L)$. (Remember that the quadratic form for $L\cap \varpi ^i L^\sharp$ is multiplied by $\varpi ^{-i}$.)

Suppose that $L\sim (1)\perp (u\varpi ^k)$. In this case, ${\mathrm {Jor}}(L)=\{0, k\}$ and $(L\cap \varpi ^i L^\sharp , \varpi ^{-i} Q)$ is expressed by $(1)\perp (u\varpi ^k)$ or $(u)\perp (\varpi ^k)$, according as $i=0$ or $i=k$. In either case, $(L\cap \varpi ^i L^\sharp , \varpi ^{-i} Q)$ is weakly equivalent to (L, Q). Hence the proposition. $\square$

For $B\in {{\mathcal {H}}}_n({{\mathfrak {o}}})$, we define ${\mathrm {EGK}}(B)^{\le 1}$ as in the main part of this paper. This is defined as follows. Let ${\mathrm {GK}}(B)=(\underbrace{0, \ldots , 0}_{m_0}, \underbrace{1, \ldots , 1}_{m_1}, a_{m_0+m_1+1}, \ldots , a_n)$, where $a_{m_0+m_1+1}>1$. If B is equivalent to a reduced form

then ${\mathrm {EGK}}(B)^{\le 1}={\mathrm {EGK}}\left( \begin{pmatrix} B_{00} &{} B_{11} \\ {}^t B_{01} &{} B_{11}\end{pmatrix}\right)$. This definition does not depend on the choice of the reduced form $B'$. If B is associated to a quadratic lattice M, we write ${\mathrm {EGK}}(M)^{\le 1}$ for ${\mathrm {EGK}}(B)^{\le 1}$.

The next proposition follows from Proposition A.10.

Proposition A.11

Suppose that $(L, Q)=({{\mathfrak {o}}}_{E,f}, {{\mathcal {N}}})$ and $i\in {\mathrm {Jor}}(L)$.

• (1) Assume that E is unramified. Then we have

  $$\begin{aligned} {\mathrm {EGK}}(L\cap \varpi ^i L^\sharp )^{\le 1}= {\left\{ \begin{array}{ll} \emptyset &{} \text { if } \,\, f <e-1, \\ (2;1;\xi ) &{} \text { if } \,\, f=0, e=1, \\ (1;1;1) &{} \text { if } \,\, f=e-1, e>1, \\ (1;0;1) &{} \text { if } \,\, f\ge e. \end{array}\right. } \end{aligned}$$

• (2) Assume that E is ramified and that $d=2g\le 2e$. Then, we have

  $$\begin{aligned} {\mathrm {EGK}}(L\cap \varpi ^i L^\sharp )^{\le 1}= {\left\{ \begin{array}{ll} \emptyset &{} \text { if } \,\, f<e-g-1, \\ (1;1;1) &{} \text { if } \,\, f=e-g-1, \\ (1;0;1) &{} \text { if } \,\, f\ge e-g, \, g<e, \\ (1;0;1) &{} \text { if } \,\, f>0, \, g=e, \\ (1,1;0,1;1,0) &{} \text { if } \,\, f=0, \, g=e. \end{array}\right. } \end{aligned}$$

• (3) Assume that E is ramified and that $d=2e+1$. Then we have

  $$\begin{aligned} {\mathrm {EGK}}(L\cap \varpi ^i L^\sharp )^{\le 1}= {\left\{ \begin{array}{ll} (1,1;0,1;1,0) &{} \text { if }\,\, f=0, \\ (1;0,1) &{} \text { if } \,\, f>0. \end{array}\right. } \end{aligned}$$

We shall show that there exist two binary quadratic lattices L and $L^{\prime }$, which satisfy the following conditions (1), (2), and (3).

• (1) ${\mathrm {GK}}(L\perp -L)={\mathrm {GK}}(L^{\prime }\perp -L^{\prime })$.

• (2) ${\mathrm {Jor}}(L)={\mathrm {Jor}}(L^{\prime })$ and ${\mathrm {EGK}}(L\cap \varpi ^i L^\sharp )^{\le 1}={\mathrm {EGK}}(L^{\prime }\cap \varpi ^i L^{\prime {\sharp }})^{\le 1}$ for each $i\in {\mathrm {Jor}}(L)$.

• (3) $\beta (L)\ne \beta (L^{\prime })$.

Example A.1

Suppose that $e=5$. Suppose also that E/F is a ramified quadratic extension with $d=2$ and $E^{\prime }/F$ is a ramified quadratic extension with $d=4$. Put $L={{\mathfrak {o}}}_{E, 2}$ and $L'={{\mathfrak {o}}}_{E^{\prime }, 1}$. Then we have

$$\begin{aligned} {\mathrm {GK}}(L\perp -L)={\mathrm {GK}}(L^{\prime }\perp -L^{\prime })=(0,3,3,6) \end{aligned}$$

by Proposition A.8. Note that ${\mathrm {Jor}}(L)={\mathrm {Jor}}(L^{\prime })=\{-2 \}$ and

$$\begin{aligned} {\mathrm {EGK}}(L\cap \varpi ^{-1} L^\sharp )^{\le 1}={\mathrm {EGK}}(L^{\prime } \cap \varpi ^{-1} L^{\prime \sharp })^{\le 1}=\emptyset \qquad (i\in {\mathrm {Jor}}(L)) \end{aligned}$$

by Proposition A.11. But we have

$$\begin{aligned} \beta (L)=q^7, \qquad \beta (L^{\prime })=2q^7 \end{aligned}$$

by Proposition A.6. Thus ${\mathrm {GL}}(L\perp -L)$ and ${\mathrm {EGK}}(L\cap \varpi ^i L^\sharp )^{\le 1}$ are not enough to determine $\beta (L)$ in the case $e>1$.