用户: 香蕉三千/一些分歧p进积分

令素数 $p>2$. 考虑 $F/F_0$ 为 $p$ 进局部域分歧二次扩张, 剩余类域是 ${\mathbb{F}}_q$. 可取 $\pi /{\pi }_0$ 为 $F/F_0$ 的参数元使得 ${\pi }^2={\pi }_0$. 局部类域论给出乘法二次特征 $\eta :F_0^{\times }\to \{\pm 1\}$. 分歧假设给出 $\eta ({\pi }_0)=1$.

1特征积分

命题 1.1. $$\eta {|}_{O_{F_0}^{\times }}=\sum _{a\in ({\mathbb{F}}_q^{\times }{)}^2}{1}_{[a]+{\pi }_0{O}_{F_0}}-\sum _{a^{\prime }\in {\mathbb{F}}_q^{\times }-({\mathbb{F}}_q^{\times }{)}^2}{1}_{[a^{\prime }]+{\pi }_0{O}_{F_0}}.$$

证明. 由于 $p>2$, $1+{\pi }_0{O}_{F_0}$ 是 $2$ 可除的, 因此 $\eta {|}_{O_{F_0}^{\times }}$ 是 ${\mathbb{F}}_q^{\times }$ 上唯一的非平凡二次特征.

$\square$

命题 1.2. 考虑 $U=a+{\pi }_0^m{O}_{F_0}$, 如果 $0\in U$, 则 $v(a)\ge m$, $U={\pi }_0^m{O}_{F_0}$; 如果 $0\notin U$, 则 $v(a)<m$, $\eta {|}_U=\eta (a)$.

考虑 $F_0$ 上 Haar 测度 $dx$ 使得 ${\int }_{F_0}{1}_{O_{F_0}}dx=1$.

定义 1.3. 对紧支光滑函数 $f\in C_c^{\infty }(F_0)$, 定义收敛的积分$$I(f,\eta )={\int }_{F_0}f(x)\eta (x)dx=\sum _{n\in \mathbb{Z}}{q}^{-n}{\int }_{O_{F_0}^{\times }}f({\pi }_0^{n}x)\eta (x)dx.$$

命题 1.4. $$I(1_{a+{\pi }_0^m{O}_{F_0}},\eta )=\{\begin{array}{ll}0& v(a)\ge m.\\ q^{-m}\eta (a)& v(a)<m.\end{array}$$

例 1.5. $$I(1_{O_{F_0}},\eta )=I(1_{{\pi }_0{O}_{F_0}},\eta )=0.$$$$I(1_{[a]+{\pi }_0{O}_{F_0}},\eta )=1,a\in ({\mathbb{F}}_q^{\times }{)}^2.$$$$I(1_{[a^{\prime }]+{\pi }_0{O}_{F_0}},\eta )=-1,a^{\prime }\in {\mathbb{F}}_q^{\times }-({\mathbb{F}}_q^{\times }{)}^2.$$

考虑 $F_0$ 上 Haar 测度 $d^{\times }x=\frac{dx}{|x|}$, 那么 ${\int }_{F_0}{1}_{O_{F_0}^{\times }}{d}^{\times }x=1-q^{-1}$.

定义 1.6. 对紧支光滑函数 $H\in C_c^{\infty }(F_0\times F_0)$ 与 $x_1,x_2\in F_0^{\times }$, 定义收敛的双重积分$$II(H,x_1,x_2)={\int }_{F_0^{\times }}H(x{x}_1,x_2{x}^{-1})\eta (x)d^{\times }x.$$

我们有 $II(H,x_0{x}_1,x_2{x}_0^{-1})=\eta (x_0)II(H,x_2{x}_1,1)$.

命题 1.7. 积分 $II(1_{a_1+{\pi }_0^{m_1}{O}_{F_0}}\times 1_{a_2+{\pi }_0^{m_2}{O}_{F_0}},x_1,x_2)$ 的积分区域非空当且仅当$$x_1{x}_2\in (a_1+{\pi }_0^{m_1}{O}_{F_0})(a_2+{\pi }_0^{m_2}{O}_{F_0}).$$假定其非空, 那么积分值为$$\{\begin{array}{ll}0,& v(a_1)\ge m_1,v(a_2)\ge m_2.\\ \eta (x_1{a}_1)q^{v(a_1)-m_1},& v(a_1)<m_1,v(a_2)\ge m_2.\\ \eta (x_2{a}_2)q^{v(a_2)-m_2},& v(a_1)\ge m_1,v(a_2)<m_2.\\ \eta (x_1{a}_1)q^{-\max \{m_1-v(a_1),m_2-v(a_2))\}},& v(a_1)<m_1,v(a_2)<m_2.\end{array}$$

证明. 积分区域是 $x{x}_1\in a_1+{\pi }_0^{m_1}{O}_{F_0},x_2{x}^{-1}\in a_2+{\pi }_0^{m_2}{O}_{F_0}$ 即$$x_1^{-1}(a_1+{\pi }_0^{m_1}{O}_{F_0})\cap (x_2^{-1}(a_2+{\pi }_0^{m_2}{O}_{F_0}){)}^{-1}.$$其非空当且仅当 $x_1{x}_2\in (a_1+{\pi }_0^{m_1}{O}_{F_0})(a_2+{\pi }_0^{m_2}{O}_{F_0})$, 特别地 $v(x_1{x}_2)\ge \min \{v(a_1),m_1\}+\min \{v(a_2),m_2\}.$

如果 $v(a_1)<m_1,v(a_2)\ge m_2$, 那么 $x_1^{-1}(a_1+{\pi }_0^{m_1}{O}_{F_0})$ 的赋值就是 $v(a_1)-v(x_1)$, $\eta$ 在其上取常值 $\eta (x_1^{-1}{a}_1)$. 而 $a_2+{\pi }_0^{m_2}{O}_{F_0}$ 就是 ${\pi }_0^{m_2}{O}_{F_0}$. 假定积分区域非空, 那么其就是 $x_1^{-1}(a_1+{\pi }_0^{m_1}{O}_{F_0})$, 积分为$${\int }_{x_1^{-1}(a_1+{\pi }_0^{m_1}{O}_{F_0})}\eta (x)\frac{dx}{|x|}=\eta (x_1^{-1}{a}_1){\int }_{x_1^{-1}(a_1+{\pi }_0^{m_1}{O}_{F_0})}\frac{dx}{|x|}=\eta (x_1^{-1}{a}_1)q^{-m_1+v(a_1)}.$$

如果 $v(a_1)\ge m_1,v(a_2)<m_2$, 那么 $(x_2^{-1}(a_2+{\pi }_0^{m_2}{O}_{F_0}){)}^{-1}$ 的赋值就是 $v(x_2)-v(a_1)$, $\eta$ 在其上取常值 $\eta (x_2^{-1}{a}_2)$. 而 $a_1+{\pi }_0^{m_1}{O}_{F_0}$ 就是 ${\pi }_0^{m_1}{O}_{F_0}$. 假定积分区域非空, 那么其就是 $(x_2^{-1}(a_2+{\pi }_0^{m_2}{O}_{F_0}){)}^{-1}$, 积分为$$\eta (x_2{a}_2^{-1}){\int }_{(x_2^{-1}(a_2+{\pi }_0^{m_2}{O}_{F_0}){)}^{-1}}\frac{dx}{|x|}=\eta (x_2{a}_2^{-1})q^{-m_2+v(a_2)}.$$

$v(a_1)<m_1,v(a_2)<m_2$ 的情况是类似且更简单的.

如果 $v(a_1)<m_1,v(a_2)<m_2$, 此时$$\begin{array}{r}(a_1+{\pi }_0^{m_1}{O}_{F_0})(a_2+{\pi }_0^{m_2}{O}_{F_0})=a_1{a}_2(1+{\pi }_0^{m_1-v(a_1)}{O}_{F_0})(1+{\pi }^{m_2-v(a_2)}{O}_{F_0})\\ =a_1{a}_2(1+{\pi }_0^{\min \{m_1-v(a_1),m_2-v(a_2))\}}{O}_{F_0}).\end{array}$$假定积分区域非空, 那么 $\eta$ 在积分区域上取常值 $\eta (x_1^{-1}{a}_1)=\eta (x_2{a}_2^{-1})$. 积分区域的体积为 $q^{-\max \{m_1-v(a_1),m_2-v(a_2))\}}$. 于是积分为 $\eta (x_1^{-1}{a}_1)q^{-\max \{m_1-v(a_1),m_2-v(a_2))\}}.$

$\square$

定义 1.8. ${\Pi }^{\prime }(H,x_1,x_2):=\eta (x_1)II(H,x_1,x_2)$.

推论 1.9. 对 $t,a,a^{\prime }\in F_0^{\times }$, 有 ${\Pi }^{\prime }(1_{{\pi }_0^{m_1}{O}_{F_0}}\times 1_{{\pi }_0^{m_2}{O}_{F_0}},t,1)=0.$$${\Pi }^{\prime }(1_{a(1+{\pi }_0{O}_{F_0})}\times 1_{{\pi }_0^m{O}_{F_0}},t,1)=\{\begin{array}{ll}\eta (a)q^{-1},& t\in a{\pi }_0^m{O}_{F_0}.\\ 0,& \text{else}.\end{array}$$$${\Pi }^{\prime }(1_{{\pi }_0^m{O}_{F_0}}\times 1_{a(1+{\pi }_0{O}_{F_0})},t,1)=\{\begin{array}{ll}\eta (at)q^{-1},& t\in a{\pi }_0^m{O}_{F_0}.\\ 0,& \text{else}.\end{array}$$$${\Pi }^{\prime }(1_{a(1+{\pi }_0{O}_{F_0})}\times 1_{a^{\prime }(1+{\pi }_0{O}_{F_0})},t,1)=\{\begin{array}{ll}\eta (a)q^{-1},& t\in a{a}^{\prime }(1+{\pi }_0{O}_{F_0}).\\ 0,& \text{else}.\end{array}$$

证明. 计算相应的 $(a_1+{\pi }_0^{m_1}{O}_{F_0})(a_2+{\pi }_0^{m_2}{O}_{F_0})$ 即可.

$\square$

例 1.10. ${\Pi }^{\prime }(1_{O_{F_0}}\times 1_{1+{\pi }_0{O}_{F_0}},t,1)=q^{-1}\eta (t)1_{O_{F_0}}$, ${\Pi }^{\prime }(1_{1+{\pi }_0{O}_{F_0}}\times 1_{O_{F_0}},t,1)=q^{-1}{1}_{O_{F_0}}$ .

定义 1.11. 对 $a\in {\mathbb{F}}_q^{\times },m\in \mathbb{Z}$, 定义开集 $U_{a,m}=[a]{\pi }_0^m(1+{\pi }_0{O}_{F_0})$. 开集 $U_{a,m}$ 两两不交, 且$$F_0^{\times }=\bigcup _{a,m}{U}_{a,m},{\pi }_0^m{O}_{F_0}-{\pi }_0^{m+1}{O}_{F_0}=\bigcup _a{U}_{a,m}.$$

推论 1.12. $${\Pi }^{\prime }(1_{U_{a,m}}\times 1_{U_{a^{\prime },m^{\prime }}})=\eta (a)q^{-1}{1}_{U_{a{a}^{\prime },m+m^{\prime }}}.$$

此推论可推出其他公式, 如$${\Pi }^{\prime }(1_{U_{a,m}},1_{{\pi }_0^{m^{\prime }}{O}_F})={\Pi }^{\prime }((1_{U_{a,m}},\sum _{a^{\prime },m^{\prime \prime }\ge m^{\prime }}{1}_{U_{a^{\prime },m^{\prime \prime }}})=\sum _{a^{\prime },m^{\prime \prime }\ge m^{\prime }}\eta (a)q^{-1}{1}_{U_{a{a}^{\prime },m+m^{\prime \prime }}}=\eta (a)q^{-1}{1}_{a{\pi }_0^{m^{\prime }}{O}_{F_0}}.$$

2零化积分元

定义 ${}^{t}H(x,y):=H(y,x)$. 那么$${\Pi }^{\prime }({}^{t}H,x_1,x_2)=\eta (x_1)II({}^{t}H,x_1,x_2)=\eta (x_1)II(H,x_2,x_1)=\eta (x_1{x}_2){\Pi }^{\prime }(H,x_2,x_1).$$

考虑如下形式的 $H$ ($c_{a,m},d_{a^{\prime },m^{\prime }}\in \mathbb{Q}$)$$H=\sum _{a,m,a^{\prime },m^{\prime }}{c}_{a,m}{d}_{a^{\prime },m^{\prime }}{1}_{U_{a,m}}\times 1_{U_{a^{\prime },m^{\prime }}}.$$

命题 2.1. $${\Pi }^{\prime }(H)=q^{-1}\sum _{a,m}(\sum _{a_1,m_1,a_2,m_2,a_1{a}_2=a,m_1+m_2=m}\eta (a_1)c_{a_1,m_1}{d}_{a_2,m_2})1_{U_{a,m}}.$$

命题 2.2. 称 $H$ 是一个零化积分元, 如果 ${\Pi }^{\prime }(H,t,1)=0$ 对所有 $t$ 成立. 这等价于$$\sum _{a_1,m_1,a_2,m_2,a_1{a}_2=a,m_1+m_2=m}\eta (a_1)c_{a_1,m_1}{d}_{a_2,m_2}=0.$$

那么 ${}^{t}H$ 也是零化积分元.

命题 2.3. ${\Pi }^{\prime }(1_U\times 1_V,t,1)={\int }_{x.t\in U,x.1\in V}\eta (x)d^{\times }x.$