On CM masses over global function fields

Abstract

The purpose of this paper is to study the masses over CM points on Drinfeld–Stuhler modular curves and definite Shimura curves over global function fields. After connecting the CM masses with the modified Hurwitz class numbers, we show that the generating function of these masses is a particular metaplectic theta series. Applying the Poisson summation formula, we are able to describe explicitly the meromorphic continuation of the Mellin transform of the generating theta series. This enables us to derive not only a Gauss-type mean value formula of the average masses in question, but also provides “new” class number relations in the positive characteristic world.

Appendices

Appendix A. Local optimal embeddings

Here we recall the needed properties of local optimal embeddings from a quadratic order into a hereditary order of a quaternion algebra over a local field. Further details are referred to [34, Chapter 2, Sect. 3] and [6, Chapter 5, Sect. 1.1].

Let $(L,|\cdot |_L)$ be a non-archimedean local field, and $O_L$ be the ring of integers in L. Given a separable quadratic algebra E over L and a quaternion algebra ${\mathcal {D}}$ over L together with a fixed embedding $\iota : E\hookrightarrow {\mathcal {D}}$, it is known that every embedding from E into ${\mathcal {D}}$ must be conjugates of $\iota$ by an element of ${\mathcal {D}}^\times$. Let ${\mathcal {O}}$ be an $O_L$-order in E and $O_{\mathcal {D}}$ a maximal $O_L$-order in ${\mathcal {D}}$. Put

$$\begin{aligned} {\mathcal {E}}({\mathcal {O}},O_{\mathcal {D}}) := \{b \in {\mathcal {D}}^\times \mid b^{-1}Eb \cap O_{\mathcal {D}}= b^{-1} {\mathcal {O}}b \}. \end{aligned}$$

Here we identify E as a subalgebra of ${\mathcal {D}}$ via $\iota$. For $\alpha \in E$, $b \in {\mathcal {E}}({\mathcal {O}},O_{\mathcal {D}})$, and $\kappa \in O_{\mathcal {D}}^\times$, one has

$$\begin{aligned} \alpha \cdot b \cdot \kappa \in {\mathcal {E}}({\mathcal {O}},O_{\mathcal {D}}). \end{aligned}$$

Moreover, the following result holds (cf. [34, Chapter 2, Theorem 3.1 and 3.2]):

Lemma A.1

• (1) Let $O_E$ be the maximal $O_L$-order in E. Then

  $$\begin{aligned} e(O_E,O_{\mathcal {D}}):= \#\left( E^\times \backslash {\mathcal {E}}(O_E,O_{\mathcal {D}})/O_{\mathcal {D}}^\times \right) = {\left\{ \begin{array}{ll} 2, &{} \text { if } {\mathcal {D}}\text { is division and } E/L \text { is inert;}\\ 0, &{} \text { if } {\mathcal {D}}\text { is division and } E/L \text { is split;}\\ 1, &{} \text { otherwise.} \end{array}\right. } \end{aligned}$$

• (2) If ${\mathcal {O}}\subsetneq O_E$, then

  $$\begin{aligned} e({\mathcal {O}},O_{\mathcal {D}}):= \#\left( E^\times \backslash {\mathcal {E}}({\mathcal {O}},O_{\mathcal {D}})/O_{\mathcal {D}}^\times \right) = {\left\{ \begin{array}{ll} 0, &{} \text { if } {\mathcal {D}}\text { is division;} \\ 1, &{} \text { otherwise.} \end{array}\right. } \end{aligned}$$

Suppose ${\mathcal {D}}$ is not division (i.e. ${\mathcal {D}}\cong {\text {Mat}}_2(L)$). Let $O_{\mathcal {D}}'$ be a hereditary $O_L$-order in $O_{\mathcal {D}}$. Put

$$\begin{aligned} {\mathcal {E}}({\mathcal {O}},O_{\mathcal {D}}') := \{b \in {\mathcal {D}}^\times \mid b^{-1}Eb \cap O_{\mathcal {D}}' = b^{-1}{\mathcal {O}}b\}. \end{aligned}$$

Then for $\alpha \in E$, $b \in {\mathcal {E}}({\mathcal {O}},O_{\mathcal {D}}')$, and $\kappa ' \in (O_{\mathcal {D}}')^\times$, one has

$$\begin{aligned} \alpha \cdot b \cdot \kappa ' \in {\mathcal {E}}({\mathcal {O}},O_{\mathcal {D}}'). \end{aligned}$$

Moreover (cf. [34, Chapter 2, Theorem 3.2]):

Lemma A.2

• (1)
  $$\begin{aligned} e(O_E,O_{\mathcal {D}}') := \#(E^\times \backslash {\mathcal {E}}(O_E,O_{\mathcal {D}}')/(O_{\mathcal {D}}')^\times ) = {\left\{ \begin{array}{ll} 0, &{} \text { if } E/L \text { is inert;}\\ 1, &{} \text { if } E/L \text { is ramified;}\\ 2, &{} \text { if } E/L \text { is split.} \end{array}\right. } \end{aligned}$$

• (2) If ${\mathcal {O}}\subsetneq O_E$, then

  $$\begin{aligned} e({\mathcal {O}},O_{\mathcal {D}}'):= \#(E^\times \backslash {\mathcal {E}}({\mathcal {O}},O_{\mathcal {D}}')/(O_{\mathcal {D}}')^\times ) = 2. \end{aligned}$$

Appendix B. Special local integrals

Given $c \in {{\mathbb {Z}}}_{\ge 0}$, put ${\mathcal {O}}(c) := O_L + \pi _L^c O_E$, where $\pi _L \in O_L$ is a uniformizer in L. For $x \in E\backslash L$, we can find a unique $c_x \in {{\mathbb {Z}}}_{\ge 0}$ so that $O_L[x] = {\mathcal {O}}(c_x)$ if $x \in O_E$; and put $c_x := -1$ if $x \notin O_E$. Let ${\mathcal {D}}^o$ be the space of pure quaternions in ${\mathcal {D}}$, i.e.

$$\begin{aligned} {\mathcal {D}}^o:= \{ b \in {\mathcal {D}}\mid {\text {Tr}}(b) = 0\}. \end{aligned}$$

Put $O_{\mathcal {D}}^o := O_{\mathcal {D}}\cap {\mathcal {D}}^o$ and $O_{{\mathcal {D}}}^{\prime , o} := O_{{\mathcal {D}}}'\cap {\mathcal {D}}^o$. We observe that:

Lemma B.1

Given $x \in E \backslash L$ with ${\text {Tr}}(x) = 0$, one has

$$\begin{aligned} {\textbf{1}}_{O_{\mathcal {D}}^o}(b^{-1}xb) = \sum _{\ell = 0}^{c_x} {\textbf{1}}_{{\mathcal {E}}({\mathcal {O}}(\ell ),O_{\mathcal {D}})}(b), \end{aligned}$$

Moreover, if ${\mathcal {D}}$ is not division, then

$$\begin{aligned} {\textbf{1}}_{O_{\mathcal {D}}^{\prime ,o}}(b^{-1}xb) = \sum _{\ell = 0}^{c_x} {\textbf{1}}_{{\mathcal {E}}({\mathcal {O}}(\ell ),O_{\mathcal {D}}')}(b). \end{aligned}$$

Proof

Notice that ${\mathcal {E}}\left( {\mathcal {O}}(\ell ),\mathcal O_{{\mathcal {D}}}\right)$ and ${\mathcal {E}}\left( \mathcal O(\ell '),{\mathcal {O}}_{{\mathcal {D}}}\right)$ are disjoint if $\ell \ne \ell '$. Thus for $b \in {\mathcal {D}}^\times$ one has

$$\begin{aligned} \sum ^{c_x}_{\ell =0}{\textbf{1}}_{{\mathcal {E}}(\mathcal O(\ell ),{\mathcal {O}}_{{\mathcal {D}}})}(b)=0\ \text {or}\ 1. \end{aligned}$$

Suppose the value is 1, i.e. $b\in {\mathcal {E}}(\mathcal O(\ell _0),{\mathcal {O}}_{{\mathcal {D}}})$ for some $0\le \ell _0\le c_x$. Then

$$\begin{aligned} x\in {\mathcal {O}}[x]=\mathcal O(c_x)\subset {\mathcal {O}}(\ell _0)\subset E\cap b{\mathcal {O}}_{{\mathcal {D}}}b^{-1}\subset b{\mathcal {O}}_{{\mathcal {D}}}b^{-1}. \end{aligned}$$

Since ${\text {Tr}}(x)=0$, we get $b^{-1}xb\in {\mathcal {O}}^\circ _{{\mathcal {D}}}$, i.e. ${\textbf{1}}_{{\mathcal {O}}^\circ _{{\mathcal {D}}}}(b^{-1}xb)=1$.

Conversely, let $b\in {\mathcal {D}}^\times$ with ${\textbf{1}}_{{\mathcal {O}}^\circ _{{\mathcal {D}}}}(b^{-1}xb)=1$. Then $x\in b{\mathcal {O}}^\circ _{{\mathcal {D}}}b^{-1}$, which implies ${\mathcal {O}}(c_x)\subset E\cap b{\mathcal {O}}_{{\mathcal {D}}}b^{-1}$. Thus there exists $\ell _0$ with $0\le \ell _0\le c_x$ such that

$$\begin{aligned} E\cap b{\mathcal {O}}_{{\mathcal {D}}}b^{-1}={\mathcal {O}}(\ell _0). \end{aligned}$$

which means that $b\in {\mathcal {E}}({\mathcal {O}}(\ell _0),{\mathcal {O}}_{{\mathcal {D}}})$. Therefore

$$\begin{aligned} \sum ^{c_x}_{\ell =0}{\textbf{1}}_{{\mathcal {E}}({\mathcal {O}}(\ell ),{\mathcal {O}}_{{\mathcal {D}}})}(b)= {\textbf{1}}_{{\mathcal {E}}({\mathcal {O}}(\ell _0),{\mathcal {O}}_{{\mathcal {D}}})}(b)=1. \end{aligned}$$

$\square$

Suppose Haar measures of ${\mathcal {D}}^\times$ and $E^\times$ are chosen, respectively. The above lemma leads to:

Corollary B.2

For $x \in E\backslash L$ with ${\text {Tr}}(x) = 0$, one has

$$\begin{aligned} \int _{E^\times \backslash {\mathcal {D}}^\times } {\textbf{1}}_{O_{\mathcal {D}}^o}(b^{-1}x b) \,d^\times b \ =\ \frac{\text { vol}(O_{\mathcal {D}}^\times )}{\text { vol}(O_E^\times )} \cdot \sum _{\ell = 0}^{c_x}\#\left( \frac{O_E^\times }{{\mathcal {O}}(\ell )^\times }\right) \cdot e({\mathcal {O}}(\ell ),O_{\mathcal {D}}). \end{aligned}$$

Suppose ${\mathcal {D}}$ is not division, then

$$\begin{aligned} \int _{E^\times \backslash {\mathcal {D}}^\times } {\textbf{1}}_{O_{\mathcal {D}}^{\prime ,o}}(b^{-1}xb)\,d^\times b \ =\ \frac{\text { vol}((O_{\mathcal {D}}')^\times )}{\text { vol}(O_E^\times )} \cdot \sum _{\ell = 0}^{c_x}\#\left( \frac{O_E^\times }{{\mathcal {O}}(\ell )^\times }\right) \cdot e({\mathcal {O}}(\ell ),O_{\mathcal {D}}'). \end{aligned}$$

Proof

Given $0\le \ell \le c_x$, one has

$$\begin{aligned} \text {vol}(E^{\times }\backslash {\mathcal {E}}({\mathcal {O}}(\ell ),{\mathcal {O}}_{{\mathcal {D}}}))&=\sum _{b\in E^{\times }\backslash {\mathcal {E}}({\mathcal {O}}(\ell ),{\mathcal {O}}_D)/{\mathcal {O}}^\times _{{\mathcal {D}}}} \frac{\text {vol}({\mathcal {O}}^\times _{{\mathcal {D}}})}{\text {vol}(E^\times \cap b{\mathcal {O}}^\times _{{\mathcal {D}}}b^{-1})}\\&=\frac{\text {vol}({\mathcal {O}}^\times _{{\mathcal {D}}})}{\text {vol}\left( {\mathcal {O}}^\times _E\right) }\cdot \#\left( \frac{{\mathcal {O}}^\times _E}{{\mathcal {O}}(\ell )^\times }\right) \cdot e\left( {\mathcal {O}}(\ell ),{\mathcal {O}}_{{\mathcal {D}}}\right) . \end{aligned}$$

Thus

$$\begin{aligned} \int _{E^\times \backslash {\mathcal {D}}^\times }{\textbf{1}}_{{\mathcal {O}}^\circ _{{\mathcal {D}}}}(b^{-1}xb)\,d^\times b&=\sum ^{c_x}_{\ell =0}\int _{E^\times \backslash {\mathcal {D}}^\times }{\textbf{1}}_{{\mathcal {E}}({\mathcal {O}}(\ell ),{\mathcal {O}}_{{\mathcal {D}}})}(b)\,d^\times b\\&=\frac{\text {vol}({\mathcal {O}}^\times _{{\mathcal {D}}})}{\text {vol}({\mathcal {O}}^\times _E)}\cdot \sum ^{c_x}_{\ell =0}\#\left( \frac{{\mathcal {O}}^\times _E}{{\mathcal {O}}(\ell )^\times }\right) \cdot e\left( {\mathcal {O}}(\ell ),{\mathcal {O}}_{{\mathcal {D}}}\right) . \end{aligned}$$

$\square$

Let $q_L$ be the cardinality of the residue field of L. Since

$$\begin{aligned} \text {vol}((O_{\mathcal {D}}')^\times ) = \frac{1}{q_L+1}\cdot \text {vol}(O_{\mathcal {D}}^\times ), \end{aligned}$$

combining Lemmas A.1, A.2, and Corollary B.2 we obtain:

Corollary B.3

Set

$$\begin{aligned} \varphi := {\left\{ \begin{array}{ll} \displaystyle \frac{1}{2}\cdot {\textbf{1}}_{O_{\mathcal {D}}^o}, &{} \text { if } D \text { is division;}\\ {\textbf{1}}_{O_{\mathcal {D}}^o} - \displaystyle \frac{q_L+1}{2} {\textbf{1}}_{O_{\mathcal {D}}^{\prime ,o}}, &{} \text { otherwise.} \end{array}\right. } \end{aligned}$$

Then for $x \in O_E\backslash O_L$ with ${\text {Tr}}(x) = 0$, one has that

$$\begin{aligned} \int _{E^\times \backslash {\mathcal {D}}^\times } \varphi (b^{-1}x b)\,d^\times b= & {} {\left\{ \begin{array}{ll} \displaystyle \frac{1}{e(E/L)} \cdot \frac{\text { vol}(O_{\mathcal {D}}^\times )}{\text { vol}(O_E^\times )}, &{}\text { if } E \text { is a field,} \\ 0, &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

Here e(E/L) is the ramification index of E/L.