Equidistribution of Gross points over rational function fields

TL;DR

This paper establishes a sparse equidistribution theorem for Gross points over rational function fields and applies it to analyze the reduction map from CM Drinfeld modules to supersingular ones, using advanced automorphic form techniques.

Contribution

It introduces a new sparse equidistribution result for Gross points over $\,\mathbb{F}_q(t)$ and connects it to the reduction theory of Drinfeld modules.

Findings

Proves a sparse equidistribution theorem for Gross points

Analyzes the reduction map from CM to supersingular Drinfeld modules

Utilizes a Lindelöf-type bound for Rankin-Selberg L-values

Abstract

In this paper we prove a sparse equidistribution theorem for Gross points over the rational function field $\mathbb{F}_q(t)$. We apply this result to study the reduction map from CM Drinfeld modules to supersingular Drinfeld modules. Our proofs rely crucially on a period formula due to M. Papikian and F.-T. Wei/J. Yu, and a Lindel\"of-type bound for central values of Rankin-Selberg $L$-functions associated to twists of automorphic forms of Drinfeld-type by ideal class group characters.

Equations

$$r_{\mathfrak{P}}:\mathcal{D}({\mathcal{O}}_D)⟶{\mathcal{D}}^{ss}({\overline{\mathbb{F}}}_{P_0}).$$

$$R_i:=\{x\in B:I_{i}x\subseteq I_i\}.$$

$$n=[{\hat{R}}^{\times }\backslash {\hat{B}}^{\times }/B^{\times }].$$

$${\hat{B}}^{\times }=\bigcup _{i=1}^n{\hat{R}}^{\times }{g}_i{B}^{\times }.$$

$$R_i=B\cap g_i^{-1}\hat{R}{g}_i.$$

$$f(K)\cap g_i^{-1}\hat{R}{g}_i=f({\mathcal{O}}_D)$$

$${\hat{R}}^{\times }\backslash {\hat{B}}^{\times }\times \mathrm{Hom}(K,B)$$

$$(g,f)\mapsto (gb,b^{-1}fb).$$

$$({\hat{R}}^{\times }\backslash {\hat{B}}^{\times }\times \mathrm{Hom}(K,B))/B^{\times }$$

$$f(K)\cap g^{-1}\hat{R}g=f({\mathcal{O}}_D).$$

$$Y(K)=\{y\in B{\otimes }_{k}K:y\ne 0,\mathrm{Tr}(y)=N(y)=0\}/K^{\times }.$$

$$X_{P_0}:=({\hat{R}}^{\times }\backslash {\hat{B}}^{\times }\times Y)/B^{\times }.$$

$$X_{P_0}\to \underset{i=1}{\overset{n}{⨆}}{X}_i$$

$$({\hat{R}}^{\times }{g}_i,y)\mathrm{mod}{B}^{\times }⟼y\mathrm{mod}{R}_i^{\times },$$

$$\text{Image}\left[{\hat{R}}^{\times }\backslash {\hat{B}}^{\times }\times Y(K)⟶X_{P_0}(K)\right]$$

$$\mathrm{Pic}({\mathcal{O}}_D)\cong {\hat{\mathcal{O}}}_D^{\times }\backslash {\hat{K}}^{\times }/K^{\times }.$$

$$x_a:=(g\hat{f}(a),y).$$

$$R_{i,\mathfrak{a}}^{\prime }=\{b\in B:R_i\mathfrak{a}b\subseteq R_i\mathfrak{a}\}.$$

$$\mathrm{Pic}(X_{P_0})=\mathbb{Z}{e}_1\oplus \cdots \oplus \mathbb{Z}{e}_n.$$

$${\mu }_{P_0}(e_i):=\frac{w_i^{-1}}{{\sum }_{j=1}^n{w}_j^{-1}}.$$

$$\sum _{j=1}^n\frac{1}{w_j}=\frac{q^{\mathrm{deg}(P_0)}-1}{q^2-1},$$

$$G_D\cdot x_{0,D}:=\{x_{0,D}^{\sigma }:\sigma \in G_D\}\subseteq \mathcal{S}.$$

$$\#(G_D\cdot x_{0,D})\to \infty$$

$$[\mathrm{Pic}({\mathcal{O}}_D):G_D]\le \Vert D{\Vert }^{\eta }$$

$$\#(G_D\cdot x_{0,D}){\gg }_{\epsilon }{q}^{-1}\Vert D{\Vert }^{(\frac{1}{2}-\eta )-\epsilon }$$

$$L({\chi }_K,1)=q\Vert D{\Vert }^{-1/2}h(D).$$

$$L({\chi }_K,1){\gg }_{\epsilon }\Vert D{\Vert }^{-\epsilon }$$

$$h(D){\gg }_{\epsilon }{q}^{-1}\Vert D{\Vert }^{\frac{1}{2}-\epsilon }.$$

$$\#(G_D\cdot x_{0,D})=|G_D|=\frac{h(D)}{[\mathrm{Pic}({\mathcal{O}}_D):G_D]}\gg \frac{h(D)}{\Vert D{\Vert }^{\eta }}{\gg }_{\epsilon }{q}^{-1}\Vert D{\Vert }^{(\frac{1}{2}-\eta )-\epsilon }.$$

$$[\mathrm{Pic}({\mathcal{O}}_D):G_D]\le \Vert D{\Vert }^{\eta },$$

Full text

This publication was made possible by the NPRP award [NPRP 9-336-1-069] from the Qatar National Research Fund (a member of The Qatar Foundation). The statements made herein are solely the responsibility of the authors.

1. Introduction and statement of results

In their ICM article [MV], Michel and Venkatesh developed a general framework concerning sparse equidistribution problems for toric orbits of special points on Shimura varieties constructed from quaternion algebras over totally real fields. These problems have been solved in a wide range of settings and have many striking applications. For example, Michel [M] proved sparse equidistribution of Gross points corresponding to supersingular elliptic curves and used this to study the supersingular reduction of CM elliptic curves.

In this paper we will prove a sparse equidistribution theorem for Gross points over the rational function field $k=\mathbb{F}_{q}(t)$; see Theorem 1.3 and Corollary 1.4. Roughly speaking, given a prime $P_{0}$, one can construct a definite Shimura curve $X_{P_{0}}$ which is a disjoint union of genus zero curves. Given a Gross point $x_{0,D}$ of discriminant $D$ on $X_{P_{0}}$ and a subgroup $G_{D}<\textrm{Pic}(\mathcal{O}_{D})$ of sufficiently small index, we prove that the orbit $G_{D}\cdot x_{0,D}$ becomes quantitatively equidistributed with respect to a natural probability measure on $X_{P_{0}}$ as $\textrm{deg}(D)\rightarrow\infty$. This result can be viewed as part of the framework developed in [MV], in which the quaternion algebra is definite and the base field is $\mathbb{F}_{q}(t)$.

In analogy with [M], we apply Theorem 1.3 to study the supersingular reduction of CM Drinfeld modules. Assume that $P_{0}$ is inert in $K=k(\sqrt{D})$. Let $\mathcal{D}(\mathcal{O}_{D})$ be the set of isomorphism classes of rank 2 Drinfeld modules over $\overline{k}$ with complex multiplication by $\mathcal{O}_{D}$ and let $\mathcal{D}^{ss}(\overline{\mathbb{F}}_{P_{0}})$ be the set of isomorphism classes of supersingular Drinfeld modules over $\overline{\mathbb{F}}_{P_{0}}$. If $\mathfrak{P}$ is a prime above $P_{0}$ in the Hilbert class field $H$ of $K$, there is a reduction map

[TABLE]

We will prove that the reduction map $r_{\mathfrak{P}}$ is surjective for all $\textrm{deg}(D)\gg\textrm{deg}(P_{0})$, in a quantitative sense which we make precise; see Theorem 1.7.

1.1. Gross points over $\mathbb{F}_{q}(t)$

We begin by developing some background we will need concerning Gross points over $\mathbb{F}_{q}(t)$ following closely the discussion in Gross [Gr], Wei and Yu [WY, §§1.1–2].

Let $q\in\mathbb{Z}$ be a power of an odd prime, $A=\mathbb{F}_{q}[t]$ be the polynomial ring in a variable $t$, and $k=\mathbb{F}_{q}(t)$ be the rational function field. We let $k_{\infty}=\mathbb{F}_{q}((t^{-1}))$ be the completion of $k$ at its infinite place. Let $P_{0}\in A$ be a monic irreducible polynomial of degree $\deg(P_{0})\geq 3$. Let $D\in A$ be irreducible of odd degree, in particular ensuring that $K=k(\sqrt{D})$ is imaginary quadratic, i.e., $\infty$ does not split in $K$. Assume further that $P_{0}$ is inert in $K$, and so $\chi_{K}(P_{0})=-1$, where $\chi_{K}$ is the quadratic character associated to $K$. Let $\mathcal{O}_{D}=A[\sqrt{D}]$ be the ring of integers, $\mathrm{Pic}(\mathcal{O}_{D})$ be the class group, and $h(D)$ be the class number of $K$, respectively.

Let $B$ be the quaternion algebra over $k$ which is ramified at $P_{0}$ and $\infty$, and fix a maximal $A$-order $R$ of $B$. There are finitely many equivalences classes of left ideals of $R$, and the number $n$ of such classes is called the class number of $R$. Let $\{I_{i}\}_{i=1}^{n}$ be a set of representatives of these equivalence classes. One associates to each $I_{i}$ the maximal right $A$-order of $B$ defined by

[TABLE]

Given a finite place $v$ of $k$, let $k_{v}$ be the completion of $k$ at $v$, let $A_{v}$ be the closure of $A$ in $k_{v}$, let $\hat{k}$ be the finite adele ring of $k$, and let $\hat{A}=\prod_{v}A_{v}$. Also, let $\hat{R}=R\otimes_{A}\hat{A}$ and $\hat{B}=B\otimes_{k}\hat{k}$.

The left ideal classes of $R$ correspond to the orbits of $B^{\times}$ acting on the right of $\hat{R}^{\times}\backslash\hat{B}^{\times}$, so that the class number $n$ is the number of double cosets,

[TABLE]

The choice of representative ideals $\{I_{i}\}_{i=1}^{n}$ corresponds to a choice of coset representatives $\{g_{i}\}_{i=1}^{n}$ in $\hat{R}^{\times}\backslash\hat{B}^{\times}$ such that

[TABLE]

The right order $R_{i}$ is given by

[TABLE]

An optimal embedding $f:\mathcal{O}_{D}\hookrightarrow R_{i}$ of $\mathcal{O}_{D}$ into $R_{i}$ is a field embedding $f:K\rightarrow B$ which satisfies

[TABLE]

in $\hat{B}$. The group $B^{\times}$ acts on the right of the set

[TABLE]

by

[TABLE]

There is a bijection between the set of all optimal embeddings of $\mathcal{O}_{D}$ into the $n$ orders $R_{i}$, modulo conjugation by $R_{i}^{\times}$, and the classes $(g,f)\mod B^{\times}$ in the quotient space

[TABLE]

satisfying

[TABLE]

The quotient space (2) can be interpreted geometrically as the set of $K$-points of a definite Shimura curve $X_{P_{0}}$ defined over $k$ as follows. There is a genus zero curve $Y$ defined over $k$ associated to $B$ whose points over $K$ are

[TABLE]

The curve $Y$ can be described explicitly as a conic in $\mathbb{P}^{2}$. The group $B^{\times}$ acts on $Y$ on the right by conjugation. Moreover, $Y(K)$ can be canonically identified with the set of embeddings $\mathrm{Hom}(K,B)$. Define the definite Shimura curve $X_{P_{0}}$ by the double coset space

[TABLE]

The decomposition (1) gives a bijection to a disjoint union of genus [math] curves,

[TABLE]

defined by

[TABLE]

where $X_{i}:=Y/R_{i}^{\times}$ and $R_{i}^{\times}=g_{i}^{-1}\hat{R}^{\times}g_{i}^{-1}\cap B^{\times}$ is a finite group for $i=1,\ldots,n$.

Definition 1.1**.**

A Gross point of discriminant $D$ on $X_{P_{0}}(K)$ is a point $x=(g,y)$ in the image

[TABLE]

such that the embedding $f\in\mathrm{Hom}(K,B)$ corresponding to the component $y\in Y(K)$ of $x$ satisfies (3). If the component $g\in\hat{R}^{\times}\backslash\hat{B}^{\times}/B^{\times}$ of $x$ is congruent to the double coset representative $g_{i}$, then $x$ lies on the component $X_{i}(K)$ of $X_{P_{0}}(K)$.

Let $\mathrm{Gr}_{D,P_{0}}$ denote the set of Gross points of discriminant $D$. There is an action of the group $\mathrm{Pic}(\mathcal{O}_{D})$ on $\mathrm{Gr}_{D,P_{0}}$ given as follows. Let $\hat{K}=K\otimes_{k}\hat{k}$ and $\hat{\mathcal{O}}_{D}=\mathcal{O}_{D}\otimes_{A}\hat{A}$. Then

[TABLE]

Let $x=(g,y)$ be a Gross point of discriminant $D$, and let $f\in\mathrm{Hom}(K,B)$ be the embedding corresponding to the component $y$. This embedding induces a homomorphism $\hat{f}:\hat{K}^{\times}\rightarrow\hat{B}^{\times}$. Let $a\in\hat{K}^{\times}$ and define

[TABLE]

This gives a free action of $\mathrm{Pic}(\mathcal{O}_{D})$ on $\mathrm{Gr}_{D,P_{0}}$ which divides the set of Gross points into two simple transitive orbits of size $h(D)$; in particular, $\#\mathrm{Gr}_{D,P_{0}}=2h(D)$ (see e.g. [Gr, p. 133] and [WY, Lem. 1.4]). We denote this action by $x\mapsto x^{\sigma}$ for $\sigma\in\mathrm{Pic}(\mathcal{O}_{D})$.

The action of $\mathrm{Pic}(\mathcal{O}_{D})$ on the set of Gross points of discriminant $D$ also translates to an action of $\mathrm{Pic}(\mathcal{O}_{D})$ on the corresponding optimal embeddings. To describe this, we follow the first paragraph of [Gr, p. 134].

Let ${\mathfrak{a}}$ be the ideal (projective module of rank 1 in $K$) which is determined by the idele $a\bmod{\hat{\mathcal{O}}_{D}}^{\times}$; specifically, ${\mathfrak{a}}=K\cap a\hat{\mathcal{O}}_{D}$. Let $R_{i,{\mathfrak{a}}}^{\prime}$ be the right order of the left $R_{i}$-module $R_{i}{\mathfrak{a}}$. More precisely, since $R_{i}$ is a maximal right order of $B$, we can consider the set of $n$ equivalence classes of left $R_{i}$-ideals in $B$. Then, viewing $R_{i}{\mathfrak{a}}$ as a left $R_{i}$-ideal, we can (as above) associate to $R_{i}{\mathfrak{a}}$ a maximal right order $R_{i,{\mathfrak{a}}}^{\prime}$ of $B$ defined by

[TABLE]

Since $\mathcal{O}_{D}$ also acts on the right of ${\mathfrak{a}}$, the optimal embedding $f:\mathcal{O}_{D}\hookrightarrow R_{i}$ corresponding to the Gross point $x$ induces an optimal embedding $f^{\prime}:\mathcal{O}_{D}\hookrightarrow R_{i,{\mathfrak{a}}}^{\prime}$ corresponding to the Gross point $x_{a}$. We denote the embedding $f^{\prime}$ by $f^{\sigma}$ and the maximal order $R_{i,{\mathfrak{a}}}^{\prime}$ by $R_{i,\sigma}^{\prime}$, where $\sigma\in\mathrm{Pic}(\mathcal{O}_{D})$ corresponds to the class of ${\mathfrak{a}}$ in $\mathrm{Pic}(\mathcal{O}_{D})$.

1.2. Equidistribution of Gross points over $\mathbb{F}_{q}(t)$

As we have seen, the Shimura curve $X_{P_{0}}$ is the disjoint union of $n$ genus zero curves $X_{i}$ defined over $k$. Let $\mathrm{Pic}(X_{P_{0}})$ denote the Picard group of $X_{P_{0}}$. If $e_{i}$ denotes the class of degree $1$ in $\mathrm{Pic}(X_{P_{0}})$ corresponding to the component $X_{i}$, then we have

[TABLE]

Since a Gross point $x\in\mathrm{Gr}_{D,{P_{0}}}$ lies on a component $X_{i}$, it determines a class $e_{x}$ in $\mathrm{Pic}(X_{P_{0}})$ which for notational convenience we continue to denote by $x$.

For each $i$, we let $w_{i}:=\#(R_{i}^{\times})/(q-1)$, and we define a probability measure on the set of divisor classes $\mathcal{S}:=\{e_{i}\}_{i=1}^{n}$ by

[TABLE]

We note that Gekeler [Ge1, Satz (5.9)] has proved the mass formula

[TABLE]

which shows how the total mass grows with the degree of $P_{0}$. Furthermore, for each $i$ we have $w_{i}$ is either $1$ or $q+1$.

Now given a Gross point $x=x_{0,D}\in\mathrm{Gr}_{D,P_{0}}$ and a subgroup $G_{D}<\mathrm{Pic}(\mathcal{O}_{D})$, define the $G_{D}$-orbit

[TABLE]

We want to study the distribution of the sequence of orbits $G_{D}\cdot x_{0,D}$ on $\mathcal{S}$ with respect to the probability measure $\mu_{P_{0}}$ as $\deg(D)\rightarrow\infty$. To make sense of this distribution problem, we must have

[TABLE]

as $\deg(D)\rightarrow\infty$. In particular, for fixed $P_{0}$, the size of the orbit $G_{D}\cdot x_{0,D}$ will eventually exceed the number $n=n(P_{0})$ of divisor classes $e_{i}$, which (as we have seen) equals the class number of the maximal order $R$ in $B$. As shown in the following result, the condition (6) is ensured by a suitable bound on the index of $G_{D}$ in $\mathrm{Pic}(\mathcal{O}_{D})$.

Proposition 1.2**.**

Fix an absolute constant $0\leq\eta<1/2$ and let $G_{D}<\mathrm{Pic}(\mathcal{O}_{D})$ be a subgroup of index satisfying

[TABLE]

where $\lVert D\rVert:=q^{\deg(D)}$. Then

[TABLE]

where the implied constant is effective. In particular, $\#(G_{D}\cdot x_{0,D})\rightarrow\infty$ as $\deg(D)\rightarrow\infty$.

Proof.

Let $L(\chi_{K},s)$ be the $L$–function of the quadratic character $\chi_{K}$. Then we have the class number formula (see e.g. [CWY, §2.2])

[TABLE]

By work of Weil, the Riemann hypothesis is known for the $L$–function $L(\chi_{K},s)$, which allows one to prove the following effective Siegel-type bound (see e.g. [AT, Lem. 3.3])

[TABLE]

for any $\varepsilon>0$. This yields the following effective lower bound for the class number,

[TABLE]

In particular, from (7) and (8) we get

[TABLE]

∎

We will prove that if $0\leq\eta<1/4$ and

[TABLE]

then the sequence of orbits $G_{D}\cdot x_{0,D}$ becomes quantitatively equidistributed on $\mathcal{S}$ with respect to $\mu_{P_{0}}$ as $\deg(D)\rightarrow\infty$. In fact, we will deduce our equidistribution results as a consequence of the following theorem.

Theorem 1.3**.**

Let $q\in{\mathbb{Z}}$ be a power of an odd prime, $A=\mathbb{F}_{q}[t]$ be the polynomial ring in a variable $t$, and $k=\mathbb{F}_{q}(t)$ be its fraction field. Let $P_{0}\in A$ be a monic irreducible polynomial satisfying $\deg(P_{0})\geq 3$. Let $D\in A$ be an irreducible polynomial of odd degree such that $P_{0}$ is inert in the imaginary quadratic field $K=k(\sqrt{D})$. Given a subgroup $G_{D}<\mathrm{Pic}(\mathcal{O}_{D})$, a Gross point $x=x_{0,D}\in\mathrm{Gr}_{D,P_{0}}$, and a divisor class $e_{i}\in\mathcal{S}$, define the counting function

[TABLE]

Then for all $\varepsilon>0$, we have

[TABLE]

where the implied constant in the error term is uniform in $e_{i}$ and effective.

The following corollary is now an immediate consequence of Theorem 1.3.

Corollary 1.4**.**

Let notation and assumptions be as in Theorem 1.3. Fix an absolute constant $0\leq\eta<1/4$ and let $G_{D}<\mathrm{Pic}(\mathcal{O}_{D})$ be a subgroup of index satisfying

[TABLE]

Then the sequence of orbits $G_{D}\cdot x_{0,D}$ becomes quantitatively equidistributed on $\mathcal{S}$ with respect to $\mu_{P_{0}}$ as $\deg(D)\rightarrow\infty$. In particular, we have

[TABLE]

Remark 1.5**.**

In their “Equidistribution of Subgroups Conjecture” [MV, Conjecture 1], Michel and Venkatesh conjecture that for any fixed $\eta>0$, if $x_{0,D}$ is a Heegner point of fundamental discriminant $D<0$ and $G_{D}<\textrm{Pic}(\mathcal{O}_{D})$ is a subgroup satisfying $|G_{D}|\geq|D|^{\eta}$, then the sequence of orbits $G_{D}\cdot x_{0,D}$ becomes equidistributed with respect to the hyperbolic measure $d\mu(z):=(3/\pi)dxdy/y^{2}$ on the modular curve $Y_{0}(1)$ as $|D|\rightarrow\infty$. Assuming the Generalized Riemann Hypothesis, this equidistribution statement holds for all $\eta>1/4$. Corollary 1.4 implies that the analogous equidistribution statement for Gross points over $\mathbb{F}_{q}(t)$ holds unconditionally for all $\eta>1/4$. This is because we are able to employ the work of Deligne and Drinfeld on the Riemann hypothesis over function fields.

Remark 1.6**.**

In our setup, we have assumed that the polynomial $D\in A$ has odd degree. This assumption is made for convenience and is not essential in our arguments. For example, the period formula of Papikian [Pa2] and Wei/Yu [WY] which we use takes a slightly different form when $D$ has even degree, but our proof works the same in this case. Simply put, if $D$ has even degree then the equidistribution statements in Theorem 1.3 and Corollary 1.4 hold with the same rate of convergence.

1.3. Supersingular reduction of CM Drinfeld modules

In this section we define the reduction map from CM Drinfeld modules to supersingular Drinfeld modules, and state our result showing that this map is surjective if $\deg(D)$ is sufficiently large compared to $\deg(P_{0})$ (in a quantitative sense which we will make precise). We first recall some definitions about Drinfeld modules (see [Go, Ch. 4], [T, Ch. 3] for more details).

Given a field $L/\mathbb{F}_{q}$, the $q$-th power Frobenius endomorphism $\tau:L\to L$ generates an $\mathbb{F}_{q}$-subalgebra of the endomorphism algebra of the additive group of $L$, which we denote by $L\{\tau\}$. The ring $L\{\tau\}$ is the ring of twisted polynomials in $\tau$ with coefficients in $L$, and it is subject to the relation $\tau c=c^{q}\tau$ for all $c\in L$. Fixing an $\mathbb{F}_{q}$-algebra homomorphism $\iota:A\to L$, a Drinfeld module of rank $r$ over $L$ is an $\mathbb{F}_{q}$-algebra homomorphism

[TABLE]

so that for all $a\in A$,

[TABLE]

For two Drinfeld modules $\phi$, $\psi$ over $L$, a morphism $u:\phi\to\psi$ over $L$ is a twisted polynomial $u\in L\{\tau\}$, so that

[TABLE]

If $\phi$ and $\psi$ have different ranks, then the only possible morphism between them is the zero morphism.

Henceforth, we will primarily focus on Drinfeld modules of rank $2$. Since a Drinfeld module $\phi$ is completely determined by its value $\phi_{t}$, we can fix a rank $2$ Drinfeld module over $L$ by setting,

[TABLE]

in which case the $j$-invariant $j(\phi):=g^{q+1}/\Delta$ is an isomorphism invariant of $\phi$ over an algebraic closure $\overline{L}$.

Let $\mathcal{D}(\mathcal{O}_{D})$ be the set of $\overline{k}$-isomorphism classes of Drinfeld modules of rank $2$ over $\overline{k}$ with complex multiplication (CM) by $\mathcal{O}_{D}$. Here we take $\iota:A\hookrightarrow\overline{k}$ to be the inclusion map, and a Drinfeld module $\phi$ of rank $2$ has complex multiplication by $\mathcal{O}_{D}$ if there is an extension of $\phi:\mathcal{O}_{D}\to L\{\tau\}$, coinciding with $\phi$ on $A$. By the theory of complex multiplication [Go, Ch. 7], each isomorphism class in $\mathcal{D}(\mathcal{O}_{D})$ is represented by a sign-normalized Drinfeld-Hayes module (of rank $1$ as a Drinfeld $\mathcal{O}_{D}$-module, see [Go, Ch. 7], [H]). Such a Drinfeld-Hayes module is defined over the Hilbert class field $H$ of $K$, i.e. the maximal abelian extension of $K$ that is everywhere unramified and totally split at the unique infinite place of $K$. The group $\mathrm{Gal}(H/K)\cong\mathrm{Pic}(\mathcal{O}_{D})$ acts simply transitively on $\mathcal{D}(\mathcal{O}_{D})$. In particular, given a Drinfeld-Hayes module $\phi\in\mathcal{D}(\mathcal{O}_{D})$, we have

[TABLE]

where $\phi^{\sigma}$ denotes the action of $\mathrm{Pic}(\mathcal{O}_{D})$ on $\phi$. There are $h(D)$ such isomorphism classes [H, Cor. 5.13].

Moreover, $\phi$ is defined over the ring of integers $\mathcal{O}_{H}$ of $H$, so that

[TABLE]

Thus if $\mathfrak{P}$ is a prime ideal of $\mathcal{O}_{H}$ and $\bar{\iota}:A\to\mathcal{O}_{H}/\mathfrak{P}$ is the map induced by the inclusion $A\hookrightarrow\mathcal{O}_{H}$, we can define the reduction $\overline{\phi}$ of $\phi$,

[TABLE]

which is a Drinfeld module of rank $2$ over $\mathcal{O}_{H}/\mathfrak{P}$. Letting $P_{0}\in A$ be the unique monic generator of $\mathfrak{P}\cap A$, we say that the reduction $\overline{\phi}$ is supersingular if

[TABLE]

which is equivalent to $\overline{\phi}_{P_{0}}$ being purely inseparable as a map on $\mathbb{G}_{a}$ (see [Ge2, §4]). An equivalent condition is simply to check that the coefficient of $\tau^{\deg(P_{0})}$ in $\phi_{P_{0}}$ is divisible by $P_{0}$, and this can be determined effectively (e.g., see [EP, Cor 8.2]). Moreover, one can determine supersingular $j$-invariants via recursive identities on Drinfeld modular forms, using a construction of Cornelissen [Co].

If $\phi$ has supersingular reduction at $P_{0}\nmid D$, then the endomorphism algebra of $\overline{\phi}$ is a maximal order in the quaternion algebra $B$ [Ge2, Thm. 2.9, Thm. 4.3]. Moreover, as this induces an embedding $\mathcal{O}_{D}\hookrightarrow B$, it follows that $P_{0}$ must be inert in $\mathcal{O}_{D}$ [Pa2, Lem. 2.2].

Fix now $P_{0}\in A$ monic, irreducible, and inert in $K$. We set $\mathbb{F}_{P_{0}}:=A/(P_{0})$ and let $\mathcal{D}^{ss}(\overline{\mathbb{F}}_{P_{0}})$ be the set of isomorphism classes of supersingular Drinfeld modules over $\overline{\mathbb{F}}_{P_{0}}$. There are $n$ such isomorphism classes

[TABLE]

and we further have bijective correspondences

[TABLE]

such that $\mathrm{End}(e_{i})\cong R_{i}$ (see [Ge2, §§3–4], [Pa2, Thm. 2.6]). Letting $\mathfrak{P}$ be a prime above $P_{0}$ in $H$, we obtain a reduction map

[TABLE]

Theorem 1.7**.**

The reduction map $r_{\mathfrak{P}}$ is surjective if $\lVert D\rVert\gg_{\varepsilon,q}\lVert P_{0}\rVert^{4+\varepsilon}$, where the implied constant in $\gg_{\varepsilon,q}$ is effective.

Remark 1.8**.**

Other aspects of CM-liftings of supersingular Drinfeld modules have been studied previously. Schweizer [S, Prop. 13] investigated how Drinfeld modules with CM by the order of conductor $P_{0}$ in $\mathbb{F}_{q^{2}}[t]$ are in bijection with supersingular Drinfeld modules modulo $P_{0}$. Cojocaru and Papikian [CP, §5.2] have shown how to perform CM-liftings for supersingular Drinfeld modules of arbitrary rank. However, in these cases one varies by orders in a fixed imaginary extension of $\mathbb{F}_{q}(t)$ rather than by maximal orders in varying imaginary quadratic extensions of $\mathbb{F}_{q}(t)$ as in Theorem 1.7.

Remark 1.9**.**

Liu, Young, and the second author [LMY, Cor. 1.3] proved a result similar to Theorem 1.7 for the reduction map from CM elliptic curves to supersingular elliptic curves.

2. Deducing Theorem 1.7 from Theorem 1.3

In this section we show how Theorem 1.7 follows from Theorem 1.3 and Corollary 1.4. To do this we show that the image $r_{\mathfrak{P}}(\mathcal{D}(\mathcal{O}_{D}))$ can be identified with the set of Gross points of discriminant $D$. Then since Corollary 1.4 implies that this image becomes equidistributed among the classes $\mathcal{D}^{ss}(\overline{\mathbb{F}}_{P_{0}})$ as $\deg(D)\rightarrow\infty$, every class $e\in\mathcal{D}^{ss}(\overline{\mathbb{F}}_{P_{0}})$ is hit at least once if $\deg(D)$ is sufficiently large, in a quantitative sense which can be made precise using (9).

Since each class in $\mathcal{D}(\mathcal{O}_{D})$ is represented by a sign-normalized Drinfeld-Hayes module $\phi$, if $r_{\mathfrak{P}}(\phi)=e$, then one obtains an embedding of the corresponding endomorphism rings,

[TABLE]

The induced embedding $f_{\mathfrak{P},\phi}:K\hookrightarrow B$ is necessarily optimal. Indeed, this amounts to having the equality of subalgebras of $B$,

[TABLE]

where $g_{e}$ is the coset representative in $\hat{R}^{\times}\backslash\hat{B}^{\times}$ corresponding to the ideal $I_{e}$; the left-hand side is always contained in the right, and by construction via sign-normalized modules, the right-hand side is contained in the left. We note that the optimality of embeddings in the context of CM elliptic curves was proved by Bertolini and Darmon [BD, Prop. 4.1].

Now recall that the $R_{e}^{\times}$-conjugacy class of the optimal embedding $f_{\mathfrak{P},\phi}$ corresponds to a Gross point $x_{\mathfrak{P},\phi}=(g_{e},f_{\mathfrak{P},\phi})$ of discriminant $D$, and using the $\mathrm{Pic}(\mathcal{O}_{D})$-action on Gross points described §1.1, the $(R_{e,\sigma})^{\times}$-conjugacy class of the optimal embedding $f_{\mathfrak{P},\phi}^{\sigma}$ for $\sigma\in\mathrm{Pic}(\mathcal{O}_{D})$ corresponds to the Gross point $x_{\mathfrak{P},\phi}^{\sigma}=(g_{e,\sigma},f_{\mathfrak{P},\phi}^{\sigma})$ of discriminant $D$. We are now led to the following crucial equivariance result.

Proposition 2.1**.**

Given a sign-normalized Drinfeld-Hayes module $\phi\in\mathcal{D}(\mathcal{O}_{D})$, we have $f_{\mathfrak{P},\phi}^{\sigma}=f_{\mathfrak{P},\phi^{\sigma}}$ for $\sigma\in\mathrm{Pic}(\mathcal{O}_{D})$.

Proof.

Bertolini and Darmon [BD, Lem. 4.2] proved this result in the context of elliptic curves. We first recall the definition of $\phi^{\sigma}$: by identifying $\sigma$ with an element of $\mathrm{Gal}(H/K)$, if $\phi_{a}=a+b_{1}\tau+\cdots+b_{\ell}\tau^{\ell}$, then

[TABLE]

It is clear that $\phi^{\sigma}$ is also a sign-normalized Drinfeld-Hayes module defined over $\mathcal{O}_{H}$.

The definition of $\phi^{\sigma}$ is compatible with an action of $\mathrm{Pic}(\mathcal{O}_{D})$ in the following way. As originally defined by Hayes [Go, §4.9], [H, §§3–5], for a fixed integral ideal $\mathfrak{a}\subseteq\mathcal{O}_{D}$ we let

[TABLE]

which is well-defined since $H\{\tau\}$ is a left principal ideal domain. Then there is a unique sign-normalized Drinfeld module $\phi^{\mathfrak{a}}$ such that $\mu_{\mathfrak{a}}:\phi\to\phi^{\mathfrak{a}}$ is a morphism, and furthermore $\mu_{\mathfrak{a}}\in\mathcal{O}_{H}\{\tau\}$ by [H, Prop. 7.5]. If the class of $\mathfrak{a}$ in $\mathrm{Pic}(\mathcal{O}_{D})$ corresponds to the Galois element $\sigma\in\operatorname{Gal}(H/K)$ via class field theory, then by [H, Prop. 8.1],

[TABLE]

Now suppose that $r_{\mathfrak{P}}(\phi)=e$. For simplicity, we will write $\mathbb{F}=\mathbb{F}_{P_{0}}$, and we will assume that $R_{e}=\mathrm{End}_{\overline{\mathbb{F}}}(e)\subseteq\mathbb{F}\{\tau\}$. Thus the embedding $f_{\mathfrak{P},\phi}:\mathcal{O}_{D}\hookrightarrow R_{e}$ defined in (10) takes values in $\mathbb{F}\{\tau\}$. If we fix again an integral ideal $\mathfrak{a}\subseteq\mathcal{O}_{D}$, we take

[TABLE]

which is a left ideal of $R_{e}$. We let

[TABLE]

and by J.-K. Yu [Y, §2], there is a unique Drinfeld module $e^{I}$ over $\overline{\mathbb{F}}$ such that $\nu_{\mathfrak{a}}:e\to e^{I}$ is a morphism. Moreover, by [Y, Prop. 2] we have

[TABLE]

the right order of $I$ in $B$, where $R_{e,\mathfrak{a}}^{\prime}$ is defined in (4).

We claim that for our fixed ideal $\mathfrak{a}\subseteq\mathcal{O}_{D}$, we have

[TABLE]

i.e., $\nu_{\mathfrak{a}}$ is obtained from $\mu_{\mathfrak{a}}$ by reducing each of its coefficients modulo $\mathfrak{P}$. Letting $\overline{\mu}_{\mathfrak{a}}$ denote reduction of $\mu_{\mathfrak{a}}$ modulo $\mathfrak{P}$, we observe that $\overline{\mu}_{\mathfrak{a}}\in e(\mathfrak{a})\subseteq I$, since $r_{\mathfrak{P}}(\phi)=e$. Now the zeros of $\mu_{\mathfrak{a}}$, as a function on $\overline{H}$, comprise the $\mathfrak{a}$-torsion of $\phi$ (see [Go, §4.9]), and as such

[TABLE]

By the same token, every element of $I$ must vanish at the $\mathfrak{a}$-torsion of $e$, and so

[TABLE]

Since $\overline{\mu}_{\mathfrak{a}}\neq 0$, it follows that $\overline{\mu}_{\mathfrak{a}}=\nu_{\mathfrak{a}}$. As a consequence, we see that

[TABLE]

Therefore,

[TABLE]

but this coincides with $f_{\mathfrak{P},\phi}^{\sigma}$ defined in §1.1. ∎

Proof of Theorem 1.7.

By the preceding discussions, we have bijective correspondences

[TABLE]

Since the components $\{X_{1},\ldots,X_{n}\}$ of the definite Shimura curve $X_{P_{0}}$, and hence the divisor classes $\mathcal{S}$, are identified with the isomorphism classes of supersingular Drinfeld modules $\{e_{1},\ldots,e_{n}\}$, and by Corollary 1.4 the $\mathrm{Pic}(\mathcal{O}_{D})$-orbit $\{x_{\mathfrak{P},\phi}^{\sigma}:\sigma\in\mathrm{Pic}(\mathcal{O}_{D})\}$ of the Gross point $x_{\mathfrak{P},\phi}$ becomes equidistributed with respect to the probability measure $\mu_{P_{0}}$ on $\mathcal{S}$ as $\deg(D)\rightarrow\infty$, it follows from the above correspondences that the reduction map $r_{\mathfrak{P}}$ is surjective if $\deg(D)$ is sufficiently large. More precisely, since

[TABLE]

and $\mu_{P_{0}}(e_{i})\asymp\lVert P_{0}\rVert^{-1}$ (see (5)), then by (9) with $\eta=0$ we have $N_{\mathrm{Pic}(\mathcal{O}_{D}),P_{0},e_{i}}>0$ (and hence that $r_{\mathfrak{P}}$ is surjective) if $\lVert D\rVert\gg_{\varepsilon,q}\lVert P_{0}\rVert^{4+\varepsilon}$, and we are done. ∎

3. Rankin-Selberg $L$–functions

In this section we deduce some facts we will need on automorphic forms of Drinfeld-type and Rankin-Selberg $L$–functions from [CWY, §§3–4].

Let $M(P_{0})$ (resp. $S(P_{0})$) be the space of automorphic forms (resp. cusp forms) of Drinfeld type for $\Gamma_{0}(P_{0})$ with Petersson inner product $\langle f,f\rangle_{P_{0}}$. Our assumption that $\textrm{deg}(P_{0})\geq 3$ ensures that the dimension of the space $S(P_{0})$ is positive (see (23)). For each place $v$ of $k$, let $T_{v}$ be the Hecke operator on $M(P_{0})$ corresponding to $v$. Let $f\in M(P_{0})$ be a normalized Hecke eigenform, and let $\lambda_{v}(f)\in{\mathbb{R}}$ be the Hecke eigenvalue corresponding to $T_{v}$. Then, we have $\lambda_{\infty}(f)=1$, and for each finite place $v$ of $k$, we have

• •

$|\lambda_{v}(f)|\leq 2q_{v}^{1/2}$ if $v\neq P_{0}$,

• •

$\lambda_{P_{0}}(f)^{2}=1$,

where $q_{v}$ is the cardinality of the residue field $\mathbb{F}_{v}$ of $k_{v}$. The first inequality is the Ramanujan bound, and the second inequality holds since $f$ has trivial central character (under our assumptions, any form $f\in M(P_{0})$ has trivial central character $w_{f}$, since $w_{f}$ can be identified with a character on the ideal class group $\mathrm{Pic}(A)$, and the latter group is trivial; see the last paragraph of [CWY, §3.1]).

Let $f\in S(P_{0})$ be a normalized newform and $\chi$ be a character of $\mathrm{Pic}(\mathcal{O}_{D})$. Define the Rankin-Selberg $L$–function

[TABLE]

where the product is over all places $v$ of $k$, and the local factors are given by

[TABLE]

where the numbers $\alpha_{v}^{(j)}(f),c_{v}^{(k)}(\chi)$ are determined as follows:

• •

when $v\nmid P_{0}\infty$, $\alpha_{v}^{(1)}(f)$ and $\alpha_{v}^{(2)}(f)$ are the two complex conjugate roots of the quadratic polynomial

[TABLE]

• •

when $v\mid P_{0}\infty$, we have $\alpha_{v}^{(1)}(f)=\lambda_{v}(f)$ and $\alpha_{v}^{(2)}(f)=0$;

• •

when $v=w_{1}w_{2}$ splits in $K/k$, $c_{v}^{(k)}(\chi)=\chi(w_{k})$ for $k=1,2$;

• •

when $v$ is inert in $K/k$, $c_{v}^{(k)}(\chi)=(-1)^{k}\sqrt{\chi(w)}$ where $w\mid v$, for $k=1,2$;

• •

when $v$ is ramified in $K/k$, $c_{v}^{(1)}(\chi)=\chi(w)$ where $w\mid v$, and $c_{v}^{(2)}(\chi)=0$.

Note that we have normalized the $L$–function $L(f\times\chi,s)$ so that the central value occurs at $s=1/2$.

Under our assumptions, the place $v=\infty$ is ramified in $K/k$, hence using that $\lambda_{\infty}(f)=1$, $\chi(K_{\infty}^{\times})=1$, and $q_{\infty}=q$, we get

[TABLE]

Similarly, under our assumptions, the place $v=P_{0}$ is inert in $K/k$, hence using that $\lambda_{P_{0}}(f)^{2}=1$, we get

[TABLE]

where $w_{P_{0}}\mid P_{0}$. Therefore, we have the Euler product

[TABLE]

4. Bounding the central value $L(f\times\chi,1/2)$

In this section we will prove the following Lindelöf-type bound for the central value $L(f\times\chi,1/2)$.

Theorem 4.1**.**

We have

[TABLE]

where $m\in{\mathbb{Z}}^{+}$ satisfies $m=O(\deg(P_{0})+\deg(D))$.

Proof.

We adapt the argument in [AT, Thm. 3.3] used to bound the degree one $L$–function $L(\chi_{K},s)$ at $s=1/2$.

By work of Deligne [De] and Drinfeld [Dr], the $L$–function $L(f\times\chi,s)$ is a polynomial of degree $m=O(\deg(P_{0})+\deg(D))$ in $q^{-s}$ and has zeros only on the critical line $\mathrm{Re}(s)=1/2$. Therefore, we can write

[TABLE]

for complex numbers $a\neq 0$ and $\alpha_{k}$ with $|\alpha_{k}|=1$. Taking logarithmic derivatives yields

[TABLE]

Define

[TABLE]

Then for $s\in{\mathbb{R}}$, we have

[TABLE]

Let $s_{0}\in{\mathbb{R}}$ be such that $1/2<s_{0}<1/2+1/\log(q)$. Then integrating from $1/2$ to $s_{0}$ gives

[TABLE]

To estimate the second term on the RHS of (13), we use the following lemma (see [AT, Lem. 3.1]).

Lemma 4.2**.**

Let $\theta$ and $0<t<1$ be real numbers. Then

[TABLE]

Since $|\alpha_{k}|=1$, we can write

[TABLE]

for some $\theta_{k}\in{\mathbb{R}}$. Make the change of variables $x=\log(q)(s-1/2)$ to get

[TABLE]

By Lemma 4.2 we have

[TABLE]

Then applying this bound in (13) gives

[TABLE]

Define $h:=\lceil\log_{q}(m/2)\rceil$. For $\mathrm{Re}(s)>0$, the integral

[TABLE]

can be computed in two different ways, first by expanding

[TABLE]

and integrating term by term, and second by analytically continuing to the left and picking up the residues at the poles. There is a double pole at $w=0$, and simple poles at the values of $w$ for which $q^{s+w}=\alpha_{k}q^{1/2}$. This yields the identity

[TABLE]

Integrate from $s_{0}$ to $\infty$, take real parts, and multiply by $\log(q)^{2}$ to obtain the identity

[TABLE]

To estimate the third term on the RHS of (15), we use the following lemma (see [AT, Lem. 3.2]).

Lemma 4.3**.**

For $s>s_{0}>1/2$ we have

[TABLE]

By Lemma 4.3 we have

[TABLE]

Then applying this bound in (15) gives

[TABLE]

We now apply (16) in (14), then use (12) to get

[TABLE]

Choose $s_{0}=1/2+1/h\log(q)$. Then

[TABLE]

and

[TABLE]

Since $F(s_{0})>0$, it follows that

[TABLE]

We next bound the coefficients $c_{f,K}(n)$. Taking the logarithmic derivative of the Euler product (11) yields

[TABLE]

Since the eigenvalues $\lambda_{v}(f)$ are real, the two complex conjugate roots $\alpha_{v}^{(j)}(f)$ of $X^{2}-\lambda_{v}(f)X+q_{v}$ have modulus $|\alpha_{v}^{(j)}(f)|=q_{v}^{1/2}$. Hence

[TABLE]

so that (say, for $\mathrm{Re}(s)>0$)

[TABLE]

Moreover, since $q_{v}=q^{\deg(v)}$ where $\deg(v)$ is the degree of the residue field $\mathbb{F}_{v}$ of $k_{v}$, a calculation yields

[TABLE]

where $a(n):=-\log(q)q^{-n/2}$,

[TABLE]

and

[TABLE]

Since $c_{f,K}(n)=a(n)+b(n)+c(n)$, then using (18) we estimate

[TABLE]

where

[TABLE]

We have

[TABLE]

Therefore, using that $s_{0}>1/2$, we apply these bounds in (17) to get

[TABLE]

Finally, since $\log_{q}(m/2)\leq h\leq\log_{q}(m/2)+1$ (so that $q^{h/2}\leq q^{1/2}(m/2)^{1/2}$), we get

[TABLE]

This completes the proof. ∎

5. The function field analog of Gross’s formula

In this section we study an analog of Gross’s formula [Gr] over rational function fields due to Papikian [Pa2], Wei and Yu [WY]. In particular, we give an alternative expression for this formula which will be useful for our calculations.

First, we recall some facts from the introduction. The Shimura curve $X_{P_{0}}$ is the disjoint union of $n$ genus zero curves $X_{i}$ defined over $k$. Hence if $\mathrm{Pic}(X_{P_{0}})$ denotes the Picard group of $X_{P_{0}}$ and if $e_{i}$ denotes the class of degree 1 in $\mathrm{Pic}(X_{P_{0}})$ corresponding to the component $X_{i}$, we have

[TABLE]

Since a Gross point $x\in\mathrm{Gr}_{D,{P_{0}}}$ lies on a component $X_{i}$, it determines a class $e_{x}$ in $\mathrm{Pic}(X_{P_{0}})$ which for notational convenience we continue to denote by $x$. We denote the action of $\mathrm{Pic}(\mathcal{O}_{D})$ on $\mathrm{Gr}_{D,{P_{0}}}$ by $x\mapsto x^{\sigma}$ for $\sigma\in\mathrm{Pic}(\mathcal{O}_{D})$.

The Gross height pairing

[TABLE]

is defined on generators by $\langle e_{i},e_{j}\rangle=w_{i}\delta_{ij}$ and extended bi-additively to $\mathrm{Pic}(X_{P_{0}})$, where $w_{i}:=\#(R_{i}^{\times})/(q-1)$.

Also, recall that $M(P_{0})$ (resp. $S(P_{0})$) denotes the space of automorphic forms (resp. cusp forms) of Drinfeld type for $\Gamma_{0}(P_{0})$ with the Petersson inner product $\langle f,f\rangle_{P_{0}}$.

Let $\mathcal{F}(P_{0})$ be an orthogonal basis for $S(P_{0})$ consisting of normalized newforms. By the Jacquet-Langlands correspondence over $k$ and the multiplicity-one theorem, for each form $f\in\mathcal{F}(P_{0})$, there is a unique one-dimensional eigenspace ${\mathbb{R}}e_{f}$ in $\mathrm{Pic}(X_{P_{0}})\otimes_{{\mathbb{Z}}}{\mathbb{R}}$ such that $\langle e_{f},e_{f}\rangle=1$ and $t_{m}e_{f}=\lambda_{m}(f)e_{f}$ for each monic polynomial $m\in A$ with $(m,P_{0})=1$, where $t_{m}$ is the Hecke correspondence and $\lambda_{m}(f)\in{\mathbb{R}}$ is the eigenvalue for $f$ associated to the Hecke operator $T_{m}$ (see e.g. [W, §2.4, §4.4.1]).

Let $e_{f}\in\mathrm{Pic}(X_{P_{0}})\otimes_{{\mathbb{Z}}}{\mathbb{R}}$ correspond to $f\in\mathcal{F}(P_{0})$ as above, and define

[TABLE]

Then an orthonormal basis for $\mathrm{Pic}(X_{P_{0}})\otimes_{{\mathbb{Z}}}{\mathbb{R}}$ is given by

[TABLE]

Given a character $\chi$ of $\mathrm{Pic}(\mathcal{O}_{D})$ and a Gross point $x\in\mathrm{Gr}_{D,P_{0}}$, define

[TABLE]

Moreover, given a form $f\in\mathcal{F}(P_{0})$, recall that $L(f\times\chi,s)$ denotes the Rankin-Selberg $L$–function associated to $f$ and $\chi$ (normalized so that the central value occurs at $s=1/2$). Then Papikian [Pa2], Wei and Yu [WY, Thm. 3.3] proved the following Gross-type formula (recall that $D$ is irreducible and $\deg(D)$ is odd)

[TABLE]

where $\langle f,f\rangle_{P_{0}}$ is the Petterson inner product.

We now give an alternative expression for (19) which will be useful for our calculations. Let $M_{B}^{{\mathbb{C}}}(P_{0})$ be the vector space of ${\mathbb{C}}$-valued functions on $\mathrm{Pic}(X_{P_{0}})\otimes_{{\mathbb{Z}}}{\mathbb{C}}$, with inner product

[TABLE]

Then the map which sends a generator $e_{i}$ to its characteristic function $\delta_{e_{i}}$ induces an isomorphism

[TABLE]

defined by

[TABLE]

Moreover, this map is an isometry of inner-product spaces, i.e., $\langle\widetilde{e},\widetilde{e}^{\prime}\rangle=\langle e,e^{\prime}\rangle$ for any $e,e^{\prime}\in\mathrm{Pic}(X_{P_{0}})\otimes_{{\mathbb{Z}}}{\mathbb{C}}$.

Let $\widetilde{f}=\widetilde{e_{f}}$ denote the image of $e_{f}$ under this isomorphism. Then an orthonormal basis for $M_{B}^{{\mathbb{R}}}(P_{0})$ is given by

[TABLE]

We can now express (19) as

[TABLE]

where we write $w_{\sigma}$ for $w_{i}$ if $x^{\sigma}$ lies in the class $e_{i}$.

6. Proof of Theorem 1.3

In this section we prove Theorem 1.3. We begin by showing that

[TABLE]

where the Weyl sum is defined by

[TABLE]

We have

[TABLE]

By decomposing the function $\widetilde{e}_{i}$ into a Hecke basis in $M_{B}^{\mathbb{R}}(P_{0})$, we get

[TABLE]

Therefore

[TABLE]

Now, recall that the probability measure $\mu_{P_{0}}$ on $\mathcal{S}$ is defined by

[TABLE]

where $w_{i}:=\#(R_{i}^{\times})/(q-1)$. Then, using that $\langle\widetilde{e}_{i},\widetilde{e}^{*}\rangle=1$ for all $i$ and $\langle\widetilde{e}^{*},\widetilde{e}^{*}\rangle=\sum_{j=1}^{n}w_{j}^{-1}$, we get

[TABLE]

Also, we compute

[TABLE]

Last, by Fourier analysis we have

[TABLE]

Then combining these calculations yields (21).

We now turn to the proof of Theorem 1.3. By Cauchy’s inequality, we have

[TABLE]

where

[TABLE]

By Bessel’s inequality, we get

[TABLE]

Next, by the period formula (20) we have

[TABLE]

which yields

[TABLE]

Theorem 4.1 gives the uniform Lindelöf bound

[TABLE]

where we recall that $m=O(\deg(P_{0})+\deg(D))$. Then applying this bound gives

[TABLE]

where

[TABLE]

To bound $C(P_{0})$, note that by [Pa1, Corollary 5.6] we have $\langle f,f\rangle_{P_{0}}\gg_{\varepsilon}\lVert P_{0}\rVert^{1-\varepsilon}$. Moreover, since the dimension of $S(P_{0})$ is equal to $h_{P_{0}}-1$, where

[TABLE]

(see for example [WY, p. 740]), we get

[TABLE]

Hence

[TABLE]

Finally, by combining the preceding estimates, it follows from (22) and the lower bound

[TABLE]

that

[TABLE]

Replacing $\varepsilon$ with $2\varepsilon/3$, we complete the proof. ∎