2.1 The GSpin Shimura variety

For a commutative ring R, let $L_{R}$ denote $L\otimes _{\mathbb {Z}}R$ , and the quadratic form Q on L induces a quadratic form Q on $L_R$ . The Clifford algebra $C(L_R)$ of $(L_R, Q)$ is the R-algebra defined as the quotient of the tensor algebra $\bigotimes L_R$ by the ideal generated by $\{(x\otimes x)-Q(x),\, x\in L_R\}$ . It has a $\mathbb {Z}/2\mathbb {Z}$ grading $C(L_R)=C(L_R)^+\oplus C(L_R)^-$ induced by the grading on $\bigotimes L_R$ . When R is a $\mathbb {Q}$ -algebra, we also denote $C(L_R)$ (respectively, $C^\pm (L_R)$ ) by $C(V_R)$ (respectively, $C^\pm (V_R)$ ) and note that $C(L)$ is a lattice in $C(V)$ .

Let $G:=\mathrm {GSpin}(V)$ be the group of spinor similitudes of V. It is the reductive algebraic group over $\mathbb {Q}$ such that

$$ \begin{align*}G(R)=\{g\in C^{+}(V_R)^\times,\, gV_R g^{-1}=V_{R}\}\end{align*} $$

for any $\mathbb {Q}$ -algebra R. We denote by $\nu :G\rightarrow \mathbb {G}_m$ the spinor similitude factor as defined in [Reference BassBas74, Section 3]. The group G acts on V via $g\bullet v=gvg^{-1}$ for $v\in V_R$ and $g\in G(R)$ . Moreover, there is an exact sequence of algebraic groups

$$ \begin{align*}1\rightarrow \mathbb{G}_{m}\rightarrow G\xrightarrow{g\mapsto g\bullet}\mathrm{SO}(V)\rightarrow 1.\end{align*} $$

Let $D_L$ be the period domain associated to $(L,Q)$ defined byFootnote ⁵

$$ \begin{align*}D_L=\{x \in \mathbb{P}(V_{\mathbb{C}})\mid (\overline{x}.x)<0,(x.x)=0\}.\end{align*} $$

It is a hermitian symmetric domain, and the group $G(\mathbb {R})$ acts transitively on $D_L$ . As in [Reference Andreatta, Goren, Howard and PeraAGHMP18, §4.1], $(G,D_L)$ defines a Shimura datum as follows: for any class $[z]\in D_{L}$ with $z\in V_{\mathbb {C}}$ , there is a morphism of algebraic groups over $\mathbb {R}$

$$ \begin{align*}h_{[z]}:\mathbb{S}=\mathrm{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_m\rightarrow G_{\mathbb{R}}\end{align*} $$

such that the induced Hodge decomposition on $V_{\mathbb {C}}$ is given by

$$ \begin{align*}V^{1,-1}=\mathbb{C} z, V^{-1,1}=\mathbb{C} \overline{z}, V_{\mathbb{C}}^{0,0}=(\mathbb{C} z\oplus \mathbb{C} \overline{z})^{\bot}.\end{align*} $$

Indeed, choose a representative $z=u+iw$ , where $u,w\in V_{\mathbb {R}}$ are orthogonal and $Q(u)=Q(w)=-1$ , then $h_{[z]}$ is the morphism such that $h_{[z]}(i)=uw\in G(\mathbb {R})\subset C^{+}(V_{\mathbb {R}})^{\times }$ . Hence $D_L$ is identified with a $G(\mathbb {R})$ -conjugacy class in $\mathrm {Hom}(\mathrm {Res}_{\mathbb {C}/\mathbb {R}}\mathbb {G}_m,G_{\mathbb {R}})$ . The reflex field of $(G,D_L)$ is equal to $\mathbb {Q}$ by [Reference AndréAnd96, Appendix 1].

Let $\mathbb {K}\subset G(\mathbb {A}_{f})$ be the compact open subgroup

$$ \begin{align*}\mathbb{K}=\mathrm{G}(\mathbb{A}_f)\cap C(\widehat{L})^{\times},\end{align*} $$

where $\widehat {L}=L\otimes _{\mathbb {Z}} \widehat {\mathbb {Z}}$ . By [Reference PeraMP16, Lemma 2.6], the image of $\mathbb {K}$ in $\mathrm {SO}(\widehat {L})$ is the subgroup of elements acting trivially on $L^{\vee }/L$ , where $L^{\vee }$ is the dual lattice of L defined by

$$ \begin{align*}L^{\vee}:=\{x\in V\mid \forall y\in L,\, (x.y)\in \mathbb{Z}\}.\end{align*} $$

By the theory of canonical models, we get a b-dimensional Deligne–Mumford stack M over $\mathbb {Q}$ , the GSpin Shimura variety associated with L, such that

$$ \begin{align*}M(\mathbb{C})=G(\mathbb{Q})\backslash D_L\times G(\mathbb{A}_f)/\mathbb{K}.\end{align*} $$

2.2 The Kuga–Satake construction and K3 type motives in characteristic $0$

The Kuga–Satake construction was first considered in [Reference Kuga and SatakeKS67] and later in [Reference DeligneDel72] and [Reference DeligneDel79]. We follow here the exposition of [Reference PeraMP16, Section 3].

Let $G\rightarrow \mathrm {Aut}(N)$ be an algebraic representation of G on a $\mathbb {Q}$ -vector space N, and let $N_{\widehat {\mathbb {Z}}}\subset N_{\mathbb {A}_f}$ be a $\mathbb {K}$ -stable lattice. Then one can construct a local system $\mathbf {N}_{B}$ on $M(\mathbb {C})$ whose fibre at a point $[[z],g]$ is identified with $N\cap g(N_{\widehat {\mathbb {Z}}})$ . The corresponding vector bundle $\mathbf {N}_{dR,M(\mathbb {C})}=\mathcal {O}_{M(\mathbb {C})}\otimes \mathbf {N}_{B}$ is equipped with a holomorphic filtration $\mathcal {F}^{\bullet }\mathbf {N}_{dR,M(\mathbb {C})}$ , which at every point $[[z],g]$ equips the fibre with the Hodge structure determined by the cocharacter $h_{[z]}$ . Hence we obtain a functor

(2.1) $$ \begin{align} (N,N_{\widehat{\mathbb{Z}}})\mapsto \left(\mathbf{N}_{B},\mathcal{F}^{\bullet}\mathbf{N}_{dR,M(\mathbb{C}))}\right) \end{align} $$

from the category of algebraic $\mathbb {Q}$ -representations of G with a $\mathbb {K}$ -stable lattice to variations of $\mathbb {Z}$ -Hodge structures over $M(\mathbb {C})$ . Applying this functor to $(V,\widehat {L})$ , we obtain a variation of $\mathbb {Z}$ -Hodge structures $\{\mathbf {V}_{B},\mathcal {F}^{\bullet }\mathbf {V}_{dR,M(\mathbb {C})}\}$ of weight $0$ over $M(\mathbb {C})$ . The quadratic form Q gives a polarisation on $(\mathbf {V}_{B},\mathcal {F}^{\bullet }\mathbf {V}_{dR,M(\mathbb {C})})$ and hence by [Reference DeligneDel79, 1.1.15], all $(\mathbf {N}_{B},\mathcal {F}^{\bullet }\mathbf {N}_{dR,M(\mathbb {C}))})$ are polarisable.

Also, if we denote by H the representation of the group G on $C(V)$ by left multiplication and $H_{\widehat {\mathbb {Z}}}=C(L)_{\widehat {\mathbb {Z}}}$ , then applying the functor (2.1) to the pair $(H,H_{\widehat {\mathbb {Z}}})$ , we obtain a polarisable variation of $\mathbb {Z}$ -Hodge structures $(\mathbf {H}_{B},\mathbf {H}_{dR,M(\mathbb {C})})$ of type $(-1,0),(0,-1)$ with a right $C(V)$ -action. Therefore, there is a family of abelian schemes $A^{\mathrm {univ}}\rightarrow M$ of relative dimension $2^{b+1}$ , the Kuga–Satake abelian scheme, such that the homology of the family $A^{\mathrm {univ},an}(\mathbb {C})\rightarrow M^{an}(\mathbb {C})$ is precisely $(\mathbf {H}_{B},\mathbf {H}_{dR,M(\mathbb {C})})$ . It is equipped with a right $C(L)$ -action and a compatible $\mathbb {Z}/2\mathbb {Z}$ -grading: $A^{\mathrm {univ}}=A^{\mathrm {univ},+}\times A^{\mathrm {univ},-}$ ; see [Reference PeraMP16, 3.5–3,7, 3.10].Footnote ⁶

Using $A^{\mathrm {univ}}$ , one descends $\mathbf {H}_{dR,M(\mathbb {C})}$ to a filtered vector bundle with an integrable connection $(\mathbf {H}_{dR},\mathcal {F}^{\bullet }\mathbf {H}_{dR})$ over M as the first relative de Rham homology with the Gauss–Manin connection ([Reference PeraMP16, 3.10]). For any prime $\ell $ , the $\ell $ -adic sheaf $\mathbb {Z}_\ell \otimes \mathbf {H}_B$ over $M(\mathbb {C})$ descends also canonically to an $\ell $ -adic étale sheaf $\mathbf {H}_{\ell ,\mathrm {\acute{e}t}}$ over M, which is canonically isomorphic to the $\ell $ -adic Tate module of $A^{\mathrm {univ}}$ ([Reference PeraMP16, 3.13]). Moreover, by Deligne’s theory of absolute Hodge cycles, one descends $\mathbf {V}_{dR,M(\mathbb {C})}$ and $\mathbb {Z}_\ell \otimes \mathbf {V}_B$ to $(\mathbf {V}_{dR},\mathcal {F}^{\bullet }\mathbf {V}_{dR})$ and $\mathbf {V}_{\ell ,\mathrm{\acute{e}t}}$ over M ([Reference PeraMP16, 3.4, 3.10–3.12]). More precisely, an idempotent

$$ \begin{align*}\pi=(\pi_{B,\mathbb{Q}}, \pi_{dR, \mathbb{Q}}, \pi_{\ell,\mathbb{Q}})\in \operatorname{\mathrm{End}}(\operatorname{\mathrm{End}}(\mathbf{H}_{B}\otimes \mathbb{Q}))\times \operatorname{\mathrm{End}}(\operatorname{\mathrm{End}}(\mathbf{H}_{dR})) \times \operatorname{\mathrm{End}}(\operatorname{\mathrm{End}}(\mathbf{H}_{\ell, \acute{\mathrm{e}}\mathrm{t}}\otimes \mathbb{Q}_\ell))\end{align*} $$

is constructed in ([Reference PeraMP16, 3.4, 3.10–3.12]) such that the fibre of $\pi $ at each closed point in M is an absolute Hodge cycle and $(\mathbf {V}_{B}\otimes \mathbb {Q}, \mathbf {V}_{dR}, \mathbf {V}_{\ell , \mathrm{\acute{e}t}}\otimes \mathbb {Q}_\ell )$ is the image of $\pi $ . In particular, $(\mathbf {V}_{B}\otimes \mathbb {Q}, \mathbf {V}_{dR}, \mathbf {V}_{\ell , \mathrm{\acute{e}t}}\otimes \mathbb {Q}_\ell )$ is a family of absolute Hodge motives over M, and we call each fibre a K3 type motive. (For a reference on absolute Hodge motives, we refer to [Reference Deligne, Milne, Ogus and ShihDMOS82, IV].)

On the other hand, let $\operatorname {\mathrm {End}}_{C(V)}(H)$ denote the endomorphism ring of H as a $C(V)$ -module with right $C(V)$ -action. Then the action of V on H as left multiplication induces a G-equivariant embedding $V\hookrightarrow \mathrm {End}_{C(V)}(H)$ , which maps $\widehat {L}\hookrightarrow H_{\widehat {\mathbb {Z}}}$ . The functoriality of (2.1) induces embeddings

$$ \begin{align*} \mathbf{V}_{B}\hookrightarrow \mathrm{End}_{C(L)}(\mathbf{H}_{B})\quad\textrm{and}\quad \mathbf{V}_{dR,M(\mathbb{C})}\hookrightarrow \mathrm{End}_{C(V)}(\mathbf{H}_{dR,M(\mathbb{C})}), \end{align*} $$

the latter being compatible with filtration. By [Reference PeraMP16, 1.2, 1.4, 3.11], these embeddings are the same as the one induced by $\pi $ above with the natural forgetful map $\operatorname {\mathrm {End}}_{C(V)}(\mathbf {H})\rightarrow \operatorname {\mathrm {End}}(\mathbf {H})$ . In particular, the embedding $\mathbf {V}_{dR}\hookrightarrow \operatorname {\mathrm {End}}_{C(V)}(\mathbf {H}_{dR})$ is compatible with filtrations and connection and $V_{\ell , \mathrm{\acute{e}t}}\hookrightarrow \operatorname {\mathrm {End}}_{C(V)}(\mathbf {H}_{\ell , \mathrm{\acute{e}t}})$ as $\mathbb {Z}_\ell $ -lisse sheaves with compatible Galois action on each fibre.Footnote ⁷

Moreover, there is a canonical quadratic form $\mathbf {Q}:\mathbf {V}_{dR}\rightarrow \mathcal {O}_M$ given on sections by $v\circ v=\mathbf {Q}(v)\cdot \mathrm {Id}$ , where the composition takes places in $\mathrm {End}_{C(V)}(\mathbf {H}_{dR})$ . Similarly, there is also a canonical quadratic form on $\mathbf {V}_{\ell ,\mathrm{\acute{e}t}}$ induced by composition in $\operatorname {\mathrm {End}}_{C(V)}(\mathbf {H}_{\ell , \mathrm{\acute{e}t}})$ and valued in the constant sheaf $\underline {\mathbb {Z}_\ell }$ .

2.3 Special divisors on M over $\mathbb {Q}$

For any vector $\lambda \in L_{\mathbb {R}}$ such that $Q(\lambda )>0$ , let $\lambda ^{\bot }$ be the set of elements of $D_{L}$ orthogonal to $\lambda $ . Let $\beta \in L^{\vee }/L$ and $m\in Q(\beta )+\mathbb {Z}$ with $m>0$ and define the complex orbifold

$$ \begin{align*} Z(\beta,m)(\mathbb{C}):=\bigsqcup_{g\in G(\mathbb{Q})\backslash G(\mathbb{A}_f)/\mathbb{K}}\Gamma_{g}\backslash\left(\bigsqcup_{\lambda\in \beta_g+L_g,\, Q(\lambda)=m}\lambda^{\bot}\right) \end{align*} $$

where $\Gamma _g=G(\mathbb {Q})\cap g\mathbb {K} g^{-1}$ , $L_g\subset V$ is the lattice determined by $\widehat {L_{g}}=g\bullet \widehat {L}$ and $\beta _g=g\bullet \beta \in L_{g}^{\vee }/L_g$ . Then $Z(\beta ,m)(\mathbb {C})$ is the set of complex points of a disjoint union of Shimura varieties associated with orthogonal lattices of signature $(b-1,2)$ and it admits a canonical model $Z(\beta ,m)$ over $\mathbb {Q}$ for $b\geq 2$ .Footnote ⁸ The natural map $Z(\beta ,m)(\mathbb {C})\rightarrow M(\mathbb {C})$ descends to a finite unramified morphism $Z(\beta ,m)\rightarrow M$ . Étally locally on $Z(\beta ,m)$ , this map is a closed immersion defined by a single equation, and hence its scheme theoretic image gives an effective Cartier divisor on M, which we will also denote by $Z(\beta ,m)$ .Footnote ⁹

2.4 Integral models

We recall the construction of an integral model of M from [Reference Andreatta, Goren, Howard and PeraAGHMP18, 4.4]; see also [Reference Howard and Madapusi PeraHM17, 6.3] and the original work of Kisin [Reference KisinKis10] and Madapusi Pera [Reference PeraMP16]. Let p be a prime number. We say that L is almost self-dual at p if either L is self-dual at p or $p=2$ , $\dim _{\mathbb {Q}}(V)$ is odd and $|L^{\vee }/L|$ is not divisible by $4$ . The following proposition is parts of [Reference Andreatta, Goren, Howard and PeraAGHMP18, Proposition 4.4.1, Theorem 4.4.6] and [Reference Howard and Madapusi PeraHM17, Remark 6.3.1].

For p such that L is self-dual at p, we now discuss the extensions of $\mathbf {V}_{dR}, \mathbf {V}_{\ell ,\mathrm{\acute{e}t}}, \ell \neq p$ over $\mathcal {M}_{\mathbb {Z}_{(p)}}$ and recall the construction of $\mathbf {V}_{cris}$ . For ease of reading, we will denote the extensions by the same notation. We will use these notions to provide an ad hoc definition of the reduction of the K3 type motives defined in Section 2.2.

By (2) in Proposition 2.1, there are natural extensions of $\mathbf {H}_{dR}, \mathbf {H}_{\ell , \mathrm{\acute{e}t}}$ as the first relative de Rham homology and the $\mathbb {Z}_\ell $ -Tate module of $\mathcal {A}^{\mathrm {univ}}$ , and we define $\mathbf {H}_{cris}$ to be the first relative crystalline homology

$$ \begin{align*}\mathrm{Hom}\left(R^{1}\pi_{cris,*}\mathcal{O}^{cris}_{\mathcal{A}_{\mathbb{F}_p}/\mathbb{Z}_p},\mathcal{O}^{cris}_{\mathcal{M}_{\mathbb{F}_{p}/\mathbb{Z}_p}}\right).\end{align*} $$

By [Reference Andreatta, Goren, Howard and PeraAGHMP18, Remark 4.2.3], there exists a canonical extension of $\mathbf {V}_{\ell , \mathrm{\acute{e}t}}\hookrightarrow \operatorname {\mathrm {End}}_{C(L)}(\mathbf {H}_{\ell , \mathrm{\acute{e}t}})$ over $\mathcal {M}_{\mathbb {Z}_{(p)}}$ . Note that $L_{\mathbb {Z}_{(p)}}$ is a $\mathbb {Z}_{(p)}$ -representation of $\mathrm {GSpin}(L_{\mathbb {Z}_{(p)}},Q)$ , then by [Reference Andreatta, Goren, Howard and PeraAGHMP18, Propositions 4.2.4, 4.2.5], there is a vector bundle with integrable connection $\mathbf {V}_{dR}$ over $\mathcal {M}_{\mathbb {Z}_{(p)}}$ and a canonical embedding into $\operatorname {\mathrm {End}}_{C(L)}(\mathbf {H}_{dR})$ extending their counterparts over M; moreover, there is an F-crystal $\mathbf {V}_{cris}$ with a canonical embedding into $\mathrm {End}_{C(L)}(\mathbf {H}_{cris})$ . Both embeddings realise $\mathbf {V}_{dR}$ and $\mathbf {V}_{cris}$ as local direct summands of $\operatorname {\mathrm {End}}_{C(L)}(\mathbf {H}_{dR})$ and $\mathrm {End}_{C(L)}(\mathbf {H}_{cris})$ and these two embeddings are compatible with the canonical crystalline-de Rham comparison.

For a point $x\in \mathcal {M}(\mathbb {F}_q)$ , where q is a power of p, we consider the fibre of $(\mathbf {V}_{\ell , \mathrm{\acute{e}t}}, \mathbf {V}_{dR},\mathbf {V}_{cris})$ at x of the ad hoc motive attached to x, where q-Frobenius $\operatorname {\mathrm {Frob}}_x$ acts on $\mathbf {V}_{\ell , \mathrm{\acute{e}t},x}$ and semi-linear crystalline Frobenius $\varphi _x$ acts on $\mathbf {V}_{cris,x}$ . Although these realisations are not as closely related as the analogous characteristic $0$ situation, there is a good notion of algebraic cycles in $(\mathbf {V}_{\ell , \mathrm{\acute{e}t}}, \mathbf {V}_{dR},\mathbf {V}_{cris})$ , namely the special endomorphisms of the Kuga–Satake abelian varieties discussed in the following subsection.

2.5 Special endomorphisms and integral models of special divisors

We recall the definition of special endomorphisms from [Reference Andreatta, Goren, Howard and PeraAGHMP18, §§4.3,4.5]. For an $\mathcal {M}$ -scheme S, we use $A_S$ to denote $\mathcal {A}^{\mathrm {univ}}_S$ , the pull-back of the universal Kuga–Satake abelian scheme to S.

By [Reference Andreatta, Goren, Howard and PeraAGHMP18, Prop.4.5.4], there is a positive definite quadratic form $Q:V(A_S)\rightarrow \mathbb {Z}$ such that for each $v\in V(A_S)$ , we have $v\circ v=Q(v)\cdot \mathrm {Id}_{A_S}$ .Footnote ¹²

For an odd prime p such that L is self-dual at p, for a point $x\in \mathcal {M}(\mathbb {F}_{p^r})$ by [Reference PeraMP15, Theorem 6.4],Footnote ¹⁴, we have isometries

$$\begin{align*}V(\mathcal{A}^{\mathrm{univ}}_x)\otimes \mathbb{Q}_\ell \cong \lim_{n\rightarrow\infty}\mathbf{V}_{\ell,\acute{\mathrm{e}}\mathrm{t},x}^{\operatorname{\mathrm{Frob}}_x^n=1},\ell\neq p, \quad V(\mathcal{A}^{\mathrm{univ}}_x)\otimes \mathbb{Q}_p \cong \lim_{n\rightarrow\infty}(\mathbb{Q}_{p^{rn}}\otimes\mathbf{V}_{cris,x})^{\varphi_x=1}.\end{align*}$$

Therefore, we view special endomorphisms of $\mathcal {A}^{\mathrm {univ}}_x$ as the algebraic cycles of the ad hoc motive $(\mathbf {V}_{\ell , \mathrm{\acute{e}t},x}, \mathbf {V}_{dR,x},\mathbf {V}_{cris,x})$ .

For $m\in \mathbb {Z}_{>0}$ , the special divisor $\mathcal {Z}(m)$ is defined as the Deligne–Mumford stack over $\mathcal {M}$ with functor of points $\mathcal {Z}(m)(S)=\{v\in V(A_S) \mid Q(v)=m\}$ for any $\mathcal {M}$ -scheme S. More generally, in [Reference Andreatta, Goren, Howard and PeraAGHMP18, §4.5], for $\beta \in L^\vee /L, m\in Q(\beta )+\mathbb {Z}, m>0$ , there is also a special cycle $\mathcal {Z}(\beta ,m)$ defined as a Deligne–Mumford stack over $\mathcal {M}$ parametrising points with certain special quasi-endomorphisms and $\mathcal {Z}(m)=\mathcal {Z}(0,m)$ . By [Reference Andreatta, Goren, Howard and PeraAGHMP18, Proposition 4.5.8], the generic fibre $\mathcal {Z}(\beta ,m)_{\mathbb {Q}}$ is equal to the divisor $Z(\beta ,m)$ defined in Section 2.3. Moreover, étale locally on the source, $\mathcal {Z}(\beta ,m)$ is an effective Cartier divisor on $\mathcal {M}$ , and we will use the same notation for the Cartier divisor on $\mathcal {M}$ defined by étale descent.

2.6 Reformulation of Theorem 1.8

Using the notion of special endomorphisms, Theorem 1.8 is a direct consequence of the following theorem.

3.1 Arithmetic special divisors and heights

Let $\rho _L:\mathrm {Mp}_{2}(\mathbb {Z})\rightarrow \mathrm {Aut_{\mathbb {C}}}(\mathbb {C}[L^\vee /L])$ denote the unitary Weil representation, where $\mathrm {Mp}_{2}(\mathbb {Z})$ is the metaplectic double cover of $\mathrm {SL}_{2}(\mathbb {Z})$ ; see, for instance, [Reference BruinierBru02, Section 1.1]. Let $k=1+\frac {b}{2}$ , and let $\mathrm {H}_{2-k}(\rho _{L}^\vee )$ be the $\mathbb {C}$ -vector space of vector valued harmonic weak Maass forms of weight $2-k$ with respect to the dual $\rho _L^\vee $ of the Weil representation as defined in [Reference Bruinier and YangBY09, 3.1].

For $\beta \in L^\vee /L$ , $m\in \mathbb {Z}+Q(\beta )$ with $m>0$ , let $F_{\beta ,m} \in \mathrm {H}_{2-k}(\rho _{L}^\vee )$ denote the Hejhal–Poincaré harmonic Maass form defined in [Reference BruinierBru02, Def. 1.8]. In fact, $F_{\beta ,m}(\tau ):=F_{\beta ,-m}(\tau , \frac {1}{2}+\frac {b}{4})$ in [Reference BruinierBru02, Def. 1.8], where $\tau $ lies in the Poincaré upper half plane $\mathbb {H}$ . Let $\Phi _{\beta ,m}$ denote the regularised theta lifting of $F_{\beta ,m}$ in the sense of Borcherds; see [Reference BruinierBru02, (2.16)] and [Reference Bruinier and FunkeBF04, §5.2]. By [Reference BorcherdsBor98, §6, Thm. 13.3] and [Reference BruinierBru02, §2.2, eqn. (3.40), Thm. 3.16], $\Phi _{\beta ,m}$ is a Green functionFootnote ¹⁶ for the divisor $\mathcal {Z}(\beta ,m)$ , and we use $\widehat {\mathcal {Z}}(\beta ,m)$ to denote the arithmetic divisor $(\mathcal {Z}(\beta ,m), \Phi _{\beta ,m})$ . Our main focus is the case when $\beta =0$ , and we set $\Phi _m:=\Phi _{0,m}, \widehat {\mathcal {Z}}(m):=\widehat {\mathcal {Z}}(0,m)$ for $m\in \mathbb {Z}_{>0}$ .

Let $\widehat {\operatorname {\mathrm {CH}}}^1(\mathcal {M})_{\mathbb {Q}}$ denote the first arithmetic Chow group of Gillet–Soulé [Reference Gillet and SouléGS90] as defined in [Reference Andreatta, Goren, Howard and PeraAGHMP17, §4.1]. Since $\mathcal {M}$ is a normal Deligne–Mumford stack, we have a natural isomorphism, as in [Reference SouléSou92, III.4],

$$ \begin{align*}\widehat{\mathrm{Pic}}(\mathcal{M})_{\mathbb{Q}}\otimes \mathbb{Q}\xrightarrow{\sim} \widehat{\mathrm{CH}^{1}}(\mathcal{M})_{\mathbb{Q}},\end{align*} $$

where $\widehat {\operatorname {\mathrm {Pic}}}(\mathcal {M})$ denotes the group of isomorphism classes of metrised line bundles and $\widehat {\operatorname {\mathrm {Pic}}}(\mathcal {M})_{\mathbb {Q}}:=\widehat {\operatorname {\mathrm {Pic}}}(\mathcal {M})\otimes \mathbb {Q}$ ; see [Reference Andreatta, Goren, Howard and PeraAGHMP17, §5.1] for more details. Since $\mathcal {Z}(\beta ,m)$ is (étale locally) Cartier, then we view $\widehat {\mathcal {Z}}(\beta ,m)\in \widehat {\operatorname {\mathrm {Pic}}}(\mathcal {M})_{\mathbb {Q}}$ .

Moreover, the line bundle $\boldsymbol {\omega }$ from Proposition 2.1 is endowed with the Petersson metric defined as follows: the fibre of $\boldsymbol {\omega }$ at a complex point $[[z],g]\in M(\mathbb {C})$ is identified with the isotropic line $\mathbb {C} z\subset V_{\mathbb {C}}$ ; then we set $||z||^2=-\frac {(z.\overline {z})}{4\pi e^{\gamma }}$ , where $\gamma =-\Gamma '(1)$ is the Euler–Mascheroni constant. Hence we get a metrised line bundle $\overline {\boldsymbol {\omega }}\in \widehat {\mathrm {Pic}}(\mathcal {M})$ .

Recall that we have a map $\mathcal {Y} \rightarrow \mathcal {M} $ , where $\mathcal {Y} = \operatorname {\mathrm {Spec}} \mathcal {O}_K$ . We now define the notion of the height of $\mathcal {M}$ with respect to the arithmetic divisors $\widehat {\mathcal {Z}}(m)$ and $\overline {\boldsymbol {\omega }}$ . As in [Reference Andreatta, Goren, Howard and PeraAGHMP17, §§5.1,5.2], [Reference Andreatta, Goren, Howard and PeraAGHMP18, §6.4], the height $h_{\widehat {\mathcal {Z}}(m)}(\mathcal {Y})$ of $\mathcal {Y}$ with respect to $\widehat {\mathcal {Z}}(m)$ (respectively, $\overline {\boldsymbol {\omega }}$ ) is defined as the image of $\widehat {\mathcal {Z}}(m)$ (respectively, $\overline {\boldsymbol {\omega }}$ ) under the composition

$$ \begin{align*}\widehat{\mathrm{CH}^{1}}(\mathcal{M})_{\mathbb{Q}}\cong \widehat{\operatorname{\mathrm{Pic}}}(\mathcal{M})_{\mathbb{Q}}\rightarrow \widehat{\operatorname{\mathrm{Pic}}}(\mathcal{Y})_{\mathbb{Q}}\xrightarrow{\widehat{\deg}} \mathbb{R},\end{align*} $$

where the middle map is the pull-back of metrised line bundles and the arithmetic degree map $\widehat {\deg }$ is the extension over $\mathbb {Q}$ of the one defined in [Reference Andreatta, Goren, Howard and PeraAGHMP18, 6.4].

Since $\mathcal {Y}$ and $\mathcal {Z}(m)$ intersect properly (recall that we assume $\mathcal {Y}_K$ is Hodge-generic), we have the following description of $h_{\widehat {\mathcal {Z}}(m)}(\mathcal {Y})$ . Let $\mathcal {A}$ denote $\mathcal {A}^{\mathrm {univ}}_{\mathcal {Y}}$ , where $\mathcal {A}^{\mathrm {univ}}$ is the Kuga–Satake abelian scheme over $\mathcal {M}$ . Using the moduli definition of $\mathcal {Z}(m)$ in Section 2.5, the $\mathcal {Y}$ -scheme $\mathcal {Y}\times _{\mathcal {M}}\mathcal {Z}(m)$ is given by

$$\begin{align*}\mathcal{Y}\times_{\mathcal{M}}\mathcal{Z}(m)(S)=\{v\in V(\mathcal{A}_S)\mid v\circ v=[m]\}\end{align*}$$

for any $\mathcal {Y}$ -scheme S. Via the natural map $\mathcal {Y}\times _{\mathcal {M}} \mathcal {Z}(m)\rightarrow \mathcal {Y}$ , and using étale descent, we view $\mathcal {Y}\times _{\mathcal {M}} \mathcal {Z}(m)$ as a $\mathbb {Q}$ -Cartier divisor on $\mathcal {Y}$ . Therefore,

(3.1) $$ \begin{align} h_{\widehat{\mathcal{Z}}(m)}(\mathcal{Y})=\sum_{\sigma:K\hookrightarrow\mathbb{C}}\Phi_{m}(\mathcal{Y}^\sigma)+\sum_{\mathfrak{P}}(\mathcal{Y}.\mathcal{Z}(m))_{\mathfrak{P}}\log|\mathcal{O}_{K}/\mathfrak{P}|, \end{align} $$

where for $\sigma :K\hookrightarrow \mathbb {C}$ , we use $\mathcal {Y}^\sigma $ to denote the point in $M(\mathbb {C})$ induced by $\operatorname {\mathrm {Spec}}(\mathbb {C})\xrightarrow {\sigma } \operatorname {\mathrm {Spec}}{\mathcal {O}_K}\xrightarrow {\mathcal {Y}} \mathcal {M}$ ; and if we denote by $\mathcal {O}_{\mathcal {Y}\times _{\mathcal {M}}\mathcal {Z}(m),v}$ the étale local ring of $\mathcal {Y}\times _{\mathcal {M}}\mathcal {Z}(m)$ at v,

(3.2) $$ \begin{align} (\mathcal{Y}.\mathcal{Z}(m))_{\mathfrak{P}}=\sum_{v\in\mathcal{Y}\times_{\mathcal{M}}\mathcal{Z}(m)(\overline{\mathbb{F}}_{\mathfrak{P}})}\operatorname{\mathrm{length}}(\mathcal{O}_{\mathcal{Y}\times_{\mathcal{M}}\mathcal{Z}(m),v}), \end{align} $$

where $\mathbb {F}_{\mathfrak {P}}$ denotes the residue field of $\mathfrak {P}$ .

3.2 Howard–Madapusi-Pera–Borcherds’ modularity theorem

Let $M_{1+\frac {b}{2}}(\rho _L)$ denote the $\mathbb {C}$ -vector space of $\mathbb {C}[L^\vee /L]$ -valued modular forms of weight $1+\frac {b}{2}$ with respect to $\rho _L$ (see see [Reference BruinierBru02, Definition 1.2]). Let $(\mathfrak {e}_\beta )_{\beta \in L^\vee /L}$ denote the standard basis of $\mathbb {C}[L^\vee /L]$ .

Theorem 3.1 [Reference Howard and Madapusi PeraHM17, Theorem 9.4.1]

Assume $b\geq 3$ , and let $q=e^{2\pi i \tau }$ . The formal generating series

$$ \begin{align*}\widehat{\Phi}_L=\overline{\boldsymbol{\omega}}^\vee\mathfrak{e}_0+\sum_{\underset{m> 0, m\in Q(\beta)+\mathbb{Z}}{\beta\in L^{\vee}/L}}\widehat{\mathcal{Z}}(\beta,m)\cdot q^m \mathfrak{e}_\beta\end{align*} $$

is an element of $M_{1+\frac {b}{2}}(\rho _L)\otimes \widehat {\mathrm {Pic}}(\mathcal {M})_{\mathbb {Q}}$ . More precisely, for any $\mathbb {Q}$ -linear map $\alpha :\widehat {\operatorname {\mathrm {Pic}}}(\mathcal {M})_{\mathbb {Q}}\rightarrow \mathbb {C}$ , we have $\alpha (\widehat {\Phi }_L)\in M_{1+\frac {b}{2}}(\rho _L)$ .

3.3 Asymptotic estimates for the global height

In this subsection, we provide asymptotic estimates for the global height. First we introduce an Eisenstein series $(\tau ,s)\rightarrow E_{0}(\tau ,s)$ for $\tau \in \mathbb {H}$ and $s\in \mathbb {C}$ with $\operatorname {\mathrm {Re}}(s)>\frac {1}{2}-\frac {b}{4}=1-\frac {k}{2}$ , which serves two purposes. First, the Fourier coefficients of its value at $s=0$ give the main term in the Fourier coefficients of $\widehat {\Phi }_L$ ; see the proof of Proposition 3.2. Second, we will use the Fourier coefficients of $E_0(\tau ,s)$ to describe $\Phi _m$ explicitly in Section 5.1.

Let $(\tau ,s)\rightarrow E_{0}(\tau ,s)$ denote the Eisenstein series defined in [Reference Bruinier and KühnBK03, Equation (1.4), (3.1) with $\beta = 0,\kappa = 1+\frac{b}{2}$ ]. It converges normally on $\mathbb {H}$ for $\mathrm {Re}(s)>1-\frac {k}{2}$ and defines a $\mathrm {Mp}_{2}(\mathbb {Z})$ -invariant real analytic function.

For a fixed $s\in \mathbb {C}$ with $\mathrm {Re}(s)>1-\frac {k}{2}$ , by [Reference Bruinier and KühnBK03, Proposition 3.1], the Eisenstein series $E_0(\cdot ,s)$ has a Fourier expansion of the form

$$ \begin{align*} E_{0}(\tau,s)=\sum_{\beta\in L^{\vee}/L}\sum_{m\in Q(\beta)+\mathbb{Z}}c_{0}(\beta,m,s,y)e^{2\pi i m x}\mathfrak{e}_{\beta}, \end{align*} $$

where we write $\tau =x+iy, x\in \mathbb {R}, y\in \mathbb {R}_{>0}$ . By [Reference Bruinier and KühnBK03, Proposition 3.2], the coefficients $c_0(\beta ,m,s,y)$ can be decomposed, for $m\neq 0$ , as

(3.3) $$ \begin{align} c_0(\beta,m,s,y)=C(\beta,m,s)\mathcal{W}_s(4\pi m y), \end{align} $$

where the function $C(\beta ,m,s)$ is independent of y (see [Reference Bruinier and KühnBK03, Equation (3.22)]) and $\mathcal {W}_s$ is defined in [Reference Bruinier and KühnBK03, (3.2)].

By [Reference Bruinier and KühnBK03, Proposition 3.1, (3.3)], the value at $s=0$ of $E_0(\tau ,s)$ is an element of $\mathrm {M}_{1+\frac {b}{2}}(\rho _L)$ . For $\beta \in L^{\vee }/L$ , $m\in Q(\beta )+\mathbb {Z}$ with $m\geq 0$ , we denote by $c(\beta ,m)$ its $(\beta ,m)$ th Fourier coefficient, and we can thus write

$$ \begin{align*} E_0(\tau):=E_0(\tau, 0)=2\mathfrak{e}_0+\sum_{\underset{m\in Q(\beta)+\mathbb{Z},m>0}{\beta\in L^{\vee}/L}}c(\beta,m)q^m \mathfrak{e}_{\beta}, \text{, where }q=e^{2\pi i \tau}. \end{align*} $$

By definition and [Reference Bruinier and KühnBK03, Proposition 3.1, (3.3)], we have $C(\beta ,n,0)=c(\beta ,n)$ . By [Reference Bruinier and KühnBK03, Prop.4.8], the coefficient $c(\beta ,m)$ encodes the degree of the special divisor $Z(\beta ,m)(\mathbb {C})$ . Moreover, [Reference Bruinier and KussBK01, Proposition 4, equation (19)] gives explicit formulas for $c(\beta ,m)$ . By [Reference Bruinier and KussBK01, Proposition 14], $c(\beta ,m)<0$ if $m\in Q(L+\beta )$ and $c(\beta ,m)=0$ if $m\notin Q(L+\beta )$ . By [Reference TayouTay20a, Example 2.3], we have that for $m\in Q(L+\beta )$ ,Footnote ¹⁷

(3.4) $$ \begin{align} |c(\beta,m)|=-c(\beta,m)\asymp m^{\frac{b}{2}}. \end{align} $$

We will henceforth focus on the case where $\beta =0$ , and we set $C(m,s):=C(0,m,s)$ and $c(m):=c(0,m)$ . We are now ready to establish asymptotics for the global height in terms of the Fourier coefficients just defined.

Proposition 3.2. For every $\epsilon>0$ and $m\in \mathbb {Z}_{>0}$ , we have

$$ \begin{align*} h_{\widehat{\mathcal{Z}}(m)}(\mathcal{Y})=\frac{-c(m)}{2}h_{\overline{\boldsymbol{\omega}}}(\mathcal{Y})+O_{\epsilon}(m^{\frac{2+b}{4}+\epsilon}). \end{align*} $$

In particular, we have $h_{\widehat {\mathcal {Z}}(m)}(\mathcal {Y})=O(m^{\frac {b}{2}})$ as $m\rightarrow \infty $ .

The second claim follows from the first claim and equation (3.4).

Proof. The proof is similar to the one in [Reference TayouTay20a, Proposition 2.5]. For $\widehat {\mathcal {Z}}\in \widehat {\operatorname {\mathrm {CH}}^1}(\mathcal {M})_{\mathbb {Q}} \cong \widehat {\operatorname {\mathrm {Pic}}}(\mathcal {M})_{\mathbb {Q}}$ , the height $h_{\widehat {\mathcal {Z}}}(\mathcal {Y})$ defines a $\mathbb {Q}$ -linear map $\widehat {\operatorname {\mathrm {Pic}}}(\mathcal {M})_{\mathbb {Q}}\rightarrow \mathbb {R}$ ; by Theorem 3.1, the following generating series

$$ \begin{align*}-h_{\overline{\boldsymbol{\omega}}}(\mathcal{Y})\mathfrak{e}_0+\sum_{\underset{m> 0, m\in Q(\beta)+\mathbb{Z}}{\beta\in L^{\vee}/L}}h_{\widehat{\mathcal{Z}}(\beta,m)}(\mathcal{Y})\cdot q^{m}\mathfrak{e}_\beta\end{align*} $$

is the Fourier expansion of an element in $M_{1+\frac {b}{2}}(\rho _L)$ . By [Reference BruinierBru02, p.27], we write

$$ \begin{align*}-h_{\overline{\boldsymbol{\omega}}}(\mathcal{Y})\mathfrak{e}_0+\sum_{\underset{m> 0, m\in Q(\beta)+\mathbb{Z}}{\beta\in L^{\vee}/L}}h_{\widehat{\mathcal{Z}}(\beta,m)}(\mathcal{Y})\cdot q^{m}\mathfrak{e}_\beta=\frac{-h_{\overline{\boldsymbol{\omega}}}(\mathcal{Y})}{2}E_0+g,\end{align*} $$

where $E_0=E_0(\tau )$ is the Eisenstein series recalled in Section 3.3 and $g\in M_{1+\frac {b}{2}}(\rho _L)$ is a cusp form; see [Reference BruinierBru02, Def.1.2] for a definition.

For $m\in \mathbb {Z}_{>0}$ , the equation for the $\mathfrak {e}_0$ -component implies that

$$ \begin{align*}h_{\widehat{\mathcal{Z}}(m)}(\mathcal{Y})=\frac{-c(m)}{2}h_{\overline{\boldsymbol{\omega}}}(\mathcal{Y})+g(m),\end{align*} $$

where $g(m)$ is the mth Fourier coefficient of the $\mathfrak {e}_0$ -component of g. We obtain the desired estimate by [Reference SarnakSar90, Prop. 1.5.5], which implies that

$$ \begin{align*}|g(m)|\leq C_{\epsilon,g} m^{\frac{2+b}{4}+\epsilon}\end{align*} $$

for all $\epsilon>0$ , some constant $C_{\epsilon ,g}>0$ and all $m\in \mathbb {Z}_{>0}$ .

4.1 Local estimates of representations by quadratic forms

Recall that $(L,Q)$ is an even quadratic lattice of signature $(b,2)$ with $b\geq 3$ , and L is maximal in $V=L\otimes \mathbb {Q}$ . Let $r=b+2$ denote the rank of L, and let $\det (L)$ denote the Gram determinant of L. Let p be a fixed prime, and let $\operatorname {\mathrm {val}}_p$ denote the p-adic valuation. For integers $m\in \mathbb {Z}$ and $n\in \mathbb {Z}_{\geq 0}$ , we define the set $\mathcal {N}_{m}(p^n)$ , its size $N_{m}(p^n)$ and the density $\mu _{p}(m,n)$ as follows:

(4.1) $$ \begin{align} \begin{array}{rcl} \displaystyle\mathcal{N}_{m}(p^n)&=&\displaystyle\{v\in L/p^{n}L \mid Q(v)\equiv m (\bmod p^n)\};\\[.1in] \displaystyle N_{m}(p^n)&=&\displaystyle |\mathcal{N}_{m}(p^n)|;\\[.1in] \displaystyle \mu_{p}(m,n)&=& \displaystyle p^{-n(r-1)}N_m(p^n). \end{array} \end{align} $$

The goal of this subsection is to study the variation of the quantity $\mu _{p}(m,n)$ . Define the quantity $w_p(m):=1+\operatorname {\mathrm {val}}_p(m)$ for $p\neq 2$ and $w_2(m):=1+2\operatorname {\mathrm {val}}_2(2m)$ . Then we prove the following result.

Proposition 4.1. If m is an integer representable by Q over $\mathbb {Z}$ , then we have

$$ \begin{align*} \left|w_p(m)-\sum_{n=0}^{w_p(m)-1}\frac{\mu_{p}(m,n)}{\mu_{p}(m,w_p(m))}\right|\ll\frac{1}{p}, \end{align*} $$

where the implied constant is independent of p and m.

We use an inductive method, due to Hanke [Reference HankeHan04], to compute the quantities $\mu _{p}(m,n)$ . Let $L_{p}:=L\otimes \mathbb {Z}_p$ be the completion of L at p. Since L is maximal in V, it follows that $L_p$ is maximal in $V\otimes \mathbb {Q}_p$ . Indeed, L being maximal is equivalent to the fact that $L^\vee /L$ has no totally isotropic subgroup; thus its p-torsion, which is equal to $L_p^\vee /L_p$ , has no totally isotropic subgroup implying that $L_p$ is maximal. It is well known that the quadratic lattice $(L_p,Q)$ admits an orthogonal decomposition

(4.2) $$ \begin{align} (L_{p},Q)=\bigoplus_j (L_j,p^{\nu_j}Q_j) \end{align} $$

with $\nu _j\geq 0$ , such that $(L_j,Q_j)$ is a $\mathbb {Z}_p$ -unimodular quadratic lattice of dimension $1$ or $2$ . (See, for example, [Reference HankeHan04, (2.3),Lemma 2.1].) Moreover, when $p\neq 2$ , then every $L_j$ is of dimension $1$ . For every $v\in L_p$ , we write $v=(v_j)_j$ , and we have

$$ \begin{align*}Q(v)=\sum_{j}p^{\nu_j}Q_j(v_j).\end{align*} $$

Note that since $L_p$ is maximal, we have $\nu _j\leq 1$ for each j. For $i\in \{0,1\}$ , let $S_i$ denote the set of indices j with $\nu _j=i$ , and let $s_i$ denote the size of $S_i$ .

Following [Reference HankeHan04, Definition 3.1], for $n\geq 1$ , $v\in \mathcal {N}_{m}(p^n)$ , we say that v is of

1. 1. zero type if $v\equiv 0\ \pmod {p}$ ,

2. 2. good type if there exists j such that $v_j\not \equiv 0\ \pmod {p}$ and $\nu _j=0$ , or

3. 3. bad type otherwise.

Let $\mathcal {N}_{m}^{\textrm {good}}(p^n)$ , $\mathcal {N}_{m}^{\textrm {bad}}(p^n)$ and $\mathcal {N}_{m}^{\textrm {zero}}(p^n)$ be the set of good type, bad type and zero type solutions, respectively, and set $N_{m}^{?}(p^n)=|\mathcal {N}_{m}^{?}(p^n)|$ and $\mu _p^{?}(m,n)=p^{-n(r-1)}N_m^{?}(p^n)$ , for $?=$ good, bad, or zero. Note also from [Reference HankeHan04, Remark 3.4.1] that we have $\mathcal {N}_m(p^n)=\mathcal {N}_m^{\mathrm {good}}(p^n)$ when $p\nmid m$ ; $\mathcal {N}_m^{\mathrm {zero}}(p^n)=\emptyset $ when $p^2\nmid m$ and $n\geq 2$ ; and $\mathcal {N}_m^{\mathrm {bad}}(p^n)=\emptyset $ when $p\nmid 2\det (L)$ .

To state the inductive result on local densities, we need to introduce the auxiliary form $Q'$ , where $Q'$ is obtained from the orthogonal decomposition given by equation (4.2) of Q by replacing $\nu _j$ with $\nu _j'=1-\nu _j$ for all j. To distinguish between the local densities of Q and $Q'$ , we use $N_{m,Q}^{?}(p^n)$ and $\mu _{p,Q}^{?}(m,n)$ to emphasis the dependence on the quadratic form. We now recall Hanke’s inductive method with the simplification that L is maximal (only used in (2)).

Proof. The base cases when $\operatorname {\mathrm {val}}_p(m)\leq 1$ or $n\leq 2$ can be checked directly by definition and Lemma 4.2. For $n>2$ and $p^2\mid m$ , by Lemma 4.2,

$$ \begin{align*} \mu_{p,Q}(m,n) & = \mu_{p,Q}^{\mathrm{good}}(m,n)+\mu_{p,Q}^{\mathrm{zero}}(m,n)+\mu_{p,Q}^{\mathrm{bad}}(m,n)\\ &= \mu_{p,Q}^{\mathrm{good}}(m,1)+p^{2-r}\mu_{p,Q}\left(\frac{m}{p^2},n-2\right)+p^{1-s_0}\mu^{\mathrm{good}}_{p,Q'}\left(\frac{m}{p},n-1\right)\\ &=\mu_{p,Q}^{\mathrm{good}}(m,1)+p^{2-r}\mu_{p,Q}\left(\frac{m}{p^2},n-2\right)+p^{1-s_0}\mu^{\mathrm{good}}_{p,Q'}\left(\frac{m}{p},1\right). \end{align*} $$

Then we conclude by induction on $\operatorname {\mathrm {val}}_p(m)$ and n.

The next lemma gives a uniform bound on $|\mu _{p}(m,n)-\mu _p(m,w_p(m))|$ for primes p, integers m and $n\in \{2,\ldots ,w_p(m)-1\}$ .

Lemma 4.4. Let p be prime and m be any integer. For $n\in \{2+\operatorname {\mathrm {val}}_2(p),\ldots ,w_p(m)-1\}$ , we have

$$\begin{align*}|\mu_{p}(m,n)-\mu_p(m,w_p(m))|\ll \frac{1}{p^{3\lfloor(n/2)\rfloor-2}},\end{align*}$$

where the implied constant is absolute.

Proof. First consider the case when p is odd. By Corollary 4.3, we have that for n odd,

$$ \begin{align*} |\mu_{p}(m,n)-\mu_p(m,w_p(m))| &\leq \displaystyle\left|\frac{\mu_{p,Q}\left(\frac{m}{p^{n-1}},1\right)}{p^{\frac{(r-2)(n-1)}{2}}}\right|\\ &\quad +\displaystyle \left|\sum_{u=\frac{(n-1)}{2}}^{\lfloor \frac{\operatorname{\mathrm{val}}_p(m)}{2}\rfloor} \frac{\mu^{\mathrm{good}}_{p,Q}\left(\frac{m}{p^{2u}},1\right)+\delta_{p,\det(L)}p^{1-s_0}\mu^{\mathrm{good}}_{p,Q'}\left(\frac{m}{p^{2u+1}},1\right)}{p^{(r-2)u}}\right|\\ &\leq \displaystyle\left| \frac{p}{p^{\frac{3(n-1)}{2}}}\right| +\sum_{u=\frac{(n-1)}{2}}^{\lfloor \frac{\operatorname{\mathrm{val}}_p(m)}{2}\rfloor}\left| \frac{p+p^2}{p^{3u}}\right|\\ &\leq \displaystyle\frac{C_1 p^2}{p^{\frac{3(n-1)}{2}}}. \end{align*} $$

Here we use the trivial bound that all $|\mu _{p,Q}\left (\frac {m}{p^{n-1}},1\right )|, |\mu ^{\mathrm {good}}_{p,Q}\left (\frac {m}{p^{2u}},1\right )|, |\mu ^{\mathrm {good}}_{p,Q'}\left (\frac {m}{p^{2u+1}},1\right )|$ are less than p by definition. The case when n is even follows by a similar argument and the trivial bound that $\mu ^{\mathrm {zero}}_{p,Q}(mp^{2-n},2)\leq p^r/p^{2(r-1)}=p^{2-r}$ .

For $p=2$ , by Lemma 4.2, we obtain analogous statements as Corollary 4.3, except that we can only reduce to $\mu ^{\mathrm {good}}_{p}(?,3)$ (instead of $\mu ^{\mathrm {good}}_{p}(?,1)$ ). The rest of the argument is the same as in Lemma 4.4, and since $p=2$ is fixed, any trivial bound on density is absorbed in the absolute constant.

We can actually show that all $\mu _{p}(m,n)$ are close to $1$ when $p\nmid 2\det (L)$ .

Lemma 4.5. There exists an absolute constant $C_2>0$ such that for all $m,n\in \mathbb {Z}_{>0}$ , all primes $p\nmid 2\det (L)$ , we have

$$\begin{align*}|\mu_{p}(m,n)-1|\leq \frac{C_2}{p}.\end{align*}$$

Proof. By Corollary 4.3(1), Lemma 4.4, we only need to show the claim for $n=1$ and $n=\operatorname {\mathrm {val}}_p(m)+1$ . For $n=1$ , we first consider the case when $p\mid m$ . Then $Q(v)\equiv 0\bmod p$ defines a smooth projective hypersurface in $\mathbb {P}^{r-1}$ ; except the solution $v=0 \bmod p$ , every $p-1$ solutions of $Q(v)\equiv 0 \bmod p$ (all these are of good type) correspond to a $\mathbb {F}_p$ -point in the hypersurface. Then by the Weil bound (see, for instance, [Reference DeligneDel74, Théorème 8.1]),Footnote ¹⁹ there exists a constant $C_3>0$ independent of p and m such that

$$ \begin{align*} |N_{m}^{\mathrm{good}}(p)-p^{r-1}|\leq C_3 p^{r-2}. \end{align*} $$

Therefore, $|\mu _{p}^{\mathrm {good}}(m,1)-1|\leq C_3/p$ .

For $p\nmid m$ , we consider the smooth projective hypersurface in $\mathbb {P}^r$ defined by $Q(v)=my^2$ . In this case, $N_m(p)=N^{\mathrm {good}}_m(p)$ is the number of $\mathbb {F}_p$ -points in the hypersurface such that $y\neq 0$ in $\mathbb {F}_p$ . By the Weil bound, the number of $\mathbb {F}_p$ -points in the hypersurface is $p^{r-1}+O_m(p^{r-2})$ ; the number of $y=0$ points on the hypersurface is $p^{r-2}+O(p^{r-3})$ by the Weil bound. Then we conclude that there exists a constant $C_4>0$ independentFootnote ²⁰ of p and m such that $|\mu _{p}^{\mathrm {good}}(m,1)-1|\leq C_4/p$ . In particular, for any m, we have

$$ \begin{align*}\mu_p^{\mathrm{good}}(m,1)\leq 1+\max\{C_3,C_4\}.\end{align*} $$

For $n=\operatorname {\mathrm {val}}_p(m)+1$ and $p|m$ , by Corollary 4.3(1), and note that $\delta _{p,\det (L)}=0$ , we have

$$ \begin{align*} |\mu_{p}(m,\operatorname{\mathrm{val}}_p(m)+1)-1|&=\left|\mu_p^{\mathrm{good}}(m,1)-1+\sum_{u=1}^{ \left\lfloor {\frac{\operatorname{\mathrm{val}}_p(m)}{2}} \right\rfloor }\frac{\mu_{p}^{\mathrm{good}}(\frac{m}{p^{2u}},1)}{p^{u(r-2)}}\right|\\ &\leq \frac{C_3}{p}+\sum_{u=1}^{ \left\lfloor {\frac{\operatorname{\mathrm{val}}_p(m)}{2}} \right\rfloor }\frac{\max\{C_3,C_4\}+1}{p^{u(r-2)}} \leq \frac{C_5}{p}, \end{align*} $$

where we take $C_2=\max \{C_3,C_4,C_5\}$ .

Due to our assumptions that $r\geq 5$ and L maximal, there is an absolute lower bound for $\mu _{p}(m,n)$ . The following lemma is well known, but we include it for the convenience of the reader.

Now we are ready to prove the main result of this subsection.

Proof of Proposition 4.1

For simplicity of notation, we denote $w_p(m)$ by $w_p$ .

First case: assume that $p\nmid 2\det (L)$ . By Lemma 4.5,

(4.3) $$ \begin{align} |\mu_{p}(m,w_p)-\mu_{p}(m,1)|\leq |\mu_{p}(m,w_p)-1|+|\mu_{p}(m,1)-1|\leq C_7/p. \end{align} $$

Note that $\mu _{p}(m,0)=1$ by definition; see 4.1. Then by Lemmas 4.4,4.5 and equation (4.3), we get

$$ \begin{align*} \left|w_p-\sum_{n=0}^{w_p-1}\frac{\mu_{p}(m,n)}{\mu_{p}(m,w_p)}\right|&\leq\frac{1}{\mu_{p}(m,w_p)}\left[\sum_{n\geq 2}\frac{C_1 p^2}{p^{3\lfloor n/2\rfloor}}+\frac{C_7}{p}+\frac{C_2}{p}\right] \leq \frac{C_8}{\mu_{p}(m,w_p)p}. \end{align*} $$

We conclude by the fact that $\mu _{p}(m,w_p)$ is uniformly bounded away from $0$ by Lemma 4.6.

Second case: assume now that $p\mid 2\det (L)$ . By Lemma 4.4, for any $n\geq 3$ ,

(4.4) $$ \begin{align} |\mu_{p}(m,w_p)-\mu_{p}(m,n)|\leq \frac{C_9p^2}{p^{3\lfloor n/2\rfloor}}. \end{align} $$

Then, as in the first case, we have

$$\begin{align*}\left|\sum_{n=3}^{w_p-1}\frac{\mu_{p}(m,w_p)-\mu_{p}(m,n)}{\mu_{p}(m,w_p)}\right|\leq \frac{C_{10}}{p}.\end{align*}$$

On the other hand, for $0\leq n\leq 2$ , we have for all $p\mid 2\det (L)$

$$\begin{align*}|\mu_{p}(m,w_p)-\mu_{p}(m,n)|\leq |\mu_{p}(m,w_p)-\mu_{p}(m,3)|+|\mu_{p}(m,3)-\mu_{p}(m,n)|\leq C_{11}/p\end{align*}$$

by equation (4.4) and the trivial bound $|\mu _{p}(m,n)|,|\mu _{p}(m,3)|\leq p^3\leq (2\det (L))^3$ . Then we conclude as in the first case.

4.2 On the number of representations of quadratic forms

Developing a new form of the circle method, Heath-Brown [Reference Heath-BrownHB96] proves a number of results pertaining to the representation of integers by quadratic forms. The purpose of this subsection is to describe the setup used in [Reference Heath-BrownHB96] and recall those results necessary for us in the sequel. We do not entirely keep the notations of [Reference Heath-BrownHB96] since we will only be concerned with homogeneous quadratic forms, which allows us to make certain simplifications in the notation.

Let $F(\mathbf {x})=F(x_1,\ldots ,x_n)$ be an integral quadratic form with nonzero discriminant in $n\geq 5$ variables. A function $\omega :\mathbb {R}^n\to \mathbb {C}$ is said to be a smooth weight function if it is infinitely differentiable of compact support. Given a set S of parameters (see [Reference Heath-BrownHB96, Page 6] for what the parameters are allowed to be), Heath-Brown defines a set of weight functions $\mathcal {C}(S)$ in [Reference Heath-BrownHB96, §2].

The reason for introducing the set $\mathcal {C}(S)$ is the following. For the quadratic form F fixed as above, an integer $m\neq 0$ and a weight function $\omega \in \mathcal {C}(S)$ for some set of parameters S, we define

$$ \begin{align*} N(F,m,\omega):=\sum_{\substack{\mathbf{x}\in\mathbb{Z}^n\\F(\mathbf{x})=m}}\omega\Bigl(\frac{\mathbf{x}}{\sqrt{m}}\Bigr). \end{align*} $$

The quantity $N(F,m,\omega )$ then is a weighted sum of representations of m by F, where the coordinates of these representations are bounded by $O_S(\sqrt {m})$ since $\omega $ has compact support. Then [Reference Heath-BrownHB96] gives asymptotics for the size of $N(F,m,\omega )$ , where the error term only depends on S.

More precisely, define the singular integral byFootnote ²¹

$$ \begin{align*} \mu_\infty(F,\omega):=\lim_{\epsilon\to 0}\frac{1}{2\epsilon}\int_{|F(\mathbf{x})-1|\leq\epsilon}\omega(\mathbf{x})d\mathbf{x}. \end{align*} $$

Recall the Siegel mass at p of the quadratic form F given byFootnote ²²

$$ \begin{align*} \mu_p(F,m):=\lim_{k\to\infty}\frac{1}{p^{(n-1)k}}|\bigl\{ \mathbf{x}\ \pmod{p^k}: F(\mathbf{x})\equiv m\ \pmod{p^k}\bigr\}| \end{align*} $$

and define the singular series by

$$ \begin{align*} \mu(F,m):=\prod_p\mu_p(F,m). \end{align*} $$

In our situation with $n\geq 5$ , the singular series always converges absolutely; see also, for instance, [Reference IwaniecIwa97, §11.5]. Now [Reference Heath-BrownHB96, Theorem 4] states the following.

Theorem 4.9 [Reference Heath-BrownHB96, Theorem 4]

Let the notation be as above, and let $\omega \in \mathcal {C}(S)$ be a weight function for some set of parameters S. Then

$$ \begin{align*} N(F,m,\omega)=\mu_\infty(F,\omega)\mu(F,m)m^{n/2-1}+O_{F,S,\epsilon}\bigl(m^{(n-1)/4+\epsilon}\bigr). \end{align*} $$

Note in particular that the error term depends only on F and S and not on the specific weight function $\omega $ or on m.

We also recall a corollary of the above theorem for positive definite quadratic forms, which will be used in Section 7.

Corollary 4.10 [Reference Heath-BrownHB96, Corollary 1]

Let the notation be as above, and assume further that F is positive definite. Then

$$\begin{align*}|\{\mathbf{x}\in \mathbb{Z}^n: F(\mathbf{x})=m\}|=\mu_{\infty}(F,1)\mu(F,m)m^{n/2-1}+O_{F,\epsilon}(m^{(n-1)/4+\epsilon}).\end{align*}$$

4.3 An application of Heath-Brown’s theorem and a result of Niedermowwe

Recall that $(L,Q)$ is an even quadratic lattice of signature $(b,2)$ with $b\geq 3$ . We will apply Heath-Brown’s result to $(L,Q)$ (here we identify L with $\mathbb {Z}^{b+2}$ ) after we construct suitable smooth weight functions. Moreover, we recall Niedermowwe’s result, which is analogous to Heath-Brown’s result but with sharp weight functions given by characteristic functions of certain expanding domains $\Omega _T$ defined below and keeps track of the dependence of the error term on T.

Similar to the definition of the singular integral $\mu _\infty (F,\omega )$ in Section 4.2, we define a measure $\mu _\infty $ on

$$ \begin{align*} L_{\mathbb{R},1}:=\{\lambda\in L_{\mathbb{R}}:Q(\lambda)=1\} \end{align*} $$

as follows. For an open bounded subset W of $L_{\mathbb {R}}$ , we set

(4.5) $$ \begin{align} \mu_\infty(W\cap L_{\mathbb{R},1}):= \lim_{\epsilon\to 0}\frac{1}{2\epsilon} \mu_L\bigl(\{\lambda\in W:|Q(\lambda)-1|<\epsilon\}\bigr), \end{align} $$

where $\mu _{L}$ is the Lebesgue measure on $L_{\mathbb {R}}$ normalised so that L has covolume $1$ .Footnote ²³

Let x denote a fixed point in the period domain $D_L$ , and let P denote the negative definite plane (with respect to Q) in $L_{\mathbb {R}}$ associated to x. Let $P^\perp $ denote the orthogonal complement of P in $L_{\mathbb {R}}$ . Given a vector $\lambda \in L_{\mathbb {R}}$ , we let $\lambda _x$ and $\lambda _{x^\bot }$ denote the projections of $\lambda $ to P and $P^\perp $ , respectively.

We introduce notation for the set of elements in $L_{\mathbb {R},1}$ with bounded value of $Q(\lambda _x)$ : for $T>0$ , define

$$ \begin{align*} \Omega_{\leq T}:=\{\lambda\in L_{\mathbb{R},1}:\,-Q(\lambda_x)\in[0,T]\}. \end{align*} $$

Then the following lemma computes the volume of the sets $\Omega _{\leq T}$ . Recall that $k=1+\frac {b}{2}$ .

Lemma 4.11. Let $T>0$ be a real number. Then

$$ \begin{align*}\mu_{\infty}(\Omega_{\leq T})=\frac{(2\pi)^{k}\left((1+T)^{\frac{b}{2}}-1\right)}{\sqrt{|L^{\vee}/L|}\Gamma(k)}.\end{align*} $$

Proof

For $\epsilon>0$ , let $U_{T,\epsilon }:=\{x\in L_{\mathbb {R}}:|Q(x)-1|<\epsilon ,\, -Q(\lambda _x)<T\}$ . Then $\Omega _{\leq T}=U_{T,\epsilon }\cap L_{\mathbb {R},1}$ and by definition

$$ \begin{align*}\mu_{\infty}(\Omega_{\leq T})=\lim_{\epsilon\rightarrow 0}\frac{\mu_{L}(U_{T,\epsilon})}{2\epsilon}.\end{align*} $$

Let $\mathcal {E}$ be an orthogonal basis of $L_{\mathbb {R}}$ adapted to the decomposition $P\oplus P^{\bot }$ and in which the bilinear form associated to Q has the intersection matrix

$$ \begin{align*}\begin{pmatrix} -1&0&0\\ 0&-1&0\\ 0&0&I_b\\ \end{pmatrix},\end{align*} $$

where $I_b$ denotes the $b\times b$ identity matrix. Let $\mu _{\mathcal {E}}$ be the associated Lebesgue measure for which the $\mathbb {Z}$ -span of $\mathcal {E}$ is of covolume $1$ . By change of variables, we have

$$ \begin{align*} \mu_{L}(U_{T,\epsilon})&=\frac{2^{1+\frac{b}{2}}}{\sqrt{|L^{\vee}/L|}}\mu_{\mathcal{E}}(U_{T,\epsilon}) \\[4pt] &=\frac{2^{1+\frac{b}{2}}}{\sqrt{|L^{\vee}/L|}}\int_{\underset{\underset{x^2_1+x_2^2<T}{|x_1^2+x_2^2-y_1^2-\cdots-y_b^2+1|<\epsilon}}{(x_1,x_2,y_1,\cdots,y_b)\in\mathbb{R}^{b+2}}} dx_1 dx_2dy_1\cdots dy_b\\[4pt] &=\frac{2^{2+\frac{b}{2}}\pi}{\sqrt{|L^{\vee}/L|}}\int_{0}^{\sqrt{T}}\left(\int_{1+r^2-\epsilon<y_1^2+\cdots+y_b^2<1+r^2+\epsilon}dy_1\cdots dy_b \right) rdr\\[4pt] &=\frac{2(2\pi)^{1+\frac{b}{2}}}{\sqrt{|L^{\vee}/L|}\Gamma\left(1+\frac{b}{2}\right)}\int_{0}^{\sqrt{T}}\left((1+r^2+\epsilon)^{\frac{b}{2}}-\left(1+r^2-\epsilon\right)^{\frac{b}{2}}\right)rdr\\[4pt] &=2\epsilon.\frac{(2\pi)^{1+\frac{b}{2}} \left((1+T)^{\frac{b}{2}}-1\right)}{\sqrt{|L^{\vee}/L|}\Gamma\left(1+\frac{b}{2}\right)}+O(\epsilon^2). \end{align*} $$

Dividing by $2\epsilon $ and letting $\epsilon $ go to zero, we get the desired result.

We now describe the desired estimates for $\bigl |\bigl \{ \lambda \in L:Q(\lambda )=m,\,\lambda /\sqrt {m}\in \Omega _{\leq T}\bigr \}\bigr |$ in two cases.

4.3.1 Assume $T\leq 1$

In this case, we only need a good upper bound, and hence we construct a suitable smooth weight function $\omega :L_{\mathbb {R}}\to \mathbb {R}$ as follows and apply Heath-Brown’s theorem.

Corollary 4.12. Given $(L,Q)$ as above, for any $m\in \mathbb {Z}_{>0}$ , any $0<T\leq 1$ , we have

$$\begin{align*}\bigl|\bigl\{ \lambda\in L:Q(\lambda)=m,\,\lambda/\sqrt{m}\in \Omega_{\leq T}\bigr\}\bigr|=O_Q(m^{\frac{b}{2}}T)+O_{Q,T,\epsilon}(m^{(b+1)/4+\epsilon}).\end{align*}$$

Proof. By Remark 4.8, there exist smooth functions $\omega _P:P\to [0,2]$ and $\omega _{P^\perp }:P^\perp \to [0,2]$ such that

1. 1. $\omega _P(\lambda _x)\geq 1$ for elements $\lambda _x\in P$ with $Q(\lambda _x)<T$ and $\omega _P(\lambda _x)=0$ if $Q(\lambda _x)>2T$ .

2. 2. $\omega _{P^\perp }(\lambda _{x^\perp })\geq 1$ if $Q(\lambda _{x^\perp })\leq 1$ and $\omega _{P^\perp }(\lambda _{x^\perp })=0$ if $Q(\lambda _{x^\perp })\geq 2$ .

We define $\omega (\lambda )=\omega _P(\lambda _x)\omega _{P^\perp }(\lambda _{x^\perp })$ , and by construction, $\omega \in \mathcal {C}(b,T)$ .

By definition, $\bigl |\bigl \{ \lambda \in L:Q(\lambda )=m,\,\lambda /\sqrt {m}\in \Omega _{\leq T}\bigr \}\bigr |\leq N(Q,m,\omega )$ . By definition and Lemma 4.11, the singular integral $\mu _\infty (Q,\omega )\ll \mu _\infty (\Omega _{\leq 2T})=O(T)$ . Then the assertion follows by applying Theorem 4.9 to $\omega $ and the fact that $\mu (Q,m)=O_F(1)$ (since $b\geq 3$ ).

4.3.2 Assume $T\geq 1$

In this case, we will need the exact main term along with an error term with explicit dependence on T, and we will apply Niedermowwe’s work [Reference NiedermowweNie10].

For the convenience of later use, for an integer $m\geq 1$ , we define the quantity

(4.6) $$ \begin{align} a(m)=\frac{-c(m)\Gamma(k)\sqrt{|L^\vee/L|}}{2(2\pi)^{k}}, \end{align} $$

where $c(m)$ is the mth Fourier coefficient of the Eisenstein series defined in Section 3.3. Note that $a(m)$ grows as $\asymp m^{\frac {b}{2}}$ . We have the following proposition, which follows from work of Niedermowwe [Reference NiedermowweNie10].

Proposition 4.13. Let $A>1$ be a positive real number. For any $m\in \mathbb {Z}_{>0}$ , $T\geq 1$ , we have

$$ \begin{align*} \bigl|\bigl\{ \lambda\in L:Q(\lambda)=m,\,\lambda/\sqrt{m}\in \Omega_{\leq T}\bigr\}\bigr|= a(m)\mu_\infty(\Omega_{\leq T})+O\bigl(m^{\frac{b}{2}}T^{\frac{b}{2}}\log(mT)^{-A}\bigr). \end{align*} $$

Proof. In [Reference NiedermowweNie10, Theorem 3.6], Niedermowwe estimates the number of lattice points with fixed norm in homogeneously expanding rectangular regions. His proof carries over without change for our region, yielding thatFootnote ²⁴

$$\begin{align*}\bigl|\bigl\{ \lambda\in L:Q(\lambda)=m,\,\lambda/\sqrt{m}\in \Omega_{\leq T}\bigr\}\bigr|= \mu_\infty(\Omega_{\leq T})m^{\frac{b}{2}}\mu(Q,m)+O\bigl(m^{\frac{b}{2}}T^{\frac{b}{2}}\log(mT)^{-A}\bigr).\end{align*}$$

We then deduce the desired formula by the explicit formula for $c(m)$ in [Reference Bruinier and KussBK01, (22),(23)], which asserts that $\displaystyle c(m)=-\frac {2(2\pi )^{\frac {b}{2}+1}m^{\frac {b}{2}}}{\sqrt {|L^\vee /L|}\Gamma (\frac {b}{2}+1)}\prod _p \mu _p(Q,m)$ .

We conclude this section by an integral computation similar to Lemma 4.11, which will be used later. For $s\in \mathbb {R}$ , consider the function

$$ \begin{align*} h_s:L_{\mathbb{R}}&\rightarrow \mathbb{R}^+\\ \lambda&\mapsto \left(\frac{1}{1-Q(\lambda_x)}\right)^{k-1+s}. \end{align*} $$

Lemma 4.14. For $s>0$ , we have

$$ \begin{align*} \int_{L_{\mathbb{R},1}}h_s(\lambda)d\mu_{\infty}(\lambda)= \frac{b}{4}\cdot \frac{|c(m)|}{s\cdot a(m)}. \end{align*} $$

Proof. As in the proof of Lemma 4.11, we define $L_\epsilon =:\{x\in L_{\mathbb {R}}:|Q(x)-1|<\epsilon \}$ . Then

$$ \begin{align*} \lim_{\epsilon\rightarrow 0} \frac{1}{2\epsilon}&\int_{L_{\epsilon}}\left(\frac{1}{1-Q(\lambda_x)}\right)^{k-1+s}d\mu_{\infty}(\lambda)\\&=\lim_{\epsilon\rightarrow 0}\frac{2^{\frac{b}{2}}}{\epsilon\sqrt{|L^\vee/L|}}\int_{(x_1,x_2)\in \mathbb{R}^2}\int_{\underset{|y_1^2+\cdots+y_b^2-x_1^2-x_2^2-1|<\epsilon}{(y_1,\dots, y_b)\in \mathbb{R}^b}}\frac{dx_1dx_2dy_1\cdots dy_b}{\left(1+x_1^2+x_2^2\right)^{k-1+s}}\\ &=\frac{2(2\pi)^{1+\frac{b}{2}}}{\Gamma\left(\frac{b}{2}\right).\sqrt{|L^\vee/L|}}\int_{0}^{+\infty}\left(\frac{1}{1+r^2}\right)^{s+1}rdr\\ &=\frac{(2\pi)^{1+\frac{b}{2}}}{\Gamma\left(\frac{b}{2}\right).\sqrt{|L^\vee/L|}}\frac{1}{s}. \end{align*} $$

The lemma now follows from the definition of $a(m)$ .

5.1 Bruinier’s explicit formula for the Green function $\Phi _m$

There is an another expression for the Green function $\Phi _m$ introduced in Section 3.1 due to Bruinier (see [Reference BruinierBru02, §2] and [Reference Bruinier and KühnBK03, §4]); this expression will allow us to make explicit computations later. As in Section 3.3, let $k=1+\frac {b}{2}$ , and $s\in \mathbb {C}$ with $\mathrm {Re}(s)>\frac {k}{2}$ . We pick a lift of $x\in M(\mathbb {C})$ to the period domain $D_L$ and still use x to denote the lift. Recall from Section 2.1 that x defines a negative definite planeFootnote ²⁵ $P_x$ of $L_{\mathbb {R}}$ and for $\lambda \in L_{\mathbb {R}}$ , we denote by $\lambda _x$ the orthogonal projection of $\lambda $ on $P_x$ . Let

$$ \begin{align*}F(s,z)=H\left(s-1+\frac{k}{2},s+1-\frac{k}{2},2s;z\right),\text{ where } H(a,b,c;z)=\sum_{n\geq 0}\frac{(a)_n(b)_n}{(c)_n}\frac{z^n}{n!}\end{align*} $$

is the Gauss hypergeometric function as in [Reference Abramowitz and StegunAS64, Chapter 15] and $(a)_n=\frac {\Gamma (a+n)}{\Gamma (a)}$ for $a,b,c,z\in \mathbb {C}$ and $|z|<1$ . Finally, letFootnote ²⁶

(5.3) $$ \begin{align} \phi_{m}(x,s)=2\frac{\Gamma(s-1+\frac{k}{2})}{\Gamma(2s)}\sum_{Q(\lambda)=m, \lambda\in L}\left(\frac{m}{m-Q(\lambda_x)}\right)^{s-1+\frac{k}{2}} F\left(s,\frac{m}{m-Q(\lambda_x)}\right). \end{align} $$

By [Reference BruinierBru02, Proposition 2.8, Theorem 2.14], the function $\phi _{m}(x,s)$ admits a meromorphic continuation to $\mathrm {Re}(s)>1$ with a simple pole at $s=\frac {k}{2}$ with residue $-c(m)$ , where $c(m)$ is the Fourier coefficient defined in Section 3.3; see also [Reference Bruinier and KühnBK03, Proposition 4.3] for the value of the residue.

We regularise $\phi _m(x,s)$ at $s=k/2$ by defining $\phi _{m}(x)$ to be the constant term at $s=\frac {k}{2}$ of the Laurent expansion of $\phi _{m}(x,s)$ . As in [Reference Bruinier and KühnBK03, Prop.4.2], for $x\in D_L$ , we have

(5.4) $$ \begin{align} \phi_m(x)=\lim_{s\rightarrow \frac{k}{2}}\left(\phi_{m}(x,s)+\frac{c(m)}{s-\frac{k}{2}}\right). \end{align} $$

To compare $\phi _m(x)$ with $\Phi _m(x)$ , we recall that $C(n,s), n\in \mathbb {Z}, s\in \mathbb {C}, \operatorname {\mathrm {Re}}(s)>1-\frac {k}{2}$ is part of the Fourier coefficient of $E_0(\tau ,s)$ defined in Section 3.3. For $s\in \mathbb {C}$ with $\mathrm {Re}(s)>1$ , defineFootnote ²⁷

(5.5) $$ \begin{align} b_m(s)=-\frac{C\left(m,s-\frac{k}{2}\right)\cdot\left(s-1+\frac{k}{2}\right)}{\left(2s-1\right)\cdot\Gamma\left(s+1-\frac{k}{2}\right)}. \end{align} $$

By [Reference BruinierBru02, Theorem 1.9], $b_m(s)$ is a holomorphic function of s in the region $\mathrm {Re}(s)>1$ .

Proposition 5.1 [Reference BruinierBru02, Proposition 2.11]

For $x\in D_{L}$ , we have

$$ \begin{align*}\Phi_m(x)=\phi_m(x)-b^{\prime}m(k/2).\end{align*} $$

This proposition shows in particular that $\phi _{m}$ is also a Green function for the arithmetic cycle $\mathcal {Z}(m)$ .

5.2 Estimating $b^{\prime }m(k/2)$

The main result of this subsection is the following.

Proposition 5.2. Let $m\in \mathbb {Z}_{>0}$ such that $\mathcal {Z}(m)\neq \emptyset $ ; and if b is odd, we further assume that for a fixed D, $\sqrt {m/D}\in \mathbb {Z}$ , as in Theorem 2.4. Then as $m\rightarrow +\infty $ , we have

$$ \begin{align*}b^{\prime}m(k/2)=|c(m)|\log m+o\left(c(m)\log m\right).\end{align*} $$

In particular, $b^{\prime }m(k/2)\asymp m^{\frac {b}{2}}\log m$ .

Lemma 5.3 below reduces the proposition into computations of certain local invariants of the lattice L at primes p.

In order to state the lemma, we recall some notations from [Reference Bruinier and KühnBK03] and Section 4.1. Recall that $r=b+2$ , and let D be the fixed integer in Theorem 2.4; let d be

$$ \begin{align*} &(-1)^{\frac{r}{2}}\det(L),\, \text{if r is even};\\ &2(-1)^{\frac{r+1}{2}}D\det(L),\, \text{otherwise}, \end{align*} $$

where $\det (L)$ denote the Gram determinant of L. Let $d_0$ denote the fundamental discriminant of number field $\mathbb {Q}(\sqrt {d})$ , and let $\chi _{d_{0}}$ be the quadratic character associated to $d_0$ . The polynomial $\mathrm {L}_m^{(p)}(t)$ is defined by

$$ \begin{align*}\mathrm{L}_m^{(p)}(t)=N_{m}(p^{w_p})t^{w_p}+(1-p^{r-1}t)\sum_{n=0}^{w_p-1}N_{m}(p^n)t^n\in\mathbb{Z}[t],\end{align*} $$

where, as in Section 4.1, $N_{m}(p^n)=\#\{v\in L/p^nL;\,Q(v)\equiv m\ \pmod {p^n}\}$ and $w_p:=w_p(m)=1+\operatorname {\mathrm {val}}_{p}(m)$ for $p\neq 2$ and $w_2:=w_2(m)=1+2\operatorname {\mathrm {val}}_2(2m)$ .Footnote ²⁸

Lemma 5.3 (Bruinier–Kühn)

Let $D\in \mathbb {Z}_{>0}$ be the fixed integer in Theorem 2.4 for all $m\in \mathbb {Z}_{>0}$ such that $\sqrt {m/D}\in \mathbb {Z}$ (and representable by $(L,Q)$ ); we have

$$\begin{align*}\frac{b^{\prime}m\left(\frac{k}{2}\right)}{b_m\left(\frac{k}{2}\right)}=\log(m)+2\frac{\sigma_{m}'(k)}{\sigma_{m}(k)}+O(1),\end{align*}$$

where for $s\in \mathbb {C}$ , $\mathrm {Re}(s)>0$ , the function $\sigma _{m}$ is given by

(5.6) $$ \begin{align} \sigma_{m}(s)=\left\{\begin{array}{ll} \underset{p\mid 2m\det(L)}{\prod}\frac{\mathrm{L}_m^{(p)}\left(p^{1-\frac{r}{2}-s}\right)}{1-\chi_{d_{0}}(p)p^{-s}},\,\textrm{if}\, r\,\textrm{is even},\\ \underset{p\mid 2m\det(L)}{\prod}\frac{1-\chi_{d_{0}}(p)p^{\frac{1}{2}-s}}{1-p^{1-2s}}\mathrm{L}_m^{(p)}\left(p^{1-\frac{r}{2}-s}\right),\, \textrm{if}\, r\,\textrm{is odd}. \end{array} \right. \end{align} $$

Proof. Taking logarithmic derivatives in equation (5.5) at $s=\frac {k}{2}$ yields

$$ \begin{align*} \frac{b^{\prime}m\left(\frac{k}{2}\right)}{b_m\left(\frac{k}{2}\right)}=\frac{C'(m,0)}{C(m,0)}-\frac{2}{b}-\Gamma'(1) \end{align*} $$

Then we conclude by [Reference Bruinier and KühnBK03, Theorem 4.11, (4.73), (4.74)], since both $d_0$ and k are independent of m. (Our definition of d above differs from the definition in [Reference Bruinier and KühnBK03] by $m/D$ , which is a square and hence yields the same $d_0$ .)

Proof of Proposition 5.2

By definition, $b_m(k/2)=-c(m)=|c(m)|$ . Therefore, by Lemma 5.3, it is enough to show that

$$ \begin{align*}\frac{\sigma_{m}'(k)}{\sigma_{m}(k)}=o(\log(m)).\end{align*} $$

Taking the logarithmic derivative in equation (5.6) at $s=k$ , we get for r even

$$ \begin{align*} \frac{\sigma_{m}'(k)}{\sigma_{m}(k)}=-\sum_{p\mid 2m\det(L)}\left(\frac{p^{1-r}\mathrm{L}_m^{(p)'}\left(p^{1-r}\right)}{\mathrm{L}_m^{(p)}\left(p^{1-r}\right)}+\frac{\chi_{d_0}(p)}{p^k-\chi_{d_0}(p)}\right)\log (p), \end{align*} $$

and for r odd

$$ \begin{align*} \frac{\sigma_{m}'(k)}{\sigma_{m}(k)}=-\sum_{p\mid 2m\det(L)}\left(\frac{p^{1-r}\mathrm{L}_m^{(p)'}\left(p^{1-r}\right)}{\mathrm{L}_m^{(p)}\left(p^{1-r}\right)}-\frac{\chi_{d_0}(p)}{p^{k-\frac{1}{2}}-\chi_{d_0}(p)}+\frac{2}{p^{2k-1}-1}\right) \log(p). \end{align*} $$

Since $k=1+\frac {b}{2}\geq \frac {5}{2}$ , we have

$$ \begin{align*} \left|\sum_{p\mid 2m\det(L)}\frac{\chi_{d_0}(p)\log(p)}{p^k-\chi_{d_0}(p)}\right|\leq \sum_{p}\frac{\log(p)}{p^{5/2}-1}<+\infty, \end{align*} $$

$$ \begin{align*} \left|\sum_{p\mid 2m\det(L)}\frac{\chi_{d_0}(p)\log(p)}{p^{k-\frac{1}{2}}-\chi_{d_0}(p)}\right|\leq \sum_{p}\frac{\log(p)}{p^{2}-1}<+\infty, \end{align*} $$

$$ \begin{align*} \left|\sum_{p\mid 2m\det(L)}\frac{2\log p}{p^{2k-1}-1}\right|\leq \sum_{p}\frac{2\log(p)}{p^{4}-1}<+\infty. \end{align*} $$

Hence it remains to treat the $L^{(p)}_m$ term. We have $\mathrm {L}^{(p)}_{m}(p^{1-r})=N_{m}(p^{w_p})p^{(1-r)w_p}$ and

$$ \begin{align*}\mathrm{L}^{(p)'}_{m}(p^{1-r})=w_pN_{m}(p^{w_p})p^{(1-r)(w_p-1)}-\sum_{n=0}^{w_p-1}N_{m}(p^n)p^{(n-1)(1-r)}.\end{align*} $$

Hence

$$ \begin{align*} \left|\frac{p^{1-r}\mathrm{L}_m^{(p)'}\left(p^{1-r}\right)}{\mathrm{L}_m^{(p)}\left(p^{1-r}\right)}\right|=\left|w_p-\sum_{n=0}^{w_p-1}\frac{N_{m}(p^n)}{N_{m}(p^{w_p})}p^{(n-w_p)(1-r)}\right|=\left|w_p-\sum_{n=0}^{w_p-1}\frac{\mu_{p}(m,n)}{\mu_{p}(m,w_p)}\right|\leq \frac{C}{p}, \end{align*} $$

where $\mu _{p}(m,n)=p^{-n(r-1)}N_{m}(p^n)$ as in Section 4.1, and the last inequality follows from Proposition 4.1 with constant C only depending on $(L,Q)$ (i.e., is independent of $m,p$ ). Thus we have

$$ \begin{align*} \left|\sum_{p\mid 2m\det(L) }\frac{p^{1-r}\mathrm{L}_m^{(p)'}\left(p^{1-r}\right)}{\mathrm{L}_m^{(p)}\left(p^{1-r}\right)}\right|\leq C\sum_{p\mid 2m\det(L)}\frac{\log(p)}{p}=O(\log\log(m)). \end{align*} $$

Here we use the fact that for $N\geq 2$ , $\sum _{p\mid N}\frac {\log (p)}{p}=O(\log \log (N))$ . Indeed, let $X=\log (N)$ , and use Mertens’ first theorem to write

$$ \begin{align*} \sum_{p\mid N}\frac{\log(p)}{p}&=\sum_{p\mid N,p<X}\frac{\log(p)}{p}+\sum_{p\mid N,p\geq X}\frac{\log(p)}{p}\leq \log(X)+\frac{1}{X}\sum_{p\mid N}\log(p)+O(1)\\ &\leq \log(X)+\frac{\log(N)}{X}+O(1)\leq \log(\log(N))+O(1). \end{align*} $$

This concludes the proof of the proposition.

5.3 Estimates on $\phi _m(x)$

Recall that $x\in M(\mathbb {C})$ is a Hodge-generic point, and we pick a lift of x to $D_L$ . We will associate the quantity $A(m,x)$ to x, which is independent of the choice of the lift. Thus, we will also denote this lift by x. Recall from Section 5.1 that for $\lambda \in L_{\mathbb {R}}$ , we use $\lambda _x$ to denote the orthogonal projection of $\lambda $ onto the negative definite plane in $L_{\mathbb {R}}$ associated to x. Define

(5.7) $$ \begin{align} A(m,x):=-2\sum_{\underset{|Q(\lambda_x)|\leq 1,Q(\lambda)=1}{\sqrt{m}\lambda\in L}}\log (|Q(\lambda_x)|). \end{align} $$

Note that since x is Hodge-generic, for any $\lambda \in L$ , $\lambda _x\neq 0$ . Hence for any $\lambda \in L_{\mathbb {R}}$ such that $\mathbb {R}\lambda \cap L\neq \{0\}$ , we also have $\lambda _x\neq 0$ . On the other hand, the conditions $|Q(\lambda _x)|\leq 1,Q(\lambda )=1$ cut out a compact region in $L_{\mathbb {R}}$ , and hence for a fixed m, $A(m,x)$ is the sum of finitely many terms. Therefore $A(m,x)$ is well-defined and non-negative.

The main purpose of this subsection is to prove the following result.

Proposition 5.4. For $m\in \mathbb {Z}_{>0}$ , we have

$$ \begin{align*}\phi_{m}(x)=A(m,x) +O(m^{\frac{b}{2}}).\end{align*} $$

Recall that $F(s,t)$ from Section 5.1. Since $F(s,0)=1$ , for $z\in \mathbb {C}$ with $|z|<1$ , we may write $F(s,z)=zG(s,z)+1$ . Recall that we set $k=1+\frac {b}{2}$ . From the definitions, we obtain the following decomposition of $\phi _{m}(x)$ :

(5.8) $$ \begin{align} \begin{array}{rcl} \phi_{m}(x)&\overset{(5.4)}{=}&\displaystyle \lim_{s\to\frac{k}{2}}\Bigl(\phi_m(x,s)+\frac{c(m)}{s-\frac{k}2} \Bigr) \\[.2in]&\overset{(5.3)}{=}&\displaystyle \lim_{\underset{\operatorname{\mathrm{Re}} s>0}{ s\to 0}}\left(\frac{c(m)}{s}+\frac{4}{b}\sum_{\underset{Q(\lambda)=m}{\lambda\in L}}\Bigl(\frac{m}{m-Q(\lambda_x)}\Bigr)^{k-1+s} F\Bigl(\frac{k}{2}+s,\frac{m}{m-Q(\lambda_x)}\Bigr)\right) \\[.3in]&=&\displaystyle \widetilde{\phi}_m(x,0)+ \lim_{\underset{\operatorname{\mathrm{Re}} s>0}{ s\to 0}}R_x(s,m), \end{array} \end{align} $$

where for $s\in \mathbb {C}$ with $\operatorname {\mathrm {Re}} s>0$ , we define

$$ \begin{align*} \begin{array}{rcl} \displaystyle\widetilde{\phi}_{m}(x,s) &=& \displaystyle\frac{4}{b}\sum_{\underset{Q(\lambda)=1}{\sqrt{m}\lambda\in L}} \Bigl(\frac{1}{1-Q(\lambda_x)}\Bigr)^{k+s}G\Bigl(\frac{k}{2}+s,\frac{1}{1-Q(\lambda_x)}\Bigr), \\[.2in] \displaystyle R_x(s,m)&=& \displaystyle\frac{c(m)}{s}+\frac{4}{b}\sum_{\underset{Q(\lambda)=1}{\sqrt{m}\lambda\in L}}\Bigl(\frac{1}{1-Q(\lambda_x)}\Bigr)^{k-1+s}; \displaystyle \end{array} \end{align*} $$

in Section 5.3.2, we will prove that the above series defining $\widetilde {\phi }_m(x,s)$ indeed converges uniformly absolutely in a small compact neighbourhood of $s=0$ in $\mathbb {C}$ , and hence it defines a function holomorphic at $0$ , and we still denote this function by $\widetilde {\phi }_m(x,s)$ . In particular, $\widetilde {\phi }_m(x,0)$ is well defined and equal to $\lim _{\operatorname {\mathrm {Re}} s>0, s\rightarrow 0}\widetilde {\phi }_m(x,s)$ . Therefore, the last equality in equation (5.8) is valid; moreover, since $\phi _m(x,s+\frac {k}{2})+\frac {c(m)}{s}$ admits a holomorphic continuation to $s=0$ (see Section 5.1), then $R_x(s,m)$ admits a holomorphic continuation to $s=0$ . We use $R_x(0,m)$ to denote $\lim _{\operatorname {\mathrm {Re}} s>0, s\to 0}R_x(s,m)=\lim _{s\in \mathbb {R}_{>0}, s\to 0}R_x(s,m)$ and rewrite equation (5.8) as

(5.9) $$ \begin{align} \phi_m(x)=\widetilde{\phi}_{m}(x,0)+R_x(0,m). \end{align} $$

Note that the second equality in equation (5.8) also uses the fact that the ratio of $\Gamma $ -functions in equation (5.3) is holomorphic and has a limit of $2/b$ .

In what follows next, we estimate $R_x(0,m)$ and $\widetilde {\phi }_{m}(x,0)$ using results from Section 4.3, where we use the work of Heath-Brown and Niedermowwe to estimate the number of the lattice points $\lambda $ in certain regions in L with $Q(\lambda )=m$ .

5.3.1 Bounding $R_x(0,m)$

We only consider $s\in \mathbb {R}_{\geq 0}$ . Recall from Section 4.3 that for $\lambda \in L_{\mathbb {R},1}=\{\lambda \in L_{\mathbb {R}}:Q(\lambda )=1\}$ , we set $\displaystyle h_s(\lambda )=\left (\frac {1}{1-Q(\lambda _x)}\right )^{k-1+s}$ ; $\displaystyle a(m)=\frac {-c(m)\Gamma (k)\sqrt {|L^\vee /L|}}{2(2\pi )^{k}}>0$ , and $\mu _\infty $ denotes the measure on $L_{\mathbb {R},1}$ defined in equation (4.5).

For $s>0$ , Lemma 4.14 yields the equality

$$ \begin{align*}R_x(s,m)=\frac{4}{b}\sum_{\underset{Q(\lambda)=1}{\sqrt{m}\lambda\in L}}h_s(\lambda)-\frac{4a(m)}{b}\int_{L_{\mathbb{R},1}}h_s(\lambda)d\mu_{\infty}(\lambda).\end{align*} $$

Our next result provides the required bound on $R_x(0,m)$ .

Proposition 5.5. For a given $x\in D_L$ Hodge-generic, we have

$$ \begin{align*}R_x(0,m)=\lim_{s\in \mathbb{R}_{>0}, s\rightarrow 0} R_x(s,m) =O(m^{\frac{b}{2}}).\end{align*} $$

Proof. Fix $\epsilon>0$ , and we only consider $s\in [0,\epsilon ]$ . As in Section 4.3, $\Omega _T=\{\lambda \in L_{\mathbb {R},1}:-Q(\lambda _x)\in [0,T]\}$ . For an integer $N\geq 0$ , define the set $\Theta _N:=\Omega _{N+1}\backslash \Omega _N$ , and note that for $\lambda \in \Theta _N$ , we have

$$ \begin{align*}h_s(\lambda)=\frac{1}{(N+1)^{s+k-1}}+O\Bigl(\frac{1}{(N+1)^{s+k}}\Bigr). \end{align*} $$

To obtain a uniformly absolutely convergent expression for $R_x(s,m)$ for $s\in ]0, \epsilon ]$ , we write

$$ \begin{align*} \frac{b}{4}R_x(s,m)= \sum_{N=0}^\infty \Bigl(\sum_{\substack{\lambda\in\Theta_N \\\sqrt{m}\lambda\in L}} h_s(\lambda)-a(m)\int_{\Theta(N)}h_s(\lambda)d\mu_{\infty}(\lambda)\Bigr). \end{align*} $$

We use the above estimate of $h_s(\lambda )$ for $\lambda \in \Theta _N$ and bound the associated error term using Proposition 4.13 and Lemma 4.11. More precisely, let $C>0$ be an absolute constant such that for all $\lambda \in \Theta _N$ , for all $s\in [0,\epsilon ]$ , we have $|h_s(\lambda )-(N+1)^{-(s+k-1)}|<C(N+1)^{-(s+k)}$ . Then the error term is bounded by

(5.10) $$ \begin{align} \begin{array}{rcl} & & \displaystyle C \sum_{N=0}^{\infty} (N+1)^{-(s+k)} \Bigl(\sum_{\substack{\lambda\in\Theta_N \\\sqrt{m}\lambda\in L}}1 + a(m)\mu_\infty(\Theta_N)\Bigr) \\& = & \displaystyle C\sum_{N=1}^\infty (N^{-(s+k)}-(N+1)^{-(s+k)})\Bigl(\sum_{\substack{\lambda\in\Omega_N \\\sqrt{m}\lambda\in L}}1 + a(m)\mu_\infty(\Omega_N)\Bigr) \\ & \ll & \displaystyle a(m) \sum_{N=1}^\infty N^{-(s+k+1)}\cdot N^{\frac{b}{2}} = a(m) \sum_{N=1}^\infty N^{-s-2} \ll a(m)=O(m^{\frac{b}{2}}), \end{array} \end{align} $$

where the first equality is a partial summation and the implicit constants above are absolute.

To bound the main term, we pick some $A>1$ in Proposition 4.13. By partial summation, we write

$$ \begin{align*} \begin{array}{rcl} &&\displaystyle \sum_{N=0}^{\infty}\frac{1}{(N+1)^{s+k-1}} \Bigl(\sum_{\substack{\lambda\in\Theta_N \\\sqrt{m}\lambda\in L}}1 -a(m)\mu_\infty(\Theta_N)\Bigr) \\[.2in]&=& \displaystyle \sum_{N=1}^{\infty} \Bigl(\frac{1}{N^{s+k-1}}-\frac{1}{(N+1)^{s+k-1}}\Bigr) \Bigl(\sum_{\substack{\lambda\in\Omega_N \\\sqrt{m}\lambda\in L}}1 -a(m)\mu_\infty(\Omega_N)\Bigr) \\[.2in]&\ll&\displaystyle \displaystyle \sum_{N=1}^{\infty}N^{-(s+k)} \cdot N^{\frac{b}{2}}m^{\frac{b}{2}}(\log mN)^{-A}\leq m^{\frac{b}{2}}\sum_{N=1}^\infty N^{-1}(\log N)^{-A}\ll m^{\frac{b}{2}}, \end{array} \end{align*} $$

which is again sufficient. The proposition follows.

5.3.2 Estimating $\widetilde {\phi }_{m}(x,0)$

We fix an $\epsilon \in ]0,1/2]$ and consider $s\in \mathbb {C}$ such that $|s|\leq \epsilon $ . Since

$$ \begin{align*}\displaystyle G\left(s+\frac{k}{2},z\right)=\sum_{n=1}^\infty\frac{\Gamma(s+k-1+n)\Gamma(s+1+n)\Gamma(2s+k)}{\Gamma(s+k-1)\Gamma(s+1)\Gamma(2s+k+n)}\frac{z^{n-1}}{n!}\end{align*} $$

converges uniformly absolutely for $|s|\leq \epsilon , |z|\leq 1/2$ , then $G\left (s+\frac {k}{2},z\right )$ is absolutely bounded for such s and z.

To show that $\widetilde {\phi }_m(x,s)$ is holomorphic in $|s|<\epsilon $ , we write

$$ \begin{align*} \begin{array}{rcl} \displaystyle\frac{b}{4}\widetilde{\phi}_m(x,s)&=& \displaystyle \sum_{\substack{\sqrt{m}\lambda\in L\\Q(\lambda)=1\\ |Q(\lambda_x)|>1}} \Bigl(\frac{1}{1-Q(\lambda_x)}\Bigr)^{k+s}G\Bigl(\frac{k}{2}+s,\frac{1}{1-Q(\lambda_x)}\Bigr) \\[.2in]&&+\displaystyle \sum_{\substack{\sqrt{m}\lambda\in L\\Q(\lambda)=1\\|Q(\lambda_x)|\leq 1}} \Bigl(\frac{1}{1-Q(\lambda_x)}\Bigr)^{k+s}G\Bigl(\frac{k}{2}+s,\frac{1}{1-Q(\lambda_x)}\Bigr), \end{array} \end{align*} $$

where, similar to the definition of $A(m,x)$ in equation (5.7), the second term is a finite sum of holomorphic functions. Thus we only need to show that the first term converges absolutely uniformly on $|s|\leq \epsilon $ . The uniform absolute convergence follows from the boundedness of $G(s+k/2,z)$ in conjunction with an argument identical to equation (5.10), which indeed implies that

$$\begin{align*}\sum_{\substack{\sqrt{m}\lambda\in L\\Q(\lambda)=1\\ |Q(\lambda_x)|>1}} \Bigl(\frac{1}{1-Q(\lambda_x)}\Bigr)^{k+s}G\Bigl(\frac{k}{2}+s,\frac{1}{1-Q(\lambda_x)}\Bigr)=O(m^{\frac{b}{2}}).\end{align*}$$

Therefore,

$$\begin{align*}\widetilde{\phi}_m(x,0)=\frac{4}{b} \sum_{\substack{\sqrt{m}\lambda\in L\\Q(\lambda)=1\\|Q(\lambda_x)|\leq 1}} \Bigl(\frac{1}{1-Q(\lambda_x)}\Bigr)^{k}G\Bigl(\frac{k}{2},\frac{1}{1-Q(\lambda_x)}\Bigr)+O(m^{\frac{b}{2}}).\end{align*}$$

Obtaining an estimate on the terms with $|Q(\lambda _x)|\leq 1$ is significantly more difficult since the function

$$ \begin{align*}\lambda\mapsto \left(\frac{1}{1-Q(\lambda_x)}\right)^{k}G\left(\frac{k}{2},\frac{1}{1-Q(\lambda_x)}\right)\end{align*} $$

has a logarithmic singularity along $\{\lambda \in L_{\mathbb {R},1},\, x\in \lambda ^{\bot }\}$ . Since $\displaystyle G(k/2,z)=\sum _{n=1}^\infty \frac {k-1}{n+k-1}z^{n-1}$ , it follows that there exists an absolute constant $C>0$ such that for $z\in [1/2, 1[$ ,

$$ \begin{align*} \left|z^{k}G\left(\frac{k}{2},z\right)+\frac{b}{2}\log(1-z)\right| \leq C. \end{align*} $$

Hence, noting that $\displaystyle \sum _{\substack {|Q(\lambda _x)|\leq 1 \\\sqrt {m}\lambda \in L}}1=O(m^{\frac {b}{2}})$ , which follows from either Corollary 4.12 or Proposition 4.13, we have

$$ \begin{align*} \frac{4}{b}\sum_{\substack{\sqrt{m}\lambda\in L\\Q(\lambda)=1\\|Q(\lambda_x)|\leq 1}} \Bigl(\frac{1}{1-Q(\lambda_x)}\Bigr)^{k}G\Bigl(\frac{k}{2},\frac{1}{1-Q(\lambda_x)}\Bigr)&=-2\sum_{\substack{\sqrt{m}\lambda\in L\\Q(\lambda)=1\\|Q(\lambda_x)|\leq 1}}\log\Bigl(\frac{-Q(\lambda_x)}{1-Q(\lambda_x)}\Bigr)+O(m^{\frac{b}{2}}) \\ &=-2\sum_{\substack{\sqrt{m}\lambda\in L\\Q(\lambda)=1\\|Q(\lambda_x)|\leq 1}}\log (|Q(\lambda_x)|)+O(m^{\frac{b}{2}}).\\ \end{align*} $$

Therefore, we have proved the following proposition.

Proposition 5.6. We have

$$ \begin{align*}\widetilde{\phi}_m(x,0)=A(m,x)+O(m^{\frac{b}{2}}).\end{align*} $$

Proposition 5.4 follows immediately from equation (5.9) and Propositions 5.5 and 5.6.

6.1 Bounding the main term using the circle method

In this section, we prove the following proposition.

Proposition 6.2. We have

$$ \begin{align*} \lim_{m\to\infty} \frac{A_{\mathrm{mt}}(m)}{m^{\frac{b}{2}}\log m}=0. \end{align*} $$

Proof. Fix a real number $T>1$ , and let $m>T$ be a positive integer. We break $A_{\mathrm {mt}}(m)$ into two terms $A_1(m)$ and $A_2(m)$ , where $A_1(m)$ is the sum over those $\lambda \in L$ such that $|Q(\lambda _x)|\geq m/T$ and $A_2(m)$ is the sum over those $\lambda \in L$ such that $|Q(\lambda _x)|< m/T$ . We start by bounding $A_1(m)$ . First note that if $\lambda \in L$ with $Q(\lambda )=m$ and $m/T\leq |Q(\lambda _x)|\leq m$ , then we have

$$ \begin{align*}\log\Bigl(\frac{m}{|Q(\lambda_x)|}\Bigr)\leq \log T.\end{align*} $$

Furthermore, by Corollary 4.12,

$$ \begin{align*} |\{\lambda \in L: Q(\lambda)=m,Q(\lambda_x)\leq m\}|\ll m^{\frac{b}{2}}. \end{align*} $$

Therefore, we obtain the bound

(6.1) $$ \begin{align} A_1(m)\ll m^{\frac{b}{2}}\log T. \end{align} $$

Next we consider $A_2(m)$ . Once again, we apply Corollary 4.12 to obtain the bound (take $\epsilon =1/4$ )

(6.2) $$ \begin{align} \left|\Bigl\{\lambda\in L, Q(\lambda)=m, |Q(\lambda_x)|<\frac{m}{T}\Bigr\}\right|\ll \frac{m^{\frac{b}{2}}}{T} +O_T(m^{\frac{b+2}{4}}). \end{align} $$

Equations (6.1) and (6.2) and the trivial bound $\log (m/|Q(\lambda _x)|)\leq \log m$ for $\lambda \in L$ with $1\leq |Q(\lambda _x)|\leq m$ yield the following estimate on $A_{\mathrm {mt}}(m)=A_1(m)+A_2(m)$ :

$$ \begin{align*} A_{\mathrm{mt}}(m)\ll m^{\frac{b}{2}}\log T+m^{\frac{b}{2}}(\log m)T^{-1}+O_{T}(m^{(b+2)/4}\log m). \end{align*} $$

Dividing by $m^{\frac {b}{2}}\log (m)$ and letting m tend to infinity, we see that

$$ \begin{align*} \limsup_{m\to\infty} \frac{A_{\mathrm{mt}}(m)}{m^{\frac{b}{2}}\log m}\ll \frac{1}{T}. \end{align*} $$

Since this is true for every T, $\displaystyle \lim _{m\rightarrow \infty } \frac {A_{\mathrm {mt}}(m)}{m^{\frac {b}{2}}\log m}$ exists and is equal to $0$ .