2.1.1 General notation

The following notational choices are largely standard.

• • The fields F and E are as fixed in the introduction unless specified otherwise; if $*$ denotes a place of $\mathbf {Q}$ or a finite set thereof, we denote by $S_{*}$ the set of places of F above $*$ .

• • We denote by $D_{F}$ , $D_{E}$ , and $D_{E/F}$ , respectively, the absolute discriminants of F and E and relative discriminant of $E/F$ ; for a finite place v of F, we denote by $d_{v}\in F_{v}$ a generator of the different ideal of F and by $D_{v}\in F_{v}$ a generator of the relative discriminant ideal.

• • We denote by $<$ the partial order on F given by $x<y$ if and only if $\tau (x)<\tau (y)$ for all $\tau \in \Sigma _{\infty }$ ; we denote $\mathbf {R}^{+}:= \{x\in \mathbf {R}\, | \, x>0\}$ and $F^{+}:=\{x\in F\, |\, x>0\}\subset F^{\times }$ .

• • $\mathbf {A}$ is the ring of adèles of F; if S is a finite set of places of a number field F, we denote $\mathbf {A}^{S}=\prod ^{\prime }_{v\notin S} F_{v}$ , and $F_{S}:= \prod _{v\in S}F_{v}$ ; when S consists of the set of places of F above some finite set of places of $\mathbf {Q}$ (for instance, the place p), we use the same notation with those places of $\mathbf {Q}$ instead of S (for instance, $F_{p}=F_{S_{p}}$ ). We denote $F_{\infty }^{+}=\{x\in F_{\infty }\ | \ x_{\tau }>0 \text { for all}\ \tau \in \Sigma _{\infty }\}$ and $\mathbf {A}^{+}:=\mathbf {A}^{\infty }\times F_{\infty }^{+}$ .

• • We denote by $\psi \colon F\backslash \mathbf {A}\to \mathbf {C}^{\times }$ the standard additive character as in Section 1.1.3.

• • If $R/R_{0}$ is a ring extension, A is an $R_{0}$ -algebra, and X is an $R_{0}$ -scheme, we denote

  $$ \begin{align*}A_{R}:=A\otimes_{R_{0}}R, \qquad X_{R}:= X\times_{\mathrm{Spec}\, R_{0}}R.\end{align*} $$

• • We denote by $G_{K}$ the absolute Galois group of a field K.

• • If K is a finite extension of F, its class number is denoted by

  $$ \begin{align*}h_{K}:= |K^{\times}\backslash \mathbf{A}_{K}^{\infty, \times}/\widehat{\mathscr{O}}^{\times}_{K}|.\end{align*} $$

• • For a place v of F, we denote by $\varpi _{v}$ a fixed uniformizer at v, and by $q_{F,v} $ the cardinality of the residue field; we denote $q_{F, p}:=(q_{F, v})_{v\in S_{p}}$ .

• • The class field theory isomorphism is normalized by sending uniformizers to geometric Frobenii; for K a number field (respectively, a local field), we will then identify characters of $G_{K}$ with characters of $K^{\times }\backslash {\mathbf {A}}_{K}^{\times }$ (respectively, $K^{\times })$ without further comment.

• • If I is a finite index set and $x=(x_{i})_{i}$ , $y=(y_{i})$ are real vectors, we define $(xy)_{i}=x_{i}y_{i}$ and $x^{y}:= \prod _{i}x_{i}^{y_{i}} $ whenever that makes sense. Moreover, we often identify an integer $w_{0}$ with the constant vector $(w_{0})_{i\in I}\in \mathbf {Z}^{I}$ .

• • We denote by $\mathbf {1}[\cdot ]$ the $\{0, 1\}$ -valued function on logical propositions such that $\mathbf {1}[\phi ]=1$ if and only if $\phi $ is true.

2.1.2 Subgroups of $\mathrm {GL}_{2}$ and special elements

We denote by Z, A, and N, respectively the center, diagonal torus, and upper unipotent subgroup of $\mathrm {G}=\mathrm {GL}_{2/F}$ ; we let $P=AN$ and $P^{1}:=P\cap \mathrm {SL}_{2/F}$ . We define a map $\mathbf {a}\colon \mathrm {GL}_{1/F}\to \mathrm {G}$ by

$$ \begin{align*}\mathbf{a}(y):= \left(\begin{array}{cc}y&\\&1\end{array}\right).\end{align*} $$

We denote

$$ \begin{align*}w:= \left(\begin{array}{cc}&1\\-1&\end{array}\right)\in \mathrm{GL}_{2}(F)\end{align*} $$

or its image in $\mathrm {GL}_{2}(R)$ for any F-algebra R. (The context will prevent any confusion with the notation for the weights of $\mathrm {G}$ .) For $r\in \mathbf {Z}_{\geq 1}^{S_{p}}$ , we define

$$ \begin{align*}w_{r_{v},v}:= \left(\begin{array}{cc}&1\\-\varpi_{v}^{r_{v}}&\end{array}\right)\in \mathrm{GL}_{2}(F_{v}), \qquad w_{r, p}:= \prod_{v\vert p} w_{r_{v},v}\in \mathrm{GL}_{2}(F_{p}),\end{align*} $$

as well as a sequence of compact subgroups

$$ \begin{align*} U_{v, r_{v}}:=U^{1}_{1}(\varpi_{v}^{r_{v}}) &:= \{ \left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right) \in \mathrm{GL}_{2}(\mathscr{O}_{F,v})\, :\, a-1\equiv d-1 \equiv c\equiv 0 \pmod{\varpi_{v}^{r_{v}}}\}\subset \mathrm{GL}_{2}(F_{v}),\\[5pt]U_{p, r}&:= \prod_{v\in S_{p}} U_{v, r_{v}}. \end{align*} $$

For $\theta \in (\mathbf {R}/2\pi \mathbf {Z})^{S_{\infty }}$ , we denote $r_{\theta }:=\left ( \left (\begin {array}{cc}\cos \theta _{v} &\sin \theta _{v}\\-\sin \theta _{v}&\cos \theta _{v}\end {array}\right )\right )_{v}\in \mathrm {SO}(2, F_{\infty }).$

2.1.3 Hecke algebras

Let S be a finite set of non-archimedean places of F, and let $U^{S}=\prod _{v\notin S}U_{v}\subset \mathrm {GL}_{2}(\mathbf {A}^{S\infty })$ be an open compact subgroup. For each finite set of finite places S, we define the Hecke algebra

$$ \begin{align*}\mathscr{H}_{U^{S}}:=C_{c}(U^{S}\backslash \mathrm{G}(\mathbf{A}^{S\infty}) /U^{S}, \mathbf{Z}).\end{align*} $$

It carries an involution

(2.1.1) $$ \begin{align}\begin{aligned} T\mapsto T^{\curlyvee}\end{aligned}\end{align} $$

arising from the map $g\mapsto g^{-1}$ on the group $\mathrm {G}$ .

Let $A_{p}:= A(F_{p})\subset \mathrm {G}(F_{p})$ be the diagonal torus, and let $A_{p}^{+}$ be the set of $t=\left (\begin {smallmatrix}t_{1}& \\&{t_{2}}\end {smallmatrix}\right ) $ such that $v(t_{1}) \geq v(t_{2})$ for all $v\vert p$ . The involution

(2.1.2) $$ \begin{align}\begin{aligned} t\mapsto t^{\curlyvee}:= \det(t)^{-1} t \end{aligned}\end{align} $$

preserves $A_{p}^{+}$ . For S a finite set of places of F disjoint from $S_{p}\cup S_{\infty }$ , we define the ordinary Hecke algebra

$$ \begin{align*}\mathscr{H}_{U^{Sp}}^{\mathrm{ord}}:= \mathscr{H}_{U^{Sp}, \mathbf{Z}_{p}}\otimes \mathbf{Z}_{p}[A_{p}]\end{align*} $$

over $\mathbf {Z}_{p}$ , which will act on spaces of ordinary modular forms (here and in the rest of the text, a subscript $Sp$ is shorthand for $S\cup S_{p}$ ). It is endowed with the involution $\curlyvee $ deduced from (2.1.1) and (2.1.2).

If $U=\prod _{v}U_{v}\subset \mathrm {G}(\mathbf {A}^{\infty })$ (respectively, $U^{p}=\prod _{v\nmid p}U_{v}\subset \mathrm {G}(\mathbf {A}^{p\infty })$ ) are open compact subgroups, and S (respectively, $S^{p}$ ) is the set of places such that $U_{v}$ is not maximal, we define

$$ \begin{align*}\mathscr{H}_{U}^{\mathrm{sph}}:= \mathscr{H}_{U^{S}}, \qquad \mathscr{H}_{U^{p}}^{\mathrm{sph,\ \mathrm{ord}}}:= \mathscr{H}_{U^{Sp}}^{\mathrm{ord}}.\end{align*} $$

(These depend on S, but their images in endomorphisms rings of spaces of modular forms do not.)

2.1.4 Measures

We use the same notation and conventions for Haar measures and integration as in [Reference Yuan, Zhang and ZhangYZZ12, Section 1.6] and [Reference DisegniDis17, Section 1.9]. In particular, we have a regularized integration functional

$$ \begin{align*}\int^{*}_{E^{\times}\backslash {\mathbf{A}}_{E}^{\times} /\mathbf{A}^{\times}} f(t) \, dt ,\end{align*} $$

which satisfies the following.

Lemma 2.1 Let f be a smooth function on $\mathbf {A}_{E}^{\times }$ that is invariant under $E_{\infty }^{\times }$ . Let $\mu \subset \mathscr {O}_{E}^{\times }$ be a finite index subgroup fixing f (under the scaling action). Then

$$ \begin{align*}\int^{*}_{E^{\times}\backslash {\mathbf{A}}_{E}^{\times} /\mathbf{A}^{\times}} \sum_{x\in E^{\times}} f(xt) \, dt = { 2L(1,\eta)\over h_{E}^{}} { [\mathscr{O}_{E}^{\times}:\mu_{}] } \int_{ \mathbf{A}_{E}^{\infty, \times}} \sum_{\alpha\in \mu_{}} f(\alpha t)\, d^{\bullet}t,\end{align*} $$

where $d^{\bullet } t $ is the Haar measure giving volume $1$ to $ \widehat {\mathscr {O}}^{\times }_{E}$ .

Proof Let $U\subset \mathbf {A}_{E}^{\infty , \times }$ be any compact open subgroup fixing f. Since both sides are independent of $\mu $ , we may assume that $\mu = \mathscr {O}_{E}^{\times }\cap U$ . By [Reference Yuan, Zhang and ZhangYZZ12, equation (1.6.1) and the following paragraphs], we have

(2.1.3)

Now, by a coset identity,

$$ \begin{align*}|E^{\times}\backslash \mathbf{A}_{E}^{\infty, \times}/U | = h_{E} { |\widehat{\mathscr{O}}^{\times}_{E}/ U| \over [\mathscr{O}_{E}^{\times}:\mu_{}]},\end{align*} $$

and by [Reference Yuan, Zhang and ZhangYZZ12, Section 1.6.3], $\mathrm {vol} ({E^{\times }\backslash {\mathbf {A}}_{E}^{\times } /\mathbf {A}^{\times }})=2L(1, \eta )$ . Hence, (2.1.3) equals

$$ \begin{align*}{2L(1, \eta)\over h_{E}} { [\mathscr{O}_{E}^{\times}:\mu_{}] \over |\widehat{\mathscr{O}}^{\times}_{E}/ U|} \sum_{t\in E^{\times}\backslash \mathbf{A}_{E}^{\infty, \times}/U} f(t).\end{align*} $$

If we compose with the operator $f(\cdot )\mapsto \sum _{x\in E^{\times }} f(x\cdot ) = \sum _{x\in \mu _{}\backslash E^{\times }} \sum _{\alpha \in \mu _{}} f(\alpha x\cdot )$ , we obtain

$$ \begin{align*}{ 2L(1,\eta)\over h_{E}^{}} { [\mathscr{O}_{E}^{\times}:\mu_{}] \over |\widehat{\mathscr{O}}^{\times}_{E}/ U|} \sum_{ t\in \mathbf{A}_{E}^{\infty, \times}/U} \sum_{\alpha\in \mu_{}}f(\alpha t) = { 2L(1,\eta)\over h_{E}^{}} { [\mathscr{O}_{E}^{\times}:\mu_{}] } \int_{ \mathbf{A}_{E}^{\infty, \times}} \sum_{\alpha\in \mu_{}} f(\alpha t)\, d^{\bullet} t .\end{align*} $$

2.2.1 Nearly holomorphic modular forms

Let $\underline {w}$ be an ( $\infty $ -adic Section 1.1.1) weight for $\mathrm {G}$ , let $U\subset \mathrm {G}(\mathbf {A}^{\infty })$ be a compact open subgroup, and let $m=(m_{\tau })\in \mathbf {Z}_{\geq 0}^{\Sigma _{\infty }}$ . A complex nearly holomorphic (Hilbert) modular form of weight $\underline {w}$ , level U, and degree $\leq m$ is a function

$$ \begin{align*}f \colon \mathrm{G}(\mathbf{A}) \to \mathbf{C}\end{align*} $$

satisfying the following two conditions:

1. (1) For all $g\in \mathrm {G}(\mathbf {A})$ , $\gamma \in \mathrm {G}(F)$ , and $k \in U C_{\infty }^{+}$ ,

   $$ \begin{align*}f(\gamma g k) = j_{\underline{w}}(k_{\infty}, \mathrm{i})^{-1}f(g),\end{align*} $$
   where for $z\in \frak h^{{\Sigma _{\infty }}}$ ,
   $$ \begin{align*}j_{\underline{w}}({\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)}, z_{}) := (ad-bc)^{(w_{0}-w)/2} (cz+d )^{w}.\end{align*} $$

2. (2) There is a Whittaker–Fourier expansion

   (2.2.1) $$ \begin{align}\begin{aligned} f\left(\left(\begin{array}{cc}y&x\\&1\end{array}\right)\right) &= |y| \sum_{a\in F} W^{\mathbf{C}}_{f,a}(y)(Y) \, {\mathbf{q}}^{a} \end{aligned}\end{align} $$
   for all $y\in {\mathbf {A}}^{+}$ , $x\in {\mathbf {A}}$ , where:

   • • we have

     $$ \begin{align*}W_{f,0}^{\mathbf{C}}(y)= y_{\infty}^{(w_{0}+w-2)/2} \mathrm{W}_{f, 0}(y) , \quad W_{f,a}^{\mathbf{C}}(y) = (ay_{\infty})^{(w_{0}+w-2)/2} \mathrm{W}_{f, a}(y) \quad (a\neq 0)\end{align*} $$
     for polynomials
     $$ \begin{align*}\mathrm{W}_{f, a}(y)\in \mathbf{C}[(T_{\tau})_{\tau\in \Sigma_{\infty}}]\end{align*} $$
     of degree $\leq m_{\tau }$ in the variables $T_{\tau }$ , evaluated at $Y:=(Y_{\tau })_{\tau \in \Sigma _{\infty }}$ with $Y_{\tau }= (4\pi y_{\tau })^{-1}$ ;

   • • we denote

     $$ \begin{align*}\mathbf{q}^{a} := \psi(ax) \psi_{\infty}(\mathrm{i} a y_{\infty}).\end{align*} $$

The polynomial $\mathrm {W}_{f,a}(y)$ only depends on the class of $ay$ modulo $\mathbf {a}^{-1}(U)$ , so that defining

(2.2.2) $$ \begin{align}\begin{aligned} \mathrm{W}_{f}(a):= W_{f, 1}(a) \end{aligned}\end{align} $$

for $a\in \mathbf {A}^{+}$ , for all $a\in F^{+}$ , and for $y\in \mathbf {A}^{+}$ , we have $W_{f,a}(y)=W_{f} (ay)$ . We say that f is cuspidal if $W_{0}(y)=0$ for all y.

If f is nearly holomorphic of degree $0$ (that is, $\leq (0, \ldots , 0)$ ), we simply say that f is a (holomorphic) modular form. If $R\subset \mathbf {C}$ is any subring, we denote by

$$ \begin{align*}S_{\underline{w}}(U, R)\subset M_{\underline{w}}(U, R)\subset N_{\underline{w}}^{ \leq m}(U, R),\end{align*} $$

respectively, the spaces of cuspidal forms and holomorphic forms of level U and weight $\underline {w}$ , and of nearly holomorphic forms of level U, weight $\underline {w}$ , and degree $\leq m=(m_{\tau })$ , such that for all $a\in \mathbf {A}^{+}$ , the polynomials $W_f({a})$ have coefficients in R. We write $N_{\underline {w}}^{}(U, R):= \varinjlim _{m} N_{\underline {w}}^{ \leq m}(U, R)$ , and $\Box (R):=\varinjlim _{U} \Box (U, R)$ if $\Box $ stands for the notation for any of the spaces of forms defined above (or below).

Finally, we define the space $S^{\mathrm {a}}_{\underline {w}}(U, \mathbf {C})$ of antiholomorphic cuspforms to be the $\mathbf {C}$ -vector space image of $S_{\underline {w}}(U, \mathbf {C})$ under complex conjugation. The formula

(2.2.3) $$ \begin{align}\begin{aligned} f\mapsto f^{\mathrm{a}}:=\left(\begin{array}{cc}-1&\\&1\end{array}\right) f \end{aligned}\end{align} $$

(where $\left (\begin {smallmatrix}-1&\\&1\end {smallmatrix}\right ) \in \mathrm {G}(F)\subset \mathrm {G}(\mathbf {A})$ acts, as usual, by right translation) defines $\mathbf {C}$ -linear bijections from $S_{\underline {w}}(U, \mathbf {C})$ to $S^{\mathrm {a}}_{\underline {w}}(U, \mathbf {C})$ and vice versa.

2.2.2 Twisted modular forms

A twisted nearly holomorphic (Hilbert) modular form of weight $\underline {w}$ , level U, and degree $\leq m=(m_{\tau })$ is a function

$$ \begin{align*}f \colon \mathrm{G}(\mathbf{A})\times \mathbf{A}^{\times} \to \mathbf{C}\end{align*} $$

satisfying the following two conditions:

1. (1) For all $g\in \mathrm {G}(\mathbf {A})$ , $\gamma \in \mathrm {G}(F)$ , and $k \in U C_{\infty }^{+}$ ,

   $$ \begin{align*}f(\gamma g k, \det(\gamma)^{-1} u) = j_{\underline{w}}(k_{\infty}, \mathrm{i})^{-1} f(g, u).\end{align*} $$

2. (2) There is a Whittaker–Fourier expansion

   (2.2.4) $$ \begin{align}\begin{aligned} f\left(\left(\begin{array}{cc}y&x\\&1\end{array}\right) , u\right) &= |y| \sum_{a\in F} W^{\mathbf{C}}_{f, a}(y, u)(Y) \, {\mathbf{q}}^{a} \end{aligned}\end{align} $$
   for all $x\in {\mathbf {A}}$ and $y, u\in {\mathbf {A}}^{\times }$ such that $(uy)_{\infty }>0$ , where:
   $$ \begin{align*} &W_{f, 0}^{\mathbf{C}}(y, u)= y_{\infty}^{(w_{0}+w-2)/2} \mathrm{W}_{f, 0}(y, u) ,\\ & W_{f, a}^{\mathbf{C}}(y, u) = (ay_{\infty})^{(w_{0}+w-2)/2} \mathrm{W}_{f, a}(y, u) \quad (a \neq 0)\end{align*} $$

   for polynomials

   $$ \begin{align*}\mathrm{W}_{f, a}(y, u)\in \mathbf{C}[(T_{\tau})_{\tau\colon F\hookrightarrow \mathbf{R}}]\end{align*} $$
   of degree $\leq m_{\tau }$ in the variables $T_{\tau }$ , evaluated at $Y:=(Y_{\tau })_{\tau \colon F\hookrightarrow \mathbf {R}}$ with $Y_{\tau }= (4\pi y_{\tau })^{-1}$ .

If $R\subset \mathbf {C}$ is any subring, we denote by $ M^{\mathrm {tw }}_{\underline {w}}(U, R)\subset N_{\underline {w}}^{\mathrm {tw}, \leq m}(U, R)$ the spaces of holomorphic and nearly holomorphic forms of level U, weight $\underline {w}$ , and degree $\leq m=(m_{\tau })$ , such that all the polynomials $\mathrm {W}_{f, a}(y, u)$ have coefficients in R.

2.2.3 Contracted product

For any open compact subgroup $U_{F}\subset \widehat {\mathscr {O}}_{F}^{\times }$ , let

(2.2.5) $$ \begin{align}\begin{aligned} \qquad \mu_{U_{F}}:= F^{\times}\cap U_{F}, \qquad \nu_{U_{F}}:=|\{\pm 1\} \cap \mu_{U_{F}}|, \qquad \qquad c_{U_{F}}= {\nu_{U_{F}} \cdot 2^{[F:\mathbf{Q}]}h_{F}\over [\mathscr{O}_{F}^{\times}:\mu_{U_{F}}^{2}]}. \end{aligned}\end{align} $$

Let $\varphi \colon \mathbf {A}^{\times }\to \mathbf {C}$ be a Schwartz function, invariant under a subgroup of the form $\mu _{U_{F}'}^{2}\subset F^{\times }$ as above. Then the sum

(2.2.6) $$ \begin{align}\begin{aligned} \sum^{\star}_{u\in F^{\times}} \varphi(u):= c_{U_{F}}\sum_{u\in \mu_{U_{F}}^{2}\backslash F^{\times}} \varphi(u) \end{aligned}\end{align} $$

is well defined independently of $U_{F}\subset U_{F}'$ , and for any such choice, the support of the sum is finite.

If $f_{1}$ , $f_{2}$ are twisted nearly holomorphic forms, we may thus define a (plain) nearly holomorphic form $f_{1} \star f_{2}$ by

(2.2.7) $$ \begin{align}\begin{aligned} f_{1} \star f_{2}(g):= \sum_{u\in F^{\times}}^{\star} f_{1}(g, u)f_{2}(g, u). \end{aligned}\end{align} $$

2.2.4 Differential operators

We attach to a nearly holomorphic (genuine or twisted) form f the function

$$ \begin{align*}\begin{aligned} f^{\frak h}\colon \mathrm{G}(\mathbf{A}^{\infty})\times \frak h^{{\Sigma_{\infty}}} &\to \mathbf{C} \\ (g^{\infty}, z=g_{\infty} \mathrm{i})& \mapsto j_{\underline{w}}(g_{\infty}, \mathrm{i}) f(g_{\infty}); \end{aligned}\end{align*} $$

the map $f\mapsto f^{\frak h}$ is injective.

The Maass–Shimura differential operators on functions on $\frak h^{{\Sigma _{\infty }}}$ are defined as follows. For $\tau \colon F\hookrightarrow \mathbf {R}$ and $w \in \mathbf {Z}$ , let

$$ \begin{align*}\delta_{w}^{\tau, \frak h}:= {1\over 2\pi i} \left({w\over 2i y_{\tau}}+ {\partial \over \partial z_{\tau}}\right), \qquad d^{\tau}:= {1\over 2\pi i} {\partial\over \partial z_{\tau}},\end{align*} $$

a differential operator on the upper half-plane $\frak h$ . For $w,k \in \mathbf {Z}_{\geq 0}^{{\Sigma _{\infty }}}$ , let

$$ \begin{align*}\delta_{w}^{k, \frak h}:=\prod_{\tau} \delta^{\tau, \frak h}_{w_{\tau}+2k_{\tau}}\circ \cdots \circ \delta_{w_{\tau}+2}^{\tau, \frak h}\circ \delta_{w_{\tau}}^{\tau, \frak h}, \qquad d^{k}:=\prod_{\tau}(d^{\tau})^{k_{\tau}}.\end{align*} $$

Then, for any ring $\mathbf {Q}\subset R\subset \mathbf {C}$ , this operator defines a map

$$ \begin{align*}\delta_{w}^{k}\colon N_{\underline{w}}^{\mathrm{(tw)} ,\leq m}(U,R)\to N_{\underline{w}+(0;2k)}^{\mathrm{(tw)}, \leq m+k }(U, R)\end{align*} $$

such that $\delta _{w}^{k}(f)^{\frak h}=\delta _{w}^{k, \frak h}(f^{\frak h})$ . (For a proof of the intuitive fact that the archimedean operator $\delta _{\underline {w}}^{k}$ indeed preserves the rationality properties of finite Whittaker–Fourier coefficients, see [Reference HidaHid91, Proposition 1.2], whose calculations also apply to the twisted case.) The subscript w will be omitted if it is clear from the context.

By [Reference ShimuraShi81, equation (1.16)], for all $k\in \mathbf {Z}_{\geq 0}^{\Sigma _{\infty }}$ , we have

(2.2.8) $$ \begin{align}\begin{aligned} \delta_{w}^{k}=\sum_{0 \leq j\leq k} \prod_{\tau\in \Sigma_{\infty}} {k_{\tau}\choose j_{\tau}} {\Gamma(w_{\tau}+k_{\tau})\over \Gamma(w_{\tau}+j_{\tau})} (-4\pi y_{\tau})^{j_{\tau}-k_{\tau}} \, d^{j_{}}. \end{aligned}\end{align} $$

If $w\geq 2m+1$ , any $f\in N_{\underline {w}}^{\mathrm {(tw), \leq m}}(U, R)$ can be written uniquely as

$$ \begin{align*}f=\sum_{0\leq r\leq m} \delta_{w-2r}^{r}f_{r}\end{align*} $$

with $f_{r}\in M_{\underline {w}+(0; -2r)}^{\mathrm {(tw) }}(U, R)$ . (The proof in [Reference ShimuraShi76, Lemma 7] carries over to our context.) Thus, the linear map

(2.2.9) $$ \begin{align}\begin{aligned} e^{\mathrm{hol}}\colon N_{\underline{w}}^{\mathrm{(tw), \leq m}}(U, R) &\to M_{\underline{w}}^{\mathrm{(tw)}}(U, R)\\ f&\mapsto f_{0}\end{aligned}\end{align} $$

is well defined.

2.3.1 Arithmetic q-expansion

Let $\underline {w}$ be a weight for $\mathrm {G}$ , and let $U\subset \mathrm {G}(\mathbf {A}^{\infty })$ be a compact open subgroup. The q-expansion map

$$ \begin{align*}f\mapsto (a\mapsto \mathrm{W}_{f}(a)=(2.2.2))\end{align*} $$

sends $S_{\underline {w}}(U,\mathbf {C})$ to $ \mathbf {C}^{\mathbf {A}^{+}/U_{F}F_{\infty }^{+}}$ , where $U_{F}=\mathbf {a}^{-1}(U)$ . By the q-expansion principle (see [Reference DisegniDis17, Proposition 2.1.1] for a version in our setting), the map is injective. We denote its image by $\mathrm {S}_{\underline {w}}(U,\mathbf {C})$ and view the map $S_{\underline {w}}(U,\mathbf {C})\to \mathrm { S}_{\underline {w}}(U,\mathbf {C})$ as an identification.

If R is any ring admitting embeddings into $\mathbf {C}$ , we denote by

$$ \begin{align*}\mathrm{S}_{\bullet}(U, R) \subset R^{\mathbf{A}^{+}/U_{F}F_{\infty}^{+}}\end{align*} $$

the set of those sequences

(2.3.1) $$ \begin{align}\begin{aligned} \mathrm{f}=(\mathrm{W}_{\mathrm{f}}(a))_{a}\end{aligned}\end{align} $$

such that for any $\iota \colon R\hookrightarrow \mathbf {C}$ , the sequence $\mathrm {f}^{\iota }:=(\iota \mathrm {W}_{\mathrm {f}}(a))_{a}$ is the q-expansion of a cuspform

$$ \begin{align*}f^{\iota}\in {S}_{\bullet}(U, \mathbf{C})=\bigoplus_{\underline{w}} S_{\underline{w}}(U, \mathbf{C}).\end{align*} $$

(In (2.3.1), the notation $\mathrm {W}_{\mathrm {f}}$ can be thought of as simply synonymous to $\mathrm {f}$ ; it is introduced in order to match the identification of the previous paragraph.) By [Reference HidaHid91, Theorem 2.2(i)] (together with a consideration of Galois actions mixing the weights), for any such ring R, we have $\mathrm {S}_{\bullet }(U, R)= \mathrm {S}_{\bullet }(U, \mathbf {Z})\otimes R$ . For more general rings, the previous equality is taken to be the definition of $\mathrm {S}_{\bullet }(U, R)$ .

2.3.2 p-adic modular forms

Let L be a finite extension of $\mathbf {Q}_{p}$ splitting F. A p-adic L-valued (cohomological) weight $\underline {w}=(w_{0}, (w_{\tau })_{\tau \colon F\hookrightarrow L})$ is a tuple of integers, all having the same parity, such that $w_{\tau }\geq 1$ for all $\tau \colon F\hookrightarrow L$ . As in Section 1.1.1, if $\underline {w}$ is an L-valued weight and $\iota \colon L\hookrightarrow \mathbf {C}$ is an embedding, we define the complex weight $\underline {w}^{\iota }=(w_{0}, (w_{\tau })_{\iota \circ \tau })$ .

Let $U\subset \mathrm {G}(\mathbf {A}^{\infty })$ be a compact open subgroup, and let $\underline {w} $ be an L-valued weight. We define $\mathrm { S}_{\underline {w}}(U, L)$ to be the set of q-expansions $\mathrm {f}$ such that for every $\iota \colon L\hookrightarrow \mathbf {C}$ , the expansion $\mathrm {f}^{\iota }$ belongs to $\mathrm {S}_{\underline {w}^{\iota }}(U,\mathbf {C})$ . The p-adic q-expansion of $\mathrm {f}=(\mathrm {W}_{\mathrm {f}}({a}))_{a}\in \mathrm {S}_{\underline {w}}(U, L)$ is the sequence

$$ \begin{align*}f=(W_{f}(a)):=(W_{\mathrm{f}}(a)),\qquad W_{\mathrm{f}}(a) := a_{p}^{(w_{0}+w-2)/2} \mathrm{W}_{\mathrm{f}}({a}),\end{align*} $$

so that

(2.3.2) $$ \begin{align}\begin{aligned} W_{f^{\iota},a}^{\mathbf{C}}(y):=(ay)_{\infty}^{(w_{0}+w^{\iota}-2)/2} \, \iota\left( (ay_{p})^{(-w_{0}-w+2)/2} W_{f}(a)\right) \end{aligned}\end{align} $$

is the Whittaker–Fourier coefficient of $f^{\iota }$ as in (2.2.1). (In other words, we have two embeddings $\mathrm {S}_{\underline {w}}(U, L)\hookrightarrow L^{\mathbf {A}^{+}/U_{F}F_{\infty }^{+}}$ : the q-expansion $\mathrm {f} \mapsto (\mathrm {W}_{\mathrm {f}}(a))_{a}$ , and the p-adic q-expansion $\mathrm {f}\mapsto (W_{\mathrm {f}}(a))_{a}$ .)

Let $U^{p}\subset \mathrm {G}(\mathbf {A}^{p\infty })$ be a compact open subgroups, let $U_{F}^{p}:=\mathbf {a}^{-1}(U^{p})$ , and for any L-valued weight $\underline {w}$ , we denote

$$ \begin{align*}\begin{aligned} \mathrm{S}_{\underline{w}}(U^{p}, L) &:=\varinjlim_{n} \mathrm{S}_{\underline{w}}(U^{p}U_{p, n}, L). \end{aligned}\end{align*} $$

The space of cuspidal p-adic modular forms

$$ \begin{align*}\mathbf{S}(U^{p}, L) \subset L^{\mathbf{A}^{\infty,\times}/U_{F}^{p}}\subset L^{\mathbf{A}^{\infty,\times}}=L^{\mathbf{A}^{+}/F^{+}_{\infty}}\end{align*} $$

is the completion of $\mathrm {S}_{\underline {w}}(U^{p}, L)$ for the norm $||\mathrm {f}||:= \sup _{a} |W_{\mathrm {f}}(a)|$ , for any $\underline {w}$ . By a fundamental result of Hida (see [Reference HidaHid91, paragraph after Theorem 3.1]), the space $\mathbf {S}(U^{p}, L)$ is independent of the choice of $\underline {w}$ . In particular, if L is Galois over $\mathbf {Q}_{p}$ , this space is stable by the action of $\mathrm {Gal}(L/\mathbf {Q}_{p})$ and so it is of the form $\mathbf {S}(U^{p}, \mathbf {Q}_{p})\otimes _{\mathbf {Q}_{p}}L$ for a space $\mathbf { S}(U^{p}, \mathbf {Q}_{p})$ .

2.3.3 Nearly holomorphic forms as p-adic modular forms

We may attach a p-adic q-expansion to a nearly holomorphic form with coefficients in a p-adic subfield of $\mathbf {C}$ .

Let L be a finite extension of $\mathbf {Q}_{p}$ , and let $\underline {w}$ be a p-adic L-valued weight. We say that

$$ \begin{align*}f=(W_{f}(a))\in L^{\mathbf{A}^{+}/F_{\infty}^{+}}\end{align*} $$

is a p-adic nearly holomorphic cuspform of weight $\underline {w}$ and level $U^{p}\subset \mathrm {G}(\mathbf {A}^{p\infty })$ if the following condition holds. For each $\iota \colon L\hookrightarrow \mathbf {C}$ , there exists a cuspidal nearly holomorphic form

$$ \begin{align*}f^{\iota}\in N_{\underline{w}^{\iota}}^{\leq \lfloor{(w+1)/2}\rfloor}(U^{p}U_{p, n},\mathbf{C})\end{align*} $$

for some $n\in \mathbf {Z}_{\geq 1}^{S_{p}}$ , whose Whittaker–Fourier polynomials have constant terms satisfying

(2.3.3) $$ \begin{align}\begin{aligned} \mathrm{W}_{f^{\iota}, 1}(a)(0) = \iota\left(a_{p}^{(-w_{0}-w+2)/2}W_{f}(a)\right). \end{aligned}\end{align} $$

The notion of a p-adic twisted nearly holomorphic cuspform is defined similarly by the identity $\mathrm {W}_{f^{\iota }, a}(y, u)(0) = \iota \left ((ay)_{p}^{(-w_{0}-w+2)/2}W_{f}(a)(y, u)\right )$ .

2.3.4 Hecke operators and ordinary projection

The space $N_{\underline {w}}(U, \mathbf {C})$ is endowed with the usual action of $\mathscr {H}_{U}$ . By writing down the effect of this action on Whittaker–Fourier coefficients of cuspforms, we may descend it to a bounded action of $\mathscr {H}_{U^{p}, L}$ on $\mathrm { S}_{\underline {w}}(U^{p},L)$ , and hence on $\mathbf {S}(U^{p}, L)$ , for any p-adic field L.

For $t\in A_{p}^{+}$ or $y\in \prod _{v\vert {p}}\mathscr {O}_{F, v}-\{0\}$ , and any $n\in \mathbf {Z}_{\geq 1} ^{S_{p}}$ , define the double coset operators

(2.3.4) $$ \begin{align}\begin{aligned} \mathrm{U}_{t}&:=[U_{p, n} \, t \, U_{p, n}], \qquad &\mathrm{U}_{t}^{\circ, \underline{w}} := t_{1}^{2-w}\det(t)^{(-w_{0}+w-2)/2} \mathrm{U}_{t}, \\ \mathrm{U}_{y}&:= \mathrm{U}_{\left(\begin{smallmatrix}y&\\&1\end{smallmatrix}\right)},\qquad & \mathrm{U}_{y}^{\circ, \underline{w}}:= y^{(-w_{0}-w+2)/2 } \mathrm{U}_{y}. \end{aligned}\end{align} $$

If L is a finite extension of $\mathbf {Q}_{p}$ , then for all $y\in \prod _{v\vert p}\mathscr {O}_{F, v}-\{0\}$ , we also define the operator

$$ \begin{align*}\begin{aligned} \mathrm{U}_{y}^{\circ}\colon \mathbf{S}(U^{p}, L) &\to \mathbf{S}(U^{p}, L)\\ W_{\mathrm{U}_{y}^{\circ}f}(c)&:= W_{f}(cy).\end{aligned}\end{align*} $$

This is compatible with the previous definition in the following sense (see [Reference HidaHid91, equation (2.2b)], where $\mathrm {U}_{y}$ is denoted by $T(y)$ ): if f is a p-adic nearly holomorphic form of weight $\underline {w}$ over L, then for all $\iota \colon L\hookrightarrow \mathbf {C}$ , we have

$$ \begin{align*}(\mathrm{U}_{y}^{\circ}f)^{\iota}= \mathrm{U}_{y}^{\circ, \underline{w}^{\iota}} f^{\iota}.\end{align*} $$

The superscript $\underline {w}$ will be omitted when understood from the context. The ordinary projector is

(2.3.5) $$ \begin{align}\begin{aligned} e^{\mathrm{ord}}:=\lim_{n\to \infty} (\mathrm{U}_{p}^{\circ})^{n!} \quad \in \quad \mathrm{End}\,_{L}(\mathbf{S}(U^{p}, L))\end{aligned}\end{align} $$

for any tame level $U^{p}$ and p-adic field L. Its image is denoted by

$$ \begin{align*}\mathbf{S}^{\mathrm{ord}}(U^{p}, L):= e^{\mathrm{ord}} \mathbf{S}(U^{p},L).\end{align*} $$

The operator $e^{\mathrm {ord}}$ preserves $\mathrm {S}_{\underline {w}}(U^{p},L)$ , and we denote $\mathrm {S}^{\mathrm {ord}}_{\underline {w}}(U^{p}, L):= e^{\mathrm {ord}} \mathbf {S}(U^{p},L)$ and $\mathrm {S}^{\mathrm {ord}}(U^{p}, L):= \bigoplus _{\underline {w}} \mathrm { S}^{\mathrm {ord}}_{\underline {w}}(U^{p}, L)$ .

If $f^{\mathbf {C}}$ is a complex modular form arising as $f^{\mathbf {C}}=f^{\iota }$ for a form $f\in \mathrm {S}(L)$ for some finite extension L of $\mathbf {Q}_{p}$ and some $\iota \colon L\hookrightarrow \mathbf {C}$ , we define

$$ \begin{align*}e^{\mathrm{ord, \iota}} (f^{\mathbf{C}}) := (e^{\mathrm{ord}} f)^{\iota}.\end{align*} $$

2.3.5 Differential operators after ordinary and holomorphic projections

Let L be a finite extension of $\mathbf {Q}_{p}$ , and let $f_{1}$ , $f_{2}$ be p-adic twisted nearly holomorphic forms over L. For any $\iota \colon L\hookrightarrow \mathbf {C}$ and $k \in \mathbf {Z}_{\geq 0}^{{\Sigma _{\infty }}}$ , we have

(2.3.6) $$ \begin{align}\begin{aligned} [e^{\mathrm{ord}} (f_{1}\star d^{k}f_{2})]^{\iota}= e^{\mathrm{ord}, \iota} [e^{\mathrm{hol}} (f_{1}^{\iota} \star \delta^{k}f_{2}^{\iota})] ;\end{aligned}\end{align} $$

the proof of [Reference HidaHid91, Proposition 7.3] carries over to the twisted case.

2.4.1 Weight space

Let $U_{F, p}^{\circ }=\prod _{v\vert p} U_{F, v}^{\circ }\subset \mathscr {O}_{F, p}^{\times }$ be a compact open subgroup (which will be fixed once and for all in Section 2.4.5). Let $U_{F}^{p}\subset \mathbf {A}^{p\infty , \times }$ be a compact open subgroup, and consider the topological groups (with the profinite topology)

$$ \begin{align*}[Z]_{U_{F}^{p}}:= Z(F)U_{F}^{p}\backslash Z(\mathbf{A}^{\infty}), \qquad [Z]_{U_{F}^{p}} \times U_{F, p}^{\circ};\end{align*} $$

the latter is isomorphic to $\Delta \times \mathbf {Z}_{p}^{1+[F:\mathbf {Q}]+\delta _{F,p}}$ , where $\Delta $ is a finite group and $\delta _{F, p}$ is the p-Leopoldt defect of F. It is embedded into $A(\mathbf {A}^{\infty })$ by

$$ \begin{align*}(z, y_{p})\mapsto \left(\begin{array}{cc}zy_{p}&\\&z\end{array}\right).\end{align*} $$

The weight space (of tame level $U_{F}^{p}$ ) is

(2.4.1)

A point $\underline {\kappa }\in \frak W$ is identified with the pair of characters

(2.4.2) $$ \begin{align}\begin{aligned} \left(\kappa_{0}:=\underline{\kappa}_{|[Z]_{U_{F}^{p}}} , \qquad \kappa= \underline{\kappa}_{|U_{F, p}^{\circ}} \right). \end{aligned}\end{align} $$

We have an involution defined as

$$ \begin{align*}\underline{\kappa}^{\curlyvee}(t):=\kappa_{0}(\det t)^{-1} \underline{\kappa}(t).\end{align*} $$

If $\underline {k}$ is a p-adic weight for $\mathrm {G}$ , we say that $\underline {\kappa } $ is classical of weight $\underline {k}$ if for all $v\vert p$ ,

$$ \begin{align*}\kappa_{0}^{\mathrm{sm}}(z_{p}):=\kappa_{0}(z_{p})z_{p}^{-k_{0}}, \qquad \kappa_{v}^{\mathrm{sm}}(y):=\kappa_{v}(y)\prod_{\tau|v} \tau(y)^{(-k_{0}-{k}_{\tau}+2)/2}\end{align*} $$

are smooth characters of $F_{p}^{\times }$ (respectively, $U_{F, v}^{\circ }$ ); in the second equation, $\kappa _{v}:=\kappa _{|U^{\circ }_{F,v}}$ , and the product runs over the $\tau \in \Sigma _{p}$ inducing the place $v\in S_{p}$ . For a classical weight $\underline {\kappa }$ , we define $\kappa ^{\mathrm {sm}}:=\otimes _{v\vert p} \kappa _{v}^{\mathrm {sm}}$ and

(2.4.3) $$ \begin{align}\begin{aligned} \kappa':= \kappa^{\mathrm{sm}}\kappa_{0}^{\mathrm{sm},-1} = \kappa^{\curlyvee, \mathrm{sm}},\end{aligned}\end{align} $$

a smooth character of $U_{F, p}^{\circ }$ .

We denote by

$$ \begin{align*}\frak W^{\mathrm{cl}}\subset \frak W\end{align*} $$

the set of points of classical weight, which has the structure of an ind-étale ind-finite scheme over $\mathbf {Q}_{p}$ . If $\underline {\kappa }$ is classical of weight $\underline {k}=(k_{0}, k)$ , then $\underline {\kappa }^{\curlyvee }$ is classical of weight $\underline {k}^{\vee }=(-k_{0}, k)$ . We let $\frak W^{\mathrm {cl}, \geq 2}$ be the set of classical points satisfying $k \geq 2$ .

2.4.2 Hida schemes

In light of the examples of the previous and following paragraphs, it will be convenient to introduce a suitable category of spaces.Footnote ⁸ Define the category of Hida rings to consist of finite flat -algebras $A^{\circ }$ (for some n) and $\mathbf {Z}_{p}$ -algebra morphisms, and the category of Hida algebras to be the image of Hida rings under the functor $\otimes _{\mathbf {Z}_{p}}\mathbf {Q}_{p}$ . Define the category of affine Hida schemes to be dual to the category of Hida algebras. A Hida scheme is an open subset of an affine Hida scheme. If $A_{i}^{\circ }$ are Hida rings (for $i=1, 2$ ) and $\mathscr {X}_{i}=\mathrm {Spec}\, (A_{i}^{\circ }\otimes _{\mathbf {Z}_{p}} \mathbf {Q}_{p})$ (for $i=1,2$ ), we define

$$ \begin{align*}\mathscr{X}_{1}\hat{\times} \mathscr{X}_{2} := \mathrm{Spec}\, (A^{\circ}_{1}\hat{\otimes}A^{\circ}_{2})_{\mathbf{Q}_{p}},\end{align*} $$

where $\hat {\otimes }$ is the completed tensor product.

2.4.3 Hida families

Let

$$ \begin{align*}\mathbf{T}_{U^{p},\mathbf{Q}_{p}}^{\mathrm{sph,\ \mathrm{ord}}}\subset \mathbf{T}_{U^{p}, \mathbf{Q}_{p}}^{\mathrm{ord}}\subset \mathrm{End}\,(\mathbf{S}^{\mathrm{ord}}(U_{p}, \mathbf{Q}_{p}))\end{align*} $$

be the images of the Hecke algebras $\mathscr {H}_{U^{p}, \mathbf {Q}_{p}}^{\mathrm {sph,\ \mathrm {ord}}}$ , $\mathscr {H}_{U^{p}, \mathbf {Q}_{p}}^{\mathrm {ord}}$ from Section 2.1.3. We let

$$ \begin{align*}\mathscr{Y}_{\mathrm{G}}=\mathscr{Y}_{\mathrm{G}, U^{p}}:= \mathrm{Spec}\, \mathbf{T}_{U^{p}, \mathbf{Q}_{p}}^{\mathrm{sph,\ \mathrm{ord}}}\end{align*} $$

be the ordinary eigenvariety for $\mathrm {G}$ of tame level $U^{p}$ . (The subscript $U^{p}$ will be omitted when unimportant or understood from the context.) The space $\mathscr {Y}_{\mathrm {G}, U^{p}}$ is a union of finitely many irreducible components, called Hida families of tame level (dividing) $U^{p}$ . It carries an involution $\curlyvee $ deduced from the one on $\mathscr {H}_{U^{p}, \mathbf {Q}_{p}}^{\mathrm {sph,\ \mathrm {ord}}}$ .

Letting $U_{F} ^{p}:= U^{p}\cap Z(\mathbf {A}^{p\infty })$ , we have a weight-character map

$$ \begin{align*}\underline{\kappa}_{\mathrm{G}}\colon \mathscr{Y}_{\mathrm{G}, U^{p}} \to \frak W_{U_{F}^{p}}\end{align*} $$

that, when identified with a pair $(\kappa _{\mathrm {G}, 0}, \kappa _{\mathrm {G}}) $ of $\mathscr {O}(\mathscr {Y}_{\mathrm {G}})^{\times }$ -valued characters as in (2.4.2), is $\kappa _{\mathrm {G}, 0}(z)=$ the Hecke operator acting by right translation by z on modular forms, $\kappa _{G}(y_{p})= \mathrm {U}_{y_{p}}^{\circ }$ . The weight map is finite and flat, and it intertwines the involutions $\curlyvee $ .

The set of classical points of $\mathscr {Y}_{\mathrm {G}} $ is

$$ \begin{align*}\mathscr{Y}_{\mathrm{G}}^{\mathrm{cl}}:=\mathscr{Y}_{\mathrm{G}} \times_{\frak W}\frak W^{\mathrm{cl}, \geq 2}\subset \mathscr{Y}_{\mathrm{G}}.\end{align*} $$

If $x_{0}\in \mathscr {Y}_{\mathrm {G}}^{\mathrm {cl}}$ , we denote by $\pi _{x_{0}}$ the automorphic representation of $\mathrm {G}(\mathbf {A})$ over $\mathbf {Q}_{p}(x_{0})$ on which $\mathscr {H}^{\mathrm {sph}}_{U^{p}}$ acts by the $\mathbf {Q}_{p}(x_{0})$ -character corresponding to $x_{0}$ . If $x\in \mathscr {Y}_{\mathrm {G}}^{\mathrm {cl}}(\mathbf {C})$ corresponds to $(x_{0}\in \mathscr {Y}_{\mathrm {G}}^{\mathrm {cl}}, \iota \colon \mathbf {Q}_{p}(x_{0})\hookrightarrow \mathbf {C})$ , we denote $\pi _{x}:= \pi _{x_{0}}^{\iota }$ .

2.4.4 Families of ordinary forms

By construction, for each $U^{p}{}'\subset U^{p}$ , the ordinary eigenvariety $\mathscr {Y}_{\mathrm {G}, U^{p}}$ (respectively, the weight space $\frak W_{U_{F}^{p}}$ ) carries a (coherent) sheaf

$$ \begin{align*}\mathscr{S}^{U^{p}{}'}\end{align*} $$

(respectively, $\mathscr {S}^{U^{p}{}'}_{\frak W}:=\kappa _{\mathrm {G}, *} \mathscr {S}^{U^{p}{}'}$ ), whose modules of global sections are $\mathbf {S}^{\mathrm {ord}}(U^{p}{}', \mathbf {Q}_{p})$ . We set $\mathscr {S}_{(\frak W)}:=\varinjlim _{U^{p}{}'}\mathscr {S}_{(\frak W)}^{U^{p}}.$ By Hida’s Control Theorem (see [Reference HidaHid91, Corollary 3.3]), the restriction of $\mathscr {S}_{\frak W}^{U^{p}}$ to $\frak W^{\mathrm {cl}}$ is the sheaf attached to $\mathrm {S}^{\mathrm {ord}}(U^{p}, \mathbf {Q}_{p})$ .

For each $x\in \mathscr {Y}_{\mathrm {G}, U^{p}}^{\mathrm {cl}}$ of weight $\underline {w}$ , there exists a unique (up to isomorphism) ordinary automorphic representation $\pi _{x}$ of $\mathrm {G}(A)$ over $L:=\mathbf {Q}_{p}(x)$ of weight $\underline {w}$ such that there is an $\mathscr {H}_{U^{p}, L}^{\mathrm {ord}}$ -isomorphism

$$ \begin{align*}\mathscr{S}_{|x}\cong \pi_{x}^{\mathrm{ord}}:= e^{\mathrm{ord}}\pi_{x}:= [\lim_{n} (\mathrm{U}_{p}^{\circ, \underline{w}})^{n!}]\, \pi_{x};\end{align*} $$

the isomorphism is unique up to scalars. This defines a bijection between $\mathscr {Y}_{\mathrm {G}, U^{p}}^{\mathrm {cl}}(\overline {\mathbf {Q}}_{p})$ and the set of isomorphism classes of ordinary automorphic representation $\pi $ of $\mathrm {G}(\mathbf {A})$ over $\overline {\mathbf {Q}}_{p}$ with $\pi ^{U^{p}}\neq 0$ .

Lemma 2.3 Let $U^{p}\subset \mathrm {G}(\mathbf {A}^{p\infty }) $ be a compact open subgroup, and let $U_{F}^{p}:= U^{p}\cap Z(\mathbf {A}^{p\infty })$ . Let $\mathscr {Z}$ be a Hida scheme endowed with a map $\varphi \colon \mathscr {Z}\to \frak {W=\frak W}_{U^{p}_{F}}$ . Then we have an $\mathscr {O}_{\mathscr {Z}}$ -linear injective q-expansion map

(2.4.4) $$ \begin{align}\begin{aligned} \mathscr{S}_{\mathscr{Z}}^{U^{p}} := \mathscr{S}^{U^{p}}_{\frak W} \otimes_{\mathscr{O}_{\frak W}} \mathscr{O}_{\mathscr{Z}}\longrightarrow \mathscr{O}_{\mathscr{Z}}^{\mathbf{A}^{\infty, \times}/U_{F}^{p}} \subset \mathscr{O}_{\mathscr{Z}}^{\mathbf{A}^{\infty, \times}} \end{aligned}\end{align} $$

characterized by the property that for every $\kappa \in \frak W^{\mathrm {cl}}$ , every closed point $z\in \varphi ^{-1}(\kappa )$ , and every $a\in \mathbf {A}^{\infty , \times }$ , we have

$$ \begin{align*}W_{\mathbf{f}}({a})(z)=W_{\mathbf{f}(z)}(a),\end{align*} $$

where the right-hand side is the p-adic q-expansion coefficient of the classical modular form ${\mathbf {f}}(z)\in (\varphi ^{*}\mathscr {S}_{\frak W})_{|z} \subset \mathrm {S}^{\mathrm {ord}}(U^{p}, \mathbf {Q}_{p}(z))$ .

Moreover, the image of (2.4.4) equals the space of those sequences $(W(a))_{a}$ for which there exists a set of closed points $\Sigma \subset \varphi ^{-1}(\frak W^{\mathrm {cl}})$ that is dense in $\mathscr {Z}$ such that for all $z\in \Sigma $ , the sequence $(W(a)(z))_{a}$ is the p-adic q-expansion of a modular form $\mathbf {f}_{z}\in (\varphi ^{*}\mathscr {S}_{\frak W})_{|z}.$

Note that the sheaf $\mathscr {S}^{U^{p}}$ on $\mathscr {Y}_{\mathrm {G}}$ is identified with $(\mathscr {S}_{\frak W}^{U^{p}}\otimes _{\mathscr {O}_{\frak W}}\mathscr {Y}_{\mathrm {G}})^{\mathscr {O}_{\mathscr {Y}_{\mathrm {G}}}}$ , the subsheaf of invariants for the diagonal $\mathscr {O}_{\frak W}$ -linear action of $\mathscr {O}_{\mathscr {Y}_{\mathrm {G}}}$ . In particular, we deduce from (2.4.4) a q-expansion map

(2.4.5) $$ \begin{align}\begin{aligned} \mathscr{S}^{U^{p}}\to \mathscr{O}_{\mathscr{Y}_{\mathrm{G}}}^{\mathbf{A}^{\infty, \times}/U_{F}^{p}}. \end{aligned}\end{align} $$

Proof It suffices to construct (2.4.4) for $\mathscr {Z}=\frak W$ as the general case follows by base change. Let , and let $\mathbf {S}^{\mathrm {ord}}(U^{p}, \mathbf {Z}_{p})$ be the space of ordinary forms with $\mathbf {Z}_{p}$ -coefficients; this is an $A^{\circ }$ -module and a $\mathbf {Z}_{p}$ -lattice in $\mathscr {S}_{\frak W}^{U^{p}}(\frak W)= \mathbf {S}^{\mathrm {ord}}(U^{p}, \mathbf {Q}_{p})$ . For $\kappa \in \frak W^{\mathrm {cl}}$ , let $\mathfrak {p}_{\kappa }\subset A^{\circ }$ be the corresponding prime ideal. Let $n\in \mathbf {N}$ , and let M range among finite subsets of $\frak W$ ; the filtered system of ideals

$$ \begin{align*}I_{n,M}:= (p^{n})+\bigcap_{\kappa \in M} \mathfrak{p}_{\kappa}\end{align*} $$

forms a fundamental system of neighborhoods of $0\in A^{\circ }$ , i.e., $A^{\circ }=\varprojlim _{n, M} A^{\circ }/I_{n, M}$ . The p-adic q-expansion maps $\mathbf {S}^{\mathrm {ord}}(U^{p}, \mathbf {Z}_{p})\otimes _{A^{\circ }}A^{\circ }/\mathfrak {p}_{\kappa }\to \mathbf {Z}_{p}(\kappa )^{\mathbf {A}^{\infty , \times }}$ yield a compatible family of maps

$$ \begin{align*}\mathbf{S}^{\mathrm{ord}}(U^{p}, \mathbf{Z}_{p})\otimes_{A^{\circ}}A^{\circ}/I_{n, M} \to (A^{\circ}/I_{n, M})^{\mathbf{A}^{\infty, \times}}\end{align*} $$

and after taking projective limits, the desired map $\mathbf {S}^{\mathrm {ord}}(U^{p}, \mathbf {Z}_{p}) \to (A^{\circ })^{\mathbf {A}^{\infty , \times }}$ . It is injective by the q-expansion principle and the preservation of injectivity under inverse limits.

We now consider the second statement. It is clear that, for any fixed $\Sigma $ as in the lemma, the space $\tilde {\mathscr {S}}_{\mathscr {Z}}^{U^{p}, \Sigma } \subset \mathscr {O}_{\mathscr {Z}}^{\mathbf {A}^{\infty , \times }}$ described contains the image of (2.4.4); we show the opposite containment. We may assume that $\mathscr {Z}=\mathrm {Spec}\, B^{\circ }_{\mathbf {Q}_{p}}$ for a Hida ring $B^{\circ }$ , and consider (2.4.4) as a map

(2.4.6) $$ \begin{align}\begin{aligned} \mathbf{S}^{\mathrm{ord}}(U^{p}, \mathbf{Z}_{p})\otimes_{A^{\circ}}B^{\circ} \to \tilde{\mathbf{S}}_{B^{\circ}}^{\mathrm{ord}}(U^{p})^{\Sigma} :=(B^{\circ})^{\mathbf{A}^{\infty, \times}}\cap \tilde{\mathscr{S}}_{\mathscr{Z}}^{U^{p}} (\mathscr{Z}). \end{aligned}\end{align} $$

For $z\in \Sigma $ , let $\mathfrak {p}_{z}\subset B^{\circ }$ be the corresponding prime ideal. Let $n\in \mathbf {N}$ , and let N range among finite subsets of $\Sigma $ ; then the filtered system of ideals $J_{n,N}:= (p^{n})+\bigcap _{z\in N} \mathfrak {p}_{z}$ forms a fundamental system of neighborhoods of $0\in B^{\circ }$ . By assumption, for each $z\in \Sigma $ , the map (2.4.6) is an isomorphism modulo $\mathfrak {p}_{z}$ ; hence, it is an isomorphism modulo $J_{n, N}$ for all $(n, N)$ , hence an isomorphism.

We call elements of $\mathscr {S}_{\mathscr {Z}}^{U^{p}}$ (respectively, $\mathscr {S}_{\mathscr {Z}}^{U^{p}}\otimes _{\mathscr {O}_{\mathscr {Z}}} \mathscr {K}(\mathscr {Z})$ ) $\mathscr {Z}$ -adic ordinary modular cuspforms (respectively, meromorphic $\mathscr {Z}$ -adic ordinary modular cuspforms) of weight $\varphi \colon \mathscr {Z}\to \frak W$ .

2.4.5 Weight-character map for $\mathrm {H}$

Let $U_{\mathrm {H}}^{p}\subset \mathrm {H}(\mathbf {A}^{p\infty })$ be an open compact subgroup, and let

as in (1.1.6). A (Hida) family for $\mathrm {H}$ is a connected component of $\mathscr {Y}_{\mathrm {H}}$ .

Fix a sufficiently small open compact subgroup $U_{F, p}^{\circ , \sqrt {}}=\prod _{v\vert p} U_{F, v}^{\circ , \sqrt {}}\subset \mathscr {O}_{F,p}^{\times }$ and an injective group homomorphism

$$ \begin{align*}j" \colon (U_{F, p}^{\circ, \sqrt{}}) \to \mathscr{O}_{E, p}^{\times,1}:= \{ t\in \mathscr{O}_{E,p}^{\times}\ |\ N_{E_{p}/F_{p}}(t)=1\},\end{align*} $$

and let

$$ \begin{align*}U_{F, p}^{\circ}:= (U_{F,p}^{\circ, \sqrt{}})^{2}\subset \mathscr{O}_{F, p}^{\times},\end{align*} $$

which is now fixed as promised in Section 2.4.1. Let $\sqrt {\ }\colon U_{F, p}^{\circ } \to U_{F, p}^{\circ , \sqrt {}} $ be the (uniquely determined, up to shrinking $U_{F, p}^{\circ , \sqrt {}}$ ) square root, and let $j, j'\colon U_{F, p}^{\circ }\to \mathscr {O}_{E , p}^{\times }$ be the mapsFootnote ⁹

(2.4.7) $$ \begin{align}\begin{aligned} j'(a):= j"(\sqrt{a})/\sqrt{a},\qquad j(a)= j'(a)a. \end{aligned}\end{align} $$

For any open compact $U_{E}^{p}\subset \mathbf {A}_{E}^{p\infty , \times }$ and $U_{F}^{p}:= U_{E}^{p}\cap \mathbf {A}^{p\infty , \times }$ , define a map

(2.4.8) $$ \begin{align}\begin{aligned} \underline{\kappa}_{\mathrm{H}}\colon \ \mathscr{Y}_{\mathrm{H}, U_{E}^{p}} &\to \frak W_{U_{F}^{p}}\\ y &\mapsto \underline{\kappa}_{\mathrm{H}}(y)=\left( \kappa_{0}=\chi_{y|[Z]_{U_{F}^{p}}}, \kappa: = \chi_{y}\circ j \right). \end{aligned}\end{align} $$

The set of classical points is

$$ \begin{align*}\mathscr{Y}_{\mathrm{H}}^{\mathrm{cl}}:= \mathscr{Y}_{\mathrm{H}} \times_{\frak W} \frak W^{\mathrm{cl}}.\end{align*} $$

Note that if $y\in \mathscr {Y}_{\mathrm {H}}$ is a classical point such that $\underline {\kappa }_{\mathrm {H}}(y)$ has weight $(l_{0}, l)$ , then $\chi _{y}$ has weight $(l_{0}, l)$ as defined in the introduction.

2.4.6 Hida families for $\mathrm {G}\times \mathrm {H}$

These are defined as in Section 1.1.4.

2.4.7 Universal automorphic sheaf on a Hida family

Let $\mathscr {X}_{\mathrm {G}}$ be a Hida family for $\mathrm {G}$ , and let $\mathscr {X}_{\mathrm {G}}^{\mathrm {cl}}:= \mathscr {X}_{\mathrm {G}} \cap \mathscr {Y}_{\mathrm {G}}^{\mathrm {cl}} $ . For each sufficiently small $U^{p}$ , we may view $\mathscr {X}\subset \mathscr {Y}_{\mathrm {G}, U^{p}}$ and we define

$$ \begin{align*}\Pi^{U^{p}}=\Pi_{\mathscr{X}_{\mathrm{G}}}^{U^{p}}:={\mathscr{S}}^{U^{p}}_{|\mathscr{X}_{\mathrm{G}}}.\end{align*} $$

For each $x\in \mathscr {X}_{\mathrm {G}}^{\mathrm {cl}}$ , by Hida’s Control Theorem (see, for instance, [Reference HidaHid91, Corollary 3.3]) and the theory of newforms, we have an isomorphism of $\mathscr {H}_{U^{p}}$ -modules,

(2.4.9) $$ \begin{align}\begin{aligned} \Pi_{|x}^{U^{p}} \cong \pi_{x}^{U^{p}, \mathrm{ord}}:= e^{\mathrm{ord}} \pi_{x}^{U^{p}}.\end{aligned}\end{align} $$

Let $U^{p}_{\mathscr {X}_{\mathrm {G}}}$ be minimal such that $\mathscr {X}_{\mathrm {G}}$ is a component of $\mathscr {Y}_{\mathrm {G}, U^{p}_{\mathscr {X}_{\mathrm {G}}}}$ . By [Reference HidaHid91, Section 3], there is a unique

(2.4.10) $$ \begin{align}\begin{aligned} \mathbf{f}_{0}=\mathbf{f}_{0,\mathscr{X}_{\mathrm{G}}}\in \Pi^{U^{p}_{\mathscr{X}_{\mathrm{G}}}}(\mathscr{X}_{\mathrm{G}}) \end{aligned}\end{align} $$

(the normalised primitive form over $\mathscr {X}_{\mathrm {G}}$ ) such that $W_{{\mathbf {f}}_{0}}(1)=1\in \mathscr {O}(\mathscr {X}_{\mathrm {G}})$ for the q-expansion map deduced from (2.4.5). Any $\mathbf {f} \in \Pi ^{U^{p}}$ can be written as $\mathbf {f} =T \mathbf {f}_{0}$ for some Hecke operator T supported at the places $v\nmid p\infty $ such that $U^{p}$ is not maximal.

2.4.8 Universal Galois sheaf on a Hida family and local–global compatibility

Let $\mathscr {X}_{\mathrm {G}}$ be a Hida family for $\mathrm {G}$ . By results of Hida and Wiles (see [Reference DisegniDis/b, Proposition 3.2.4]), there exist an open subset $\mathscr {X}_{\mathrm {G}}'\subset \mathscr {X}_{\mathrm {G}}$ containing $\mathscr {X}_{\mathrm {G}}^{\mathrm {cl}}$ and a locally free sheaf $\mathscr {V}_{\mathrm {G}}$ of rank $2$ , endowed with a Galois action

$$ \begin{align*}G_{F}\to \mathrm{End}\,_{\mathscr{O}_{\mathscr{X}_{\mathrm{G}}'}} (\mathscr{V}_{\mathrm{G}})\end{align*} $$

such that for all $x\in \mathscr {X}_{\mathrm {G}}^{\mathrm {cl}}$ , the fiber $\mathscr {V}_{\mathrm {G}|x}$ is the Galois representation attached to $\pi _{x}$ by the global Langlands correspondence.

Let S be a finite set of finite places of F, disjoint from $S_{p}$ , such that for all $v\notin S$ , the tame level $U^{p}=U^{Sp}U_{S}$ of $\mathscr {X}_{\mathrm {G}}$ is maximal at v. We define

(2.4.11) $$ \begin{align}\begin{aligned} \Pi_{\mathscr{X}_{\mathrm{G}}}^{U^{Sp}}:= \varinjlim_{U^{\prime}_{S}} \Pi_{\mathscr{X}_{\mathrm{G}}}^{U^{Sp}U^{\prime}_{S}}, \end{aligned}\end{align} $$

which is a finitely generated $\mathscr {O}_{\mathscr {X}_{\mathrm {G}}}[\mathrm {G}(F_{S})]$ -module. On the other hand, [Reference DisegniDis20, Theorem 4.4.1] attaches to the restriction $\mathscr {V}_{\mathrm {G}, v}:= \mathscr {V}_{\mathrm {G}|G_{F_{v}}}$ an $\mathscr {O}_{\mathscr {X}_{\mathrm {G}}'}[\mathrm {G}(F_{S})]$ -module

$$ \begin{align*}\Pi(\mathscr{V}_{ \mathrm{G}, v}),\end{align*} $$

which is torsion-free and co-Whittaker in the sense of [Reference DisegniDis20, Definition 4.2.2].

Proposition 2.4 After possibly replacing $\mathscr {X}_{\mathrm {G}}'\subset \mathscr {X}_{\mathrm {G}}$ with a smaller open subset still containing $\mathscr {X}_{\mathrm {G}}^{\mathrm {cl}}$ , there exists a line bundle $\Pi ^{\circ }_{\mathscr {X}_{\mathrm {G}}'} $ over $\mathscr {X}_{\mathrm {G}}'$ with trivial $\mathrm {G}(F_{S})$ -action, such that

$$ \begin{align*}\Pi^{U^{Sp}}_{|\mathscr{X}_{\mathrm{G}}'} \cong \Pi^{\circ}_{\mathscr{X}_{\mathrm{G}}'} \otimes \bigotimes_{v\in S} \Pi(\mathscr{V}_{\mathrm{G},v})\end{align*} $$

as $\mathscr {O}_{\mathscr {X}^{\prime }_{\mathrm {G}}}[\mathrm {G}(F_{S})]$ -modules.

3.1.1 Local case

Let $V=(V,q)$ be a quadratic space of even dimension over a local field F of characteristic not 2. Fix a nontrivial additive character $\psi $ of F. For $u\in F^\times $ , we denote by $V_u$ the quadratic space $(V,uq)$ . We let $\mathrm {GL}_2(F)\times \mathrm {GO}(V)$ act on the usual space of Schwartz functions ${\mathcal {S}}'(V\times F^\times )$ as follows (here, $\nu \colon \mathrm {GO}(V)\rightarrow \mathbf {G}_m$ denotes the similitude character):

• • $r(h)\phi (x,u)=\phi (h^{-1}x,\nu (h)u)$ for $h\in \mathrm {GO}(V)$ ;

• • $r(n(b))\phi (x,u)=\psi (buq(x))\phi (x,u)$ for $n(b)\in N(F)\subset \mathrm {GL}_2(F)$ ;

• • $r\left (\begin {pmatrix}a&\\& d\end {pmatrix}\right )\phi (x,u)=\chi _{V_{u}}(a)|{a\over d}|^{\dim V\over 4}\phi (at, d^{-1}a^{-1}u)$ ;

• • $r(w)\phi (x,u)=\gamma (V_u)\hat {\phi }(x,u)$ for $w=\begin {pmatrix}&1\\-1 &\end {pmatrix}.$

Here, $\chi _{V}=\chi _{(V,q)}$ is the quadratic character attached to V, $\gamma (V,q)$ is a fourth root of unity, and $\hat {\phi }$ denotes Fourier transform in the first variable with respect to the self-dual measure for the character $\psi _{u}(x)=\psi (ux)$ . We will need to note the following facts (see, for instance, [Reference Jacquet and LanglandsJL70]): $\chi _{V}$ is trivial if V is a quaternion algebra over F or $V=F\oplus F$ , and $\chi _{V}=\eta $ if V is a separable quadratic extension E of F with associated character $\eta $ .

3.1.2 Fock model and reduced Fock model

Assume that $F=\mathbf {R}$ and V is positive definite. Then we will prefer to consider a modified version of the previous setting. Let the Fock model $\mathcal {S}(V\times \mathbf {R}^{\times }, \mathbf {C})$ be the space of functions spanned by those of the form

$$ \begin{align*}H(u)P(x) e^{-2\pi |u| q(x)},\end{align*} $$

where H is a compactly supported smooth function on $\mathbf {R}^{\times }$ and P is a complex polynomial function on V. This space is not stable under the action of $\mathrm {GL}_{2}(\mathbf {R})$ , but it is so under the restriction of the induced $(\mathfrak {gl}_{2, \mathbf {R}},\mathbf {O}_{2}(\mathbf {R}))$ -action on the usual Schwartz space (see [Reference Yuan, Zhang and ZhangYZZ12, Section 2.1.2]).

We will also need to consider the reduced Fock space $\overline {\mathcal {S}}(V\times \mathbf {R}^{\times })$ spanned by functions of the form

$$ \begin{align*}\phi(x,u)=(P_{1}(uq(x))+\mathrm{sgn}(u)P_{2}(uq(x))) e^{- 2\pi |u| q(x)},\end{align*} $$

where $P_{1}$ , $P_{2}$ are polynomial functions with rational coefficients.

By [Reference Yuan, Zhang and ZhangYZZ12, Sections 3.4.1 and 4.4.1], there is a surjective quotient map

(3.1.1)

We let $\mathcal {S}(V\times \mathbf {R}^{\times } ) \subset \mathcal {S}(V\times \mathbf {R}^{\times } , \mathbf {C})$ be the preimage of $\overline {\mathcal {S}}(V\times \mathbf {R}^{\times })$ . For the sake of uniformity, when F is non-archimedean, we set $\overline {\mathcal {S}}(V\times F^{\times })=\mathcal {S}(V\times F^{\times }):= \mathcal {S}'(V\times F^{\times })$ .

3.1.3 Global case

Let $(\mathbf {V},q)$ be an even-dimensional quadratic space over the adèles $\mathbf {A}$ of a totally real number field F, and suppose that $\mathbf {V}_{\infty }$ is positive definite; we say that $\mathbf {V}$ is coherent if it has a model over F and incoherent otherwise. Given an $\widehat {\mathscr {O}}_{F}$ -lattice $\mathbf {{V}}^{\circ }\subset \mathbf {V}$ , we define the space $\mathcal {S}(\mathbf {V}\times \mathbf {A}^{\times })$ as the restricted tensor product of the corresponding local spaces, with respect to the spherical elements

$$ \begin{align*}\phi_{v}(x,u)=\mathbf{1}_{\mathbf{V}^{\circ}_{v}}(x)\mathbf{1}_{\varpi_{v}^{n_{v}}}(u),\end{align*} $$

if $\psi _{v}$ has level $n_{v}$ . We call such $\phi _{v}$ the standard Schwartz function at a non-archimedean place v. We define similarly the reduced space $\overline {\mathcal {S}}(\mathbf {V}\times \mathbf {A}^{\times })$ , which admits a quotient map

(3.1.2) $$ \begin{align} \mathcal{S}(\mathbf{V}\times \mathbf{A}^{\times})\to\overline{\mathcal{S}}(\mathbf{V}\times \mathbf{A}^{\times}) \end{align} $$

defined by the product of the maps (3.1.1) at the infinite places and of the identity at the finite places. The Weil representation of $ \mathrm {GO}(\mathbf {V})\times \mathrm {G}(\mathbf {A}^{\infty })\times (\mathfrak {gl}_{2,F_{\infty }},\mathbf {O}(\mathbf {V}_{\infty }))$ is the restricted tensor product of the local representations.

For a quadratic space $\mathbf {V}=(\mathbf {V}, q) $ over $\mathbf {A}$ , we define $\varepsilon (\mathbf {V})=+1$ (respectively, $-1$ ) if and only if there exists (respectively, does not exist) a quadratic space V over F such that $V\otimes _{F}\mathbf {A}=\mathbf {V}$ .

3.1.4 The quadratic spaces of interest

Let us go back to our usual notation: thus, F is our chosen totally real field and E its chosen CM quadratic extension. In this paper, we will consider the quadratic spaces $\mathbf {V}=(\mathbf {B}, q)$ , where $\mathbf {B}$ is a quaternion algebra over $\mathbf {A}$ , definite at all the archimedean places and split at p, and endowed with an $\mathbf {A}$ -embedding $\mathbf {A}_{E}\hookrightarrow \mathbf {B}$ , and $q\colon \mathbf {B}=\mathbf {V}\to \mathbf {A}$ is its reduced norm. It has a decomposition

$$ \begin{align*}\mathbf{V}=\mathbf{V}_{1}\oplus \mathbf{V}_{2},\end{align*} $$

where $\mathbf {V}_{1}=\mathbf {A}_{E}$ (on which the restriction of q coincides with $N_{E/F}$ ) and $\mathbf {V}_{2}$ is the q-orthogonal complement. Thus, $\varepsilon (\mathbf {V})=\varepsilon (\mathbf {V}_{2})$ . We denote by $r_{1}$ the restriction of r to a representation of $\mathbf {A}_{E}^{\times }=\mathrm {GO}(\mathbf {V}_{1})$ on $\overline {\mathcal {S}}(\mathbf {V}_{1}\times \mathbf {A}^{\times })$ .

For each place v, we have

(3.1.3) $$ \begin{align}\begin{aligned} \varepsilon({\mathbf{B}}_{v})=\varepsilon(\mathbf{V}_{v})= \begin{cases} +1, &\text{if}\ \mathbf{B}_{v}\cong M_{2}(F_{v}), \\ -1, &\text{if}\ \mathbf{B}_{v}\ \text{is a division algebra.} \end{cases} \end{aligned}\end{align} $$

We have $\varepsilon (\mathbf {V}):=\prod _{v} \varepsilon (\mathbf {V}_{v}) = (-1)^{[F:\mathbf {Q}]} \prod _{v\nmid p} \varepsilon (\mathbf {V}_{v})$ .

3.3.1 Schwartz function at $\infty $

Let $P_{k_{0}, k} \in \mathbf {R}[X]$ be the (rescaled) Laguerre polynomial

(3.3.2) $$ \begin{align}\begin{aligned} P^{}_{k_{0}, k}(X) &:=(2\pi i)^{-k_{0}} (4\pi)^{-k} (k+k_{0})! \sum_{j=0}^{k} {k\choose j} {(-X)^{j}\over j!}. \\ \end{aligned}\end{align} $$

For $k=(k_{0}, (k_{v}))\in \mathbf {Z}\times \mathbf {Z}_{\geq 0}^{{\Sigma _{\infty }}}$ such that $k_{v}+k_{0}\geq 0$ for all v, define

$$ \begin{align*}{E}^{}_{r} (g, u , \phi_{2}^{\infty} ;\xi, k) = {E}^{}_{r}(g, u , \phi_{2}^{\infty}\phi^{}_{2, \infty, k}; \xi),\end{align*} $$

where $\phi ^{}_{2, \infty , k}=\otimes _{v\vert \infty }\phi ^{}_{2, v, k_{v}}$ with

(3.3.3) $$ \begin{align}\begin{aligned} \phi^{}_{2,v, k_{v}}(x, u) = \mathbf{1}_{\mathbf{R}^{+}}(u) P_{k_{0},k_{v}}(4\pi uq(x))e^{-2\pi uq(x)}.\end{aligned}\end{align} $$

The series ${E}_{r}(\phi _{2}^{\infty }; \xi , k)$ belongs to $N_{\mathrm {tw}, (-k_{0}, k+k_{0})}^{\leq k}( \mathbf {C})$ .

3.3.2 Whittaker–Fourier expansion

The following standard result is essentially [Reference DisegniDis17, Proposition 3.2.1].

Proposition 3.3 We have

$$ \begin{align*}{E}_{r}(\bigl(\begin{smallmatrix}yx{}1&,\\u&,\end{smallmatrix}\bigr) \phi_{2}; \xi)= \sum_{a\in F} W_{a,r}( \bigl(\begin{smallmatrix}y{}{}1&,\\ &u\end{smallmatrix}\bigr), \phi_{2}; \xi)\psi(ax),\end{align*} $$

where

$$ \begin{align*}W_{a,r}(g,u, \phi_{2}; \xi)= \prod_{v} W_{a,r,v}(g,u, \phi_{2,v};\xi_{v})\end{align*} $$

with, for each v and $a\in F_{v}$ ,

$$ \begin{align*} & W_{a,r,v}(g,u, \phi_{2,v};\xi_{v})\\ & \quad = { L^{(p\infty)}(1, \eta_{v}\xi_{v}) \over L^{(p\infty)}(1, \eta_{v})} \int_{F_{v}}\delta_{\xi,r, v}(wn(b)gw_{r,v}) r(wn(b)g ) \phi_{2,v }(0,u)\psi_{v}(-ab) \, db. \end{align*} $$

Here, $L^{(p\infty )}(s, \xi ^{\prime }_{v}):=L(s, \xi ^{\prime }_{v})$ if $v\nmid p\infty $ and $L^{(p \infty )}(s,\xi ^{\prime }_{v}):=1$ if $v\vert p\infty $ , and we use the convention that $r_{v}=0$ if $v\nmid p$ .

(Note that the functions $W_{a,r}(\phi _{2}, \xi )$ correspond to the $W^{\mathbf {C}}_{E_{r}(\phi _{2}, \xi ), a}$ of Section 2.2.2. We prefer to use lighter notation in this section.)

We choose convenient normalizations for the local Whittaker functions: let $\gamma _{u, v}=\gamma (\mathbf {V}_{2,v}, uq)$ be the Weil index, and for $a\in F_{v}^{\times }$ , set

$$ \begin{align*}W^{\circ}_{a,r,v}(g,u, \phi_{2,v}; \xi_{v}):= \gamma_{u,v}^{-1} L^{(p)}(1, \eta_{v}) W_{a,r,v}(g,u, \phi_{2,v}; \xi_{v}). \end{align*} $$

Then, for the global Whittaker functions, we have

(3.3.4) $$ \begin{align} W_{a,r}(g,u, \phi_{2}; \xi)={-\varepsilon(\mathbf{V}_{2}) \over L^{(p)}(1, \eta)} \prod_{v}W^{\circ}_{a,r,v}(g,u, \phi_{2,v}; \xi_{v}) \end{align} $$

if $a\in F^{\times }$ , where $\varepsilon (\mathbf {V}_{2})=\prod _{v}\gamma _{u, v}$ equals $-1$ if $\mathbf {V}_{2}$ is coherent or $+1$ if $\mathbf { V}_{2}$ is incoherent. We similarly define $W_{0, r}^{\circ }(g, u, \phi _{2}, \xi ) $ by the identity

(3.3.5) $$ \begin{align} W_{0, r}^{}(g, u, \phi_{2}; \xi) = {-\varepsilon(\mathbf{V}_{2}) \over L^{(p)}(1, \eta)} W_{0, r}^{\circ}(g, u, \phi_{2}; \xi). \end{align} $$

A simple calculation shows that for all v and $a\neq 0$ ,

(3.3.6) $$ \begin{align}\begin{aligned} \qquad\qquad W_{a,r,}^{\circ}( \left(\begin{smallmatrix}y&\\&1\end{smallmatrix}\right),u, \phi_{2,v}; \xi_{v})= \eta\xi^{-1}(y)|y|^{1/2}W_{ay,r,}^{\circ}( 1, y^{-1}u, \phi_{2,v}; \xi_{v}).\end{aligned}\end{align} $$

We will sometimes drop $\phi _{2, v}$ from the notation.

The following sufficient condition for cuspidality will simplify matters a little later on.

Lemma 3.4 Assume that there is a place $v\nmid p\infty $ , at which $\xi _{v}$ is unramified, such that

(3.3.7) $$ \begin{align}\begin{aligned} \phi_{2,v}(0, u)=0\end{aligned}\end{align} $$

for all u. Then, for all $g={\left (\begin {smallmatrix}y&x\\&1\end {smallmatrix}\right )}$ with $y\in \mathbf {A}^{+}, x\in \mathbf {A}$ , we have

$$ \begin{align*}W_{0,r}(g, u, \phi_{2}; \xi)=0.\end{align*} $$

3.3.3 Archimedean Whittaker functions

We compute them explicitly based on our explicit choice of Schwartz function.

Lemma 3.5 Let $v\vert \infty $ , let $\xi _{v}(x)=x^{k_{0}}$ for some $k_{0}\in \mathbf {Z}$ , and let $\phi _{2,v}(x, u):= \mathbf {1}_{\mathbf {R}^{+}}(u) (uq(x))^{k}e^{-2\pi uq(x)}$ for some $k\in \mathbf {Z}_{\geq 0}$ with $k\geq -k_{0}$ . Let $a\in \mathbf {R}^{\times }$ . Then

$$ \begin{align*} W_{a,v}^{\circ}( 1,u, \phi_{2,v, k_{v}};\xi_{v}) = \begin{cases} \displaystyle 2 (2\pi i)^{k_{0}} {k! \over (k+k_{0})!} a^{k+k_{0}} e^{-2\pi a}, & \text{if}\ a,u>0,\\ 0, & \text{if}\ a<0\ \text{or}\ u<0. \end{cases} \end{align*} $$

Proof We drop the subscript v and write $\delta _{\xi , s}$ for $\delta _{\xi , r,s}$ . We have

$$ \begin{align*}wn(b) = \left(\begin{array}{@{}cc@{}}(1 +b^{2})^{-1/2} &-b(1 +b^{2})^{-1/2 }\\& {(1 +b^{2})^{1/2} } r_{\theta_{b}}\end{array}\right) \end{align*} $$

with $e^{ i \theta _{b}} = (i-b)/ {(1 +b^{2})^{1/2} }$ . Then $\delta _{\xi ,s} (wn(b)) = i^{k_{0}} (1+b^{2})^{-s/2} (1- ib)^{-k_{0}}$ . Since $L(1,\eta )=\pi ^{-1} $ and $\gamma _{v}=i$ , we have

(3.3.8) $$ \begin{align}\begin{aligned} & i^{-k_{0}}W_{a}^{\circ}(s, 1,u, \phi_{2, k})\\ & \quad := i^{-1} \pi^{-1}\int_{\mathbf{R}} \delta_{\xi, s}(w n(b) ) r(wn(b))\phi_{2, k}(0, u)\psi(-ab) \, db\\ & \quad = \pi^{-1}\int_{\mathbf{R}} (1+b^{2})^{-s/2} (1- ib)^{-k_{0}} \int_{\mathbf{C}} \phi_{2, k}(x, u) \psi(ubq(x)) d_{u}x \, \psi(-ab) db\\ & \quad = \pi^{-1} \int_{\mathbf{R}} (1+b^{2})^{-s/2} (1- ib)^{-k_{0}} \int_{\mathbf{C}} u^{k+1} q(x)^{k} e^{-2\pi u q(x)} \psi(ubq(x)) d_{1}x\, \psi(-ab) db, \end{aligned}\end{align} $$

where we recall that $d_{u}x= |u| d_{1}x$ and $d_{1}x$ is twice the usual Lebesgue measure. The integral over $\mathbf {C}$ is

$$ \begin{align*}2u^{k+1}\sum_{j=0}^{k} \int_{\mathbf{R}} \int_{\mathbf{R}}{k\choose j} x_{1}^{2j}x_{2}^{k-j} e^{-2\pi u (1-ib) (x_{1}^{2}+x_{2}^{2})} dx_{1}dx_{2}.\end{align*} $$

Since $\int _{\mathbf {R}} x^{2j}e^{-Ax^{2}} \, dx = {A^{-j-1/2}\Gamma (j+1/2)}$ , this equals

$$ \begin{align*}\begin{aligned} & 2 u^{k} \sum_{j=0}^{k} {k \choose j} \Gamma(j+1/2) \Gamma(k-j+1/2)\cdot (2\pi u )^{-k-1} (1-ib)^{-k-1} = 2^{-k} \pi^{-k} k! \, (1-ib)^{-k-1} \end{aligned}\end{align*} $$

by the combinatorial identity (see Appendix A)

(3.3.9) $$ \begin{align}\begin{aligned} \sum_{j=0}^{k} {k \choose j} \Gamma(j+1/2) \Gamma(k-j+1/2) = \pi k!.\end{aligned}\end{align} $$

Therefore, when $u>0$ ,

$$ \begin{align*}\begin{aligned} W_{a}^{\circ}(s, 1,u, \phi_{2, k})& = i 2^{-k} \pi^{-k} k! \int_{\mathbf{R}} (1+ib)^{-s/2} (1-ib)^{-(s+2k+2k_{0}+2)/2 } e^{-2\pi i a b} db.\end{aligned}\end{align*} $$

The integral is the same one appearing in [Reference Yuan, Zhang and ZhangYZZ12, bottom of p. 55] with $d=2+2k+2k_{0}$ . By [Reference Yuan, Zhang and ZhangYZZ12, Proposition 2.11] (whose normalization differs from ours by $L(1, \eta _{v}\xi _{v})=\pi i $ ), we find

$$ \begin{align*}W_{a}^{\circ}(0, 1,u, \phi_{2, k}) &= i^{k_{0}} 2^{-k} \pi^{-k-1} k! {(2\pi)^{1+k+k_{0}}\over \Gamma(1+k+k_{0})} a^{k+k_{0}} e^{-2\pi a}\\ &= 2 (2\pi i)^{k_{0}} {k! \over (k+k_{0})!} a^{k+k_{0}} e^{-2\pi a}\end{align*} $$

if $a, u>0$ , as well as simpler formulas implying the desired ones in the other cases.

We deduce the following. Let

(3.3.10) $$ \begin{align}\begin{aligned} Q_{k_{0}, k}(X)= \sum_{j=0}^{k} {k\choose j} {(k+k_{0})! \over (j+k_{0})!} (-X)^{k-j},\end{aligned}\end{align} $$

which satisfies $Q_{k_{0}, k}(0)=1$ .

Proposition 3.6 Let $v\vert \infty $ , let $a\in \mathbf {R}$ , and let $k_{0}\in \mathbf {Z}$ and $k\in \mathbf {Z}_{\geq 0}$ with $k\geq -k_{0}$ . Then, for $a\neq 0$ , we have

$$ \begin{align*}W_{a,v}( \left(\begin{smallmatrix}y&\\&1\end{smallmatrix}\right),u, \phi_{2,v ,k_{v}}; \xi_{v}) = (-1)^{k} \eta_{v}\xi_{v}^{-1}(y) |y|^{1/2} \cdot 2 (ay)^{k+k_{0}} Q_{k_{0},k}((4\pi ay)^{-1}) e^{-2 \pi a y} \end{align*} $$

if $ay>0$ , $uy>0$ , and $W_{a,v}(0, \left (\begin {smallmatrix}y&\\&1\end {smallmatrix}\right ),u, \phi _{2,v ,k_{v}}; \xi _{v})=0$ otherwise.

Proof After recalling the definition of $P_{k_{0}, k}$ in (3.3.2), by Lemma 3.5 and (3.3.6), we find the asserted vanishing and that for $ay, uy>0$ we have (dropping subscripts v):

$$ \begin{align*} & W_{a,v}( \left(\begin{smallmatrix}y&\\&1\end{smallmatrix}\right),u, \phi_{2, k}; \xi)\\ & \quad = \eta\xi^{-1}(y) |y|^{1/2} \cdot 2 (4\pi)^{-k} (-4\pi)^{-k_{0}} (k+k_{0})! \sum_{j=0}^{k} {k\choose j} {(k+k_{0})! \over (j+k_{0})!} (-4\pi ay)^{j+k_{0}} e^{-2\pi ay} ,\end{align*} $$

which is equal to the asserted formula.

Corollary 3.7 Let $\xi \colon F^{\times }\backslash \mathbf {A}^{\times }\to \mathbf {C}^{\times }$ with $\xi (x_{\infty })=x_{\infty }^{k_{0}}$ for some $k_{0}\in \mathbf {Z}$ . For each $k\in \mathbf {Z}_{\geq 0}^{\Sigma _{\infty }}$ with $k\geq -k_{0}$ , we have

$$ \begin{align*}{E}_{r}(g, u, \phi_{2}^{\infty}; \xi, k) = (-1)_{\infty}^{k} \delta^{k} {E}_{r}(g, u, \phi_{2}^{\infty}; \xi, 0).\end{align*} $$

3.3.4 Schwartz function at p

Let $U_{F,p}^{\circ } \subset \mathscr {O}_{F, p}^{\times }$ be as fixed in Section 2.4.5, and let $\kappa _{2}'\colon U_{F,p}^{\circ }\to \mathbf {C}^{\times }$ be a smooth character. We define

(3.3.11) $$ \begin{align}\begin{aligned} \phi_{2,p,\kappa_{2}'}(x, u):= \mathbf{1}_{\mathbf{V}^{\circ}_{2,p}}(x)\cdot \delta_{U_{F,p}^{\circ}}(u) \kappa_{2}'(u),\end{aligned}\end{align} $$

where $\mathbf {V}^{\circ }_{2,p}\subset \mathbf {V}_{2,p}$ is a fixed $\mathscr {O}_{F, p}$ -lattice and $\displaystyle \delta _{U_{F,p}^{\circ }}(u):= {\mathrm {vol}(\mathscr {O}_{F, p}^{\times })\over \mathrm {vol}(U_{F,p}^{\circ })}\cdot \mathbf {1}_{U_{F,p}^{\circ }}(u)$ . It is invariant under $N(\mathscr {O}_{F,p})$ .

Let

(3.3.12) $$ \begin{align}\begin{aligned} E_{r}( \phi_{2}^{p\infty}; \xi, \kappa_{2}', k):= E_{r}(\phi_{2}^{p\infty}\phi_{2,p,\kappa_{2}' }\phi_{2, \infty, k};\xi), \end{aligned}\end{align} $$

and denote its normalized Whittaker functions by

$$ \begin{align*}W_{a, r, v}^{\circ}(g, u, \phi_{2}^{p\infty}; \xi,\kappa_{2}', k);\end{align*} $$

depending on the place v, we will drop the unnecessary elements from the notation.

3.3.5 Non-archimedean Whittaker functions

We study the functions $W_{a,v}^{\circ }$ and the q-expansion of $E_{r}$ .

Proof See [Reference DisegniDis17, Proposition 3.2.3 and Lemma 3.2.4] for parts 1 and 2. For part 3, we drop subscripts v and compute

$$ \begin{align*}\delta_{\xi, r}(wn(b) w_{r})=\xi(-1)\mathbf{1}_{\mathscr{O}_{F}}(b), \quad \gamma_{u}^{-1} r(wn(b))\phi_{2, \kappa_{2}'}(0, u) = |d_{v}| |D_{v}|^{1/2} \mathbf{1}_{\mathscr{O}_{F}^{\times}}(u)\kappa_{2}'(u)\end{align*} $$

(the latter if $b\in \mathscr {O}_{F}$ ), so that

$$ \begin{align*}W_{a}^{\circ}(1,u;\xi, \kappa_{2}') = |d_{v}| |D_{v}|^{1/2} \xi(-1)\mathbf{1}_{\mathscr{O}_{F}^{\times}}(u)\kappa_{2}'(u) \int_{\mathscr{O}_{F}}\psi(-ab)\, db,\end{align*} $$

which gives the asserted value.

Corollary 3.9 For $g=\left (\begin {smallmatrix}y&x\\&1\end {smallmatrix}\right ) $ with $y\in \mathbf {A}^{+}$ and $x\in \mathbf {A}$ , we have

$$ \begin{align*} &{E}_{r} (g,u,\phi_{2}^{\infty}; \xi, k) = {-\varepsilon(\mathbf{V}_{2}) \over L^{(p)}(1, \eta)} \eta\xi^{-1}(y) |y|^{1/2} \\ &\cdot \left(W_{0}^{\circ}(y, u)+ \sum_{a\in F^{+}} 2^{[F:\mathbf{Q}]} (ay_{\infty})^{k+k_{0}} W_{a,r}^{\circ , \infty}(0, 1, y^{-1}u, \phi_{2}, \xi) Q_{k_{0}, k}((4\pi a y_{\infty})^{-1}) \, \mathbf{q}^{a}\right), \end{align*} $$

where $W_{a,r}^{\circ , \infty }(0, 1, u, \phi _{2}; \xi ) = \prod _{v\nmid \infty }W_{a,r,v}^{\circ }(0, 1, y^{-1}u, \phi _{2, v}; \xi )$ .

Let $\phi _{2}^{p\infty }$ satisfy (3.3.7), so that by Lemma 3.4, the corresponding Eisenstein series is cuspidal. For $\xi $ a locally algebraic p-adic character of $\mathbf {A}^{\times }$ of weight $k_{0}$ , consider the (bounded) sequence of coefficients in $\mathbf {Q}_{p}(\xi )$

$$ \begin{align*}\mathrm{E}_{r}(u, \phi^{\infty}_{2}; \xi, k) = \left( \lambda \cdot \eta\xi^{-1}(y) 2^{[F:\mathbf{Q}]} (by_{p})^{k+k_{0}} |D_{F}|^{1/2}|D_{E}|^{1/2}W^{\circ, \infty}_{by}(1, y^{-1}u, \phi_{2}^{\infty};\xi) \right)_{b\in F^{+}}, \end{align*} $$

where $\lambda = {-\varepsilon (\mathbf {V}_{2}) / |D_{E/F}|^{1/2} L^{(p)}(1, \eta )} $ . This is the p-adic q-expansion attached to $E_{r}$ . Analogously to Corollary 3.7, we have

(3.3.13) $$ \begin{align}\begin{aligned} \mathrm{E}_{r}(u, \phi_{2}; \xi, k) = (-1)_{\infty}^{k} d^{k} \mathrm{E}_{r}(u, \phi_{2}; \xi, 0).\end{aligned}\end{align} $$

3.3.6 p-adic interpolation of Whittaker functions

For a place $v\nmid p\infty $ of F, denote $\mathscr {Y}_{v}:= \mathrm {Spec}\, \mathbf {Q}_{p}[F_{v}^{\times }]$ and (for a later use) $\mathscr {Y}_{\mathrm {H}, v}:=\mathrm {Spec}\, \mathbf {Q}_{p}[E_{v}^{\times }]$ , the spaces of characters of $F_{v}^{\times }$ (respectively, $E_{v}^{\times }$ ). We say that a meromorphic function $\Phi $ on an integral scheme has poles controlled by the (nonzero) meromorphic function $\Phi '$ if $\Phi /\Phi '$ is regular.

Proposition 3.10 Let $v\nmid p\infty $ . For each $a\in F_{v}^{\times }$ , $y\in F_{v}^{\times }$ , and $\phi _{2,v}\in \mathcal {S}(\mathbf { V}_{2}^{p\infty }\times \mathbf {A}^{p\infty , \times }, \mathbf {Q}_{p})$ , there is a meromorphic function

$$ \begin{align*}\mathscr{W}_{a,v}^{\circ}(y,u, \phi_{2,v})\in \mathscr{K}(\mathscr{Y}_{v}),\end{align*} $$

regular if $\phi _{2, v}$ is standard and otherwise with poles controlled by $L(1,\eta _{v}\xi _{v})$ , satisfying

$$ \begin{align*}\mathscr{W}_{a,v}^{\circ}(y, u, \phi_{2,v};\xi_{v}) = |d_{v}|^{-3/2}|D_{v}|^{-1/2} W_{a,r,v}^{\circ}(\left(\begin{smallmatrix}y_{v}&\\&1\end{smallmatrix}\right), u,\phi_{2,v}(\xi_{v}); \xi_{v})\end{align*} $$

for all $\xi _{v}\in \mathscr {Y}_{v}(\mathbf {C})$ whose underlying scheme point is not a pole.