Celebratio Mathematica

In the con­text of Drin­feld mod­u­lar vari­et­ies there are two dif­fer­ent con­cepts that gen­er­al­ize the clas­sic­al mod­u­lar forms. The first are the Drin­feld mod­u­lar forms, which are \( \mathbb{C}_\infty \)-val­ued holo­morph­ic func­tions on \( \Omega^r \). The second are the Drin­feld auto­morph­ic forms, which are \( \mathbb{C} \)-val­ued func­tions on cer­tain ad­ele groups; these auto­morph­ic forms have a com­bin­at­or­i­al in­ter­pret­a­tion as har­mon­ic co­chains on the Bruhat–Tits build­ing \( \mathcal{B}^r \) of \( \operatorname{PGL}_r(F_\infty) \). Both of these con­cepts are im­port­ant in the study of Drin­feld mod­u­lar vari­et­ies.

Our main ref­er­ences are [e17], [e26], [e29], [e92], [e116]. Let \( r\geq 2 \) and let \( \Gamma \) be a con­gru­ence sub­group of \( \mathrm{GL}_r(A) \), i.e., \( \Gamma(\mathfrak{n})\subseteq \Gamma\subseteq \mathrm{GL}_r(A) \) for some \( \mathfrak{n}\neq 0 \). If we nor­mal­ize the pro­ject­ive co­ordin­ates of \( \boldsymbol{z}=(z_1, \dots, z_r)\in \Omega^r \) by \( z_r=1 \), then \( \gamma=(\gamma_{m,n})\in \Gamma \) acts on \( \Omega^r \) as \[ \gamma \boldsymbol{z} = j(\gamma, \boldsymbol{z})^{-1} (\,\dots, \sum_n \gamma_{m, n} z_n, \dots ), \] where \( j(\gamma, \boldsymbol{z}) = \sum_{n=1}^r \gamma_{r, n}z_n \). For ex­ample, if \( r=2 \) and we identi­fy \( \Omega^2 \) with \( \mathbb{C}_\infty - F_\infty \), then for \[ \gamma = \begin{pmatrix}a&b\\c&d\end{pmatrix}\in\Gamma \] we have \[ j(\gamma, \boldsymbol{z})=j(\gamma,z) = cz+d\quad\text{ and }\quad \gamma \boldsymbol{z} = \gamma z = \dfrac{az+b}{cz+d}. \]

Con­di­tion (ii) is tech­nic­ally com­plic­ated to ex­plain when \( r\geq 3 \), so we will ex­plain it only for \( r=2 \), and refer to [e116] for \( r\geq 2 \). Be­cause \( \Gamma \) is a con­gru­ence group, it con­tains the sub­group \( U_b=\left(\begin{smallmatrix} 1 & bA \\ 0 & 1\end{smallmatrix}\right) \) for some nonzero \( b\in A \). Con­di­tion (i) im­plies that \( f(z+b)=f(z) \), which it­self im­plies that \( f(z) \) can be ex­pan­ded as \[ f(z)=\sum_{n\in \mathbb{Z}} a_n (1/e_{bA}(z))^n, \qquad a_n\in \mathbb{C}_\infty, \] as­sum­ing \( \Im(z):= \inf_{\alpha\in F_\infty}\lvert z-\alpha \rvert\gg 0 \) (for sim­pli­city, we will omit this con­di­tion in what fol­lows). We say that \( f(z) \) is holo­morph­ic at the cusp \( [\infty] \) if in the above ex­pan­sion \( a_n=0 \) for all \( n < 0 \) (this van­ish­ing of coef­fi­cients with neg­at­ive in­dices does not de­pend on the choice of \( b \)). Next, for \( g\in \mathrm{GL}_2(A) \), put \( f|_g (z) = (\det g)^m j(g,z)^{-k}f(g z) \). This \( f|_g \) sat­is­fies (i) for any \( \gamma \in g^{-1}\Gamma g \), which is again a con­gru­ence group. Con­di­tion (ii) means that \( f|_g \) is holo­morph­ic at \( [\infty] \) for all \( g\in \mathrm{GL}_2(A) \). (Note that \( f|_g=f \) for \( g\in \Gamma \), so for this last con­di­tion to hold it suf­fices that \( f|_g \) is holo­morph­ic at \( [\infty] \) for left coset rep­res­ent­at­ives of \( \Gamma \) in \( \mathrm{GL}_2(A) \).)

Now we spe­cial­ize to \( r=2 \) and \( \Gamma=\Gamma_0(\mathfrak{n}) \). To sim­pli­fy the nota­tion, we will omit the su­per­script \( r \), so for ex­ample \( \Omega:=\Omega^2 \) and \( M_{k,m}(\mathfrak{n}) := M^2_{k,m}(\Gamma_0(\mathfrak{n})) \). Since \( \Gamma_0(\mathfrak{n}) \) con­tains the group of scal­ar matrices \( \left(\begin{smallmatrix} \alpha & 0 \\ 0 & \alpha\end{smallmatrix}\right) \), \( \alpha\in \mathbb{F}_q^\times \), ap­ply­ing con­di­tion (i) to such matrices one con­cludes that if \( M_{k, m}(\mathfrak{n})\neq 0 \), then \( k\equiv 2m(\operatorname{mod} q-1) \). Hence, if \( q \) is odd and \( M_{k, m}(\mathfrak{n})\neq 0 \), then \( k \) is ne­ces­sar­ily even and \( m=k/2 \) or \( m=k/2+(q-1)/2 \) mod­ulo \( q-1 \). Next, a simple cal­cu­la­tion shows that the dif­fer­en­tial \( \mathrm{d} z \) on \( \Omega \) sat­is­fies \[ \mathrm{d}(\gamma z) = \frac{\det(\gamma)}{(cz+d)^2}\,\mathrm{d} z \quad \text{for all }\gamma\in \mathrm{GL}_2(F_\infty). \] Hence, if \( f(z)\in M_{2k, k}(\mathfrak{n}) \), then \( f(z)(\mathrm{d} z)^k \) can be iden­ti­fied with a \( k \)-fold dif­fer­en­tial form on the Drin­feld mod­u­lar curve \( X_0(\mathfrak{n}) \).

Since \( \Gamma_0(\mathfrak{n}) \) con­tains the sub­group \( U_1 \), the ex­pan­sion of a Drin­feld mod­u­lar form \( f \) at \( [\infty] \) is \( f(z)=\sum_{n\geq 0} a_n (1/e_A(z))^n \). In­stead we will now use the para­met­er \[ t(z) = \frac{1}{\pi_C e_A(z)} = \frac{1}{e_C(\pi_C z)} = \frac{1}{\pi_C} \sum_{a\in A} \frac{1}{z+a}, \] which plays the role of the clas­sic­al para­met­er \( q = e^{2i\pi z} \) (this renor­mal­iz­a­tion is chosen so that the ex­pan­sions of the mod­u­lar forms \( \pi_C^{1-q} g_1 \) and \( \pi_{C}^{1-q^2} g_2 \) at \( [\infty] \) have coef­fi­cients in \( A \)). Then we have \( f = \sum_{n\geq 0} a^{\prime}_n t^n \) with \( a^{\prime}_n \in \mathbb{C}_\infty \). Since \( t(\alpha z)=\alpha^{-1}t(z) \) for any \( \alpha \in \mathbb{F}_q^\times \), the coef­fi­cients \( a^{\prime}_n \) are zero un­less \( n\equiv m (\operatorname{mod}q-1) \) so the ex­pan­sion of \( f \) is of the form \[ f = \sum_{i\geq 0} b_i t^{m+i(q-1)}. \] A Drin­feld mod­u­lar form is said to be cuspid­al (resp. doubly cuspid­al) at the cusp \( [\infty] \) if \( a^{\prime}_0=0 \) (resp. \( a^{\prime}_0 = 0 \) and \( a^{\prime}_1=0 \)). If for all \( g\in \mathrm{GL}_2(A) \), \( f|_g \) is cuspid­al (resp. doubly cuspid­al) at \( [\infty] \), we say that the Drin­feld mod­u­lar form \( f \) is cuspid­al (resp. doubly cuspid­al). Let \( M_{k,m}^0(\mathfrak{n}) \) (resp. \( M_{k,m}^{0,0}(\mathfrak{n}) \)) de­note the sub­spaces of such func­tions. Since \( \mathrm{d} e_A(z) / \mathrm{d} z=1 \), we have \( \mathrm{d} t = -\pi_C t^2 \mathrm{d} z \) so doubly cuspid­al Drin­feld mod­u­lar forms play a role sim­il­ar to clas­sic­al cusp forms. Namely the map \( f(z) \mapsto f(z) \mathrm{d} z \) is an iso­morph­ism between \( M_{2,1}^{0,0}(\mathfrak{n}) \) and the space of holo­morph­ic dif­fer­en­tial forms on the curve \( X_0(\mathfrak{n}) \). In par­tic­u­lar the di­men­sion of \( M_{2,1}^{0,0}(\mathfrak{n}) \) is equal to the genus of \( X_0(\mathfrak{n}) \).

Hecke op­er­at­ors on Drin­feld mod­u­lar forms can be defined as double coset op­er­at­ors for \( \Gamma_0(\mathfrak{n}) \). Let \( \mathfrak{m} \lhd A \) be a nonzero ideal of \( A \). For the double coset \( \Gamma_0(\mathfrak{n}) \setminus (\Gamma_0(\mathfrak{n}) \left(\begin{smallmatrix}\mathfrak{m}&0\\0&1\end{smallmatrix}\right) \Gamma_0(\mathfrak{n})) \), a set of rep­res­ent­at­ives is \begin{equation}\label{eq-Sm} S_{\mathfrak{m}}=\bigl\{ \left(\begin{smallmatrix} a & b \\ 0 & d \end{smallmatrix}\right) \in M_2(A) : a,d \text{ monic}, (ad) = \mathfrak{m}, (a)+\mathfrak{n} = A, \deg b < \deg d\bigr\}.\tag{4.2} \end{equation} For \( f \in M_{k,m}(\mathfrak{n}) \), we define \( f|T_\mathfrak{m} = \sum_{g\in S_{\mathfrak{m}}} f|_g \). In more con­crete terms \[ f|T_{\mathfrak{m}}\, (z) = \frac{1}{\mathfrak{m}^{k-m}} \sum_{\left(\begin{smallmatrix}a&b\\0&d\end{smallmatrix}\right) \in S_\mathfrak{m}} f\left( \frac{az+b}{d} \right). \] The \( \mathfrak{m} \)-th Hecke op­er­at­or \( T_\mathfrak{m} \) is a \( \mathbb{C}_\infty \)-lin­ear trans­form­a­tion of the space \( M_{k,m}(\mathfrak{n}) \). It sta­bil­izes the cuspid­al and doubly-cuspid­al sub­spaces. The Hecke op­er­at­ors \( (T_\mathfrak{m})_{\mathfrak{m}\lhd A} \) gen­er­ate a com­mut­at­ive \( \mathbb{C}_\infty \)-sub­al­gebra of \( M_{k,m}(\mathfrak{n}) \), called the Hecke al­gebra for \( M_{k,m}(\mathfrak{n}) \). On Drin­feld mod­u­lar forms, Hecke op­er­at­ors are com­pletely mul­ti­plic­at­ive, i.e., for all \( \mathfrak{m},\mathfrak{m}^{\prime} \lhd A \), we have \( T_\mathfrak{m} T_{\mathfrak{m}^{\prime}} = T_{\mathfrak{m} \mathfrak{m}^{\prime}}=T_{\mathfrak{m}^{\prime}} T_\mathfrak{m} \). This prop­erty dis­tin­guishes them from Hecke op­er­at­ors on clas­sic­al mod­u­lar forms, where they are only mul­ti­plic­at­ive.

An­oth­er im­port­ant dif­fer­ence is that, the char­ac­ter­ist­ic be­ing pos­it­ive, the space \( M_{k,m}^{0,0}(\mathfrak{n}) \) has no in­ner product. Hence there is no guar­an­tee that \( T_\mathfrak{m} \) is di­ag­on­al­iz­able. Goss was the first to point out this prob­lem [e16]. Since then, ex­amples of non­di­ag­on­al­iz­able Hecke op­er­at­ors have been ob­tained in spe­cial cases (see [e70] for the sub­groups \( \Gamma_1(T) \) and \( \Gamma(T) \)) but in gen­er­al, the ques­tion is still wide open. In par­tic­u­lar we know no nat­ur­al bases of Drin­feld mod­u­lar forms which are sim­ul­tan­eously ei­gen­forms for the Hecke op­er­at­ors.

Let \( r\geq 2 \) and let \( V \) be an \( r \)-di­men­sion­al vec­tor space over \( F_\infty \). De­note by \( \mathcal{O}_\infty \) the ring of in­tegers of \( F_\infty \) and let \( \pi_\infty \) be a uni­form­izer of \( \mathcal{O}_\infty \). An \( \mathcal{O}_\infty \)-lat­tice in \( V \) is a free \( \mathcal{O}_\infty \)-mod­ule of rank \( r \) which con­tains a basis of \( V \). Two lat­tices \( L_1 \) and \( L_2 \) are ho­mothet­ic if there ex­ists \( \alpha\in F_\infty^\times \) with \( \alpha\cdot L_1=L_2 \); this defines an equi­val­ence re­la­tion on the set of lat­tices in \( V \). We de­note the equi­val­ence class of \( L \) by \( [L] \). The Bruhat–Tits build­ing of \( \operatorname{PGL}_r(F_\infty) \) is the \( (r-1) \)-di­men­sion­al sim­pli­cial com­plex \( \mathscr{B}^r \) with set of ver­tices \[ \mathrm{Ver}(\mathscr{B}^r)=\{[L]\mid L \text{ is a lattice in }V\}, \] in which the ver­tices \( [L_0], \dots, [L_n] \) form an \( n \)-sim­plex if and only if there is \( L_i^{\prime}\in [L_i] \), \( 1\leq i\leq n \), such that \[ L_0^{\prime}\supsetneq L_1^{\prime}\supsetneq \cdots \supsetneq L_n^{\prime}\supsetneq \pi_\infty L_0^{\prime}. \] Since \( \mathrm{GL}_r(F_\infty) \) acts trans­it­ively on \( \mathrm{Ver}(\mathscr{B}^r) \) and the sta­bil­izer of a ver­tex is con­jug­ate to \( F_\infty^\times \mathrm{GL}_r(\mathcal{O}_\infty) \), we have a bijec­tion \[ \mathrm{Ver}(\mathscr{B}^r) \cong \mathrm{GL}_r(F_\infty)/F_\infty^\times \mathrm{GL}_r(\mathcal{O}_\infty). \] Sim­il­ar bijec­tions ex­ist between the sets of high­er di­men­sion­al sim­plices of \( \mathscr{B}^r \) of vari­ous types and left cosets of parahor­ic sub­groups of \( \mathrm{GL}_r(\mathcal{O}_\infty) \). When \( r=2 \), \( \mathscr{B}^2 \) is an in­fin­ite tree in which every ver­tex is ad­ja­cent to ex­actly \( q+1 \) oth­er ver­tices.

Har­mon­ic co­chains on sim­pli­cial com­plexes nat­ur­ally arise in the study of com­bin­at­or­i­al lapla­cians; see [e8]. A vari­ant of har­mon­ic \( i \)-co­chains on \( \mathscr{B}^r \) was defined by de Shalit [e54]: these are func­tions on poin­ted \( i \)-sim­plices of \( \mathscr{B}^r \), \( 1\leq i\leq r-1 \), sat­is­fy­ing cer­tain con­di­tions (in [e54] the build­ing \( \mathscr{B}^r \) is defined over an ar­bit­rary loc­al field). The sig­ni­fic­ance of the group \( \operatorname{Har}^i(\mathscr{B}^r, \mathbb{Q}_\ell) \) of \( \mathbb{Q}_\ell \)-val­ued har­mon­ic \( i \)-co­chains is that it is iso­morph­ic to the \( \ell \)-ad­ic co­homo­logy group \( H^i_{\mathrm{et}}(\Omega^r, \mathbb{Q}_\ell) \) of \( \Omega^r \); see [e32], [e54]. Also, \( \operatorname{Har}^i(\mathscr{B}^r, \mathbb{Q}_\ell) \) can be in­ter­preted as a space of auto­morph­ic forms on the ad­ele group \( \mathrm{GL}_r(\mathbb{A}_F) \) with spe­cial re­stric­tion at \( \infty \); see [e68] (one trans­forms func­tions on \( \mathrm{GL}_r(\mathbb{A}_F) \) to func­tions on \( \mathrm{GL}_r(F_\infty) \) us­ing the strong ap­prox­im­a­tion the­or­em and uses the bijec­tions between the sets of sim­plices of \( \mathscr{B}^r \) and cosets in \( \operatorname{PGL}_r(F_\infty) \) men­tioned earli­er).

The most rel­ev­ant for our pur­poses are the har­mon­ic 1-co­chains, which for \( r=2 \) were already defined by van der Put in [e21].

From an­oth­er per­spect­ive, \( \mathscr{B}^r \) is a com­bin­at­or­i­al “skel­et­on” of \( \Omega^r \). In fact, there is an ad­miss­ible cov­er­ing of \( \Omega^r \) through af­fin­oid sub­spaces and a \( \mathrm{GL}_r(F_\infty) \)-equivari­ant map \( \Omega^r\to \mathscr{B}^r \) that sends af­fin­oids to sim­plices; see [e10]. The fol­low­ing im­port­ant res­ult relates the group of holo­morph­ic in­vert­ible func­tions \( \mathcal{O}(\Omega^r)^\times \) on \( \Omega^r \) to \( \mathbb{Z} \)-val­ued har­mon­ic 1-co­chains on \( \mathscr{B}^r \): \begin{equation}\label{eqGvdP} 0 \longrightarrow \mathbb{C}_\infty^\times \longrightarrow \mathcal{O}(\Omega^r)^\times \xrightarrow{\mathrm{dlog}} \mathrm{Har}^1(\mathscr{B}^r, \mathbb{Z})\longrightarrow 0, \tag{4.3} \end{equation} where \( \mathrm{dlog} \) is some sort of a log­ar­ithmic de­riv­at­ive. The ex­ist­ence of \eqref{eqGvdP} was proved by van der Put for \( r=2 \) and ex­ten­ded to ar­bit­rary \( r\geq 2 \) by Gekel­er [e99].

From now on, the dis­cus­sion will fo­cus mostly on har­mon­ic 1-co­chains on the Bruhat–Tits tree \( \mathscr{B}^2 \). To sim­pli­fy the nota­tion, we de­note \( \mathscr{T}=\mathscr{B}^2 \). The group \( \mathrm{GL}_2(F_\infty) \) acts on \( \mathscr{T} \) from the left. Let \[ \mathcal{H}(\mathfrak{n}, R):= \operatorname{Har}^1(\mathscr{T}, R)^{\Gamma_0(\mathfrak{n})} \] be the sub­mod­ule of \( R \)-val­ued har­mon­ic 1-co­chains such that \( f(\gamma e)=f(e) \) for all \( \gamma\in \Gamma_0(\mathfrak{n}) \) and \( e\in \mathrm{Ed}(\mathscr{T}) \). The mod­ule of \( R \)-val­ued \( \Gamma_0(\mathfrak{n}) \)-in­vari­ant cuspid­al har­mon­ic co­chains, de­noted \( \mathcal{H}_0(\mathfrak{n}, R) \), is the sub­mod­ule of \( \mathcal{H}(\mathfrak{n}, R) \) con­sist­ing of func­tions which have com­pact sup­port as func­tions on \( \Gamma_0(\mathfrak{n})\setminus \mathscr{T} \), i.e., func­tions which have value 0 on all but fi­nitely many edges of \( \Gamma_0(\mathfrak{n})\setminus \mathscr{T} \). We de­note by \( \mathcal{H}_{00}(\mathfrak{n}, R) \) the im­age of \( \mathcal{H}_0(\mathfrak{n},\mathbb{Z})\otimes_{\mathbb{Z}} R \) in \( \mathcal{H}_0(\mathfrak{n}, R) \). (It is easy to con­struct ex­amples where the in­clu­sion \( \mathcal{H}_{00}(\mathfrak{n}, R)\subseteq \mathcal{H}_0(\mathfrak{n}, R) \) is strict; see ([e85], Sec­tion 1.1).) It is known that the quo­tient graph \( \Gamma_0(\mathfrak{n})\setminus \mathscr{T} \) is the edge dis­joint uni­on \[ \Gamma_0(\mathfrak{n})\setminus \mathscr{T} = (\Gamma_0(\mathfrak{n})\setminus \mathscr{T})^0\cup \bigcup_{s\in \Gamma_0(\mathfrak{n})\setminus \mathbb{P}^1(F)} h_s \] of a fi­nite graph \( (\Gamma_0(\mathfrak{n})\setminus \mathscr{T})^0 \) with a fi­nite num­ber of half-lines \( h_s \), called cusps. The cusps are in bijec­tion with the or­bits of the nat­ur­al (left) ac­tion of \( \Gamma_0(\mathfrak{n}) \) on \( \mathbb{P}^1(F) \). It is clear that \( f\in \mathcal{H}(\mathfrak{n}, R) \) is cuspid­al if and only if it even­tu­ally van­ishes on each \( h_s \). One can show that \( \mathcal{H}_0(\mathfrak{n}, \mathbb{Z}) \) and \( \mathcal{H}(\mathfrak{n}, \mathbb{Z}) \) are fi­nitely gen­er­ated free \( \mathbb{Z} \)-mod­ules of rank \( g(\mathfrak{n}) \) and \( g(\mathfrak{n})+c(\mathfrak{n})-1 \), re­spect­ively, where \( g(\mathfrak{n}) \) is the genus of the curve \( X_0(\mathfrak{n}) \) and \( c(\mathfrak{n}) \) is the num­ber of its cusps.

Har­mon­ic 1-co­chains in­vari­ant un­der con­gru­ence groups have Four­i­er ex­pan­sions. This the­ory for \( r=2 \) was first de­veloped by Weil in ad­el­ic lan­guage, and re­cast in­to more ex­pli­cit for­mu­las by Gekel­er [e43] and Pál [e66]. The the­ory was ex­ten­ded to ar­bit­rary \( r\geq 2 \) in [e112]. We briefly dis­cuss the \( r=2 \) case.

The edges of \( \mathscr{T} \) are in bijec­tion with \( \mathrm{GL}_2(F_\infty)/F_\infty^\times \mathcal{I}_\infty \), where \( \mathcal{I}_\infty \) is the Iwahori group: \[ \mathcal{I}_\infty:=\left\{\begin{pmatrix} a & b\\ c & d\end{pmatrix}\in \mathrm{GL}_2(\mathcal{O}_\infty)\ \bigg|\ c\in \pi_\infty\mathcal{O}_\infty\right\}. \] Let \[ \mathrm{Ed}(\mathscr{T})^+:=\left\{\begin{pmatrix} \pi_\infty^k & u \\ 0 & 1\end{pmatrix}\ \bigg|\ \begin{matrix} k\in \mathbb{Z}\\ u\in F_\infty,\ u\mod{\pi_\infty^k\mathcal{O}_\infty}\end{matrix}\right\}. \] It is not hard to show that \[ \mathrm{Ed}(\mathscr{T})=\mathrm{Ed}(\mathscr{T})^+\sqcup \mathrm{Ed}(\mathscr{T})^+ \begin{pmatrix} 0 & 1\\ \pi_\infty & 0\end{pmatrix}. \]

As­sume that \( R \) is a ring such that \( p\in R^\times \) and \( R \) con­tains the \( p \)-th roots of unity. A func­tion \( f\in \mathcal{H}(\mathfrak{n}, R) \) has a Four­i­er ex­pan­sion \[ f \left(\begin{pmatrix} \pi_\infty^k & u \\ 0 &1\end{pmatrix}\right) = f^0(\pi_\infty^k)+ \sum_{0\leq j\leq k-2} q^{-k+2+j}\sum_{\substack{\mathfrak{m}\in A\text{, monic}\\ \deg(\mathfrak{m})=j}}f^\ast(\mathfrak{m})\, \nu(\mathfrak{m} u), \] where \( \nu(x):=-1 \) if \( x \) has a term of or­der \( \pi_\infty \) in its \( \pi_\infty \)-ex­pan­sion; \( \nu(x):=q-1 \) oth­er­wise. Here the Four­i­er coef­fi­cients \( f^0(\pi_\infty^k) \) and \( f^\ast(\mathfrak{m}) \) are cer­tain fi­nite sums of val­ues of \( f \) twis­ted by a char­ac­ter \( \chi\colon F_\infty\to \mathbb{C}^\times \) tak­ing val­ues in the \( p \)-th roots of unity. Moreover, the con­stant Four­i­er coef­fi­cient \( f^0(\pi_\infty^k) \) is equal to 0 for cuspid­al har­mon­ic co­chains.

Fol­low­ing Weil, one can at­tach a Di­rich­let \( L \)-series to a cuspid­al har­mon­ic co­chain \( f \in \mathcal{H}_0(\mathfrak{n},\mathbb{C}) \). Put \[ L (f,s) = \sum_{\mathfrak{m}} f^*(\mathfrak{m}) |\mathfrak{m} |^{-s-1}, \] where the sum is over all non­neg­at­ive di­visors on \( F \). This \( L \)-series is defined for \( s\in\mathbb{C} \), is a poly­no­mi­al in \( q^{-s} \), and sat­is­fies a func­tion­al equa­tion when sub­sti­tut­ing \( s \mapsto 2-s \).

Fix a nonzero ideal \( \mathfrak{n}\lhd A \). Giv­en a nonzero ideal \( \mathfrak{m}\lhd A \), define an \( R \)-lin­ear trans­form­a­tion of the space of \( R \)-val­ued func­tions on \( \mathrm{Ed}(\mathscr{T}) \) by \begin{equation}\label{eqDefTm} f|T_\mathfrak{m}:=\sum_{g\in S_{\mathfrak{m}}} f|_g, \tag{4.4} \end{equation} where the set \( S_{\mathfrak{m}} \) has been defined in \eqref{eq-Sm}. This trans­form­a­tion is the Hecke op­er­at­or \( T_\mathfrak{m} \). Fol­low­ing a com­mon con­ven­tion, for a prime di­visor \( \mathfrak{p} \) of \( \mathfrak{n} \) one some­times writes \( U_\mathfrak{p} \) in­stead of \( T_\mathfrak{p} \).

The group-the­or­et­ic proofs of the prop­er­ties of Hecke op­er­at­ors act­ing on clas­sic­al mod­u­lar forms also work in this set­ting. In par­tic­u­lar, the Hecke op­er­at­ors pre­serve \( \mathcal{H}_0(\mathfrak{n}, R) \), and sat­is­fy the re­curs­ive for­mu­las \begin{align*} T_{\mathfrak{m}\mathfrak{m}^{\prime}}&= T_\mathfrak{m} T_{\mathfrak{m}^{\prime}}&\quad \text{if}\quad \mathfrak{m}+\mathfrak{m}^{\prime}=A,\\ T_{\mathfrak{p}^i} &= T_{\mathfrak{p}^{i-1}}T_\mathfrak{p}-|\mathfrak{p}|T_{\mathfrak{p}^{i-2}}\quad \text{if}\quad \mathfrak{p}\nmid \mathfrak{n}, \\ T_{\mathfrak{p}^i} &= T_\mathfrak{p}^i\quad \text{if}\quad \mathfrak{p}| \mathfrak{n}. \end{align*} Let \( \mathbb{T}(\mathfrak{n}) \) be the com­mut­at­ive \( \mathbb{Z} \)-sub­al­gebra of \( \operatorname{End}_\mathbb{Z}(\mathcal{H}_0(\mathfrak{n}, \mathbb{Z})) \) gen­er­ated by all Hecke op­er­at­ors.

An ex­pli­cit for­mula can be giv­en for the ac­tion of \( T_\mathfrak{m} \) on the Four­i­er ex­pan­sion of \( f\in \mathcal{H}_0(\mathfrak{n}, R) \). This for­mula im­plies that \begin{equation}\label{eqfTm} (f|T_\mathfrak{m})^\ast(1)=|\mathfrak{m}|f^\ast(\mathfrak{m}). \tag{4.5} \end{equation} Thus, the pair­ing \begin{equation}\label{eqPairing} \begin{array}{ccc} \mathbb{T}(\mathfrak{n})\times \mathcal{H}_0(\mathfrak{n}, \mathbb{Z}) & \longrightarrow & \mathbb{Z} \\ ( T, f) & \longmapsto & (f|T_\mathfrak{m})^\ast(1) \end{array}\tag{4.6} \end{equation} is nonde­gen­er­ate and be­comes per­fect after tensor­ing with \( \mathbb{Z}[p^{-1}] \).

The ana­logue of the Petersson in­ner product in this set­ting is the pair­ing on \( \mathcal{H}_0(\mathfrak{n}, \mathbb{C}) \) defined by \[ (f, g) =\sum_{e\in \mathrm{Ed}(\Gamma_0(\mathfrak{n})\setminus \mathscr{T})} f(e)\overline{g(e)} \mu(e)^{-1}, \] where \( \mu(e)=\frac{q-1}{2}\# \operatorname{Stab}_{\Gamma_0(\mathfrak{n})}(\tilde{e}) \) and \( \tilde{e} \) is a preim­age of \( e \) in \( \mathscr{T} \). (A Haar meas­ure on \( \mathrm{GL}_2(F_\infty) \) in­duces a push-for­ward meas­ure on \( \mathrm{Ed}(\Gamma_0(\mathfrak{n})\setminus \mathscr{T}) \), which, up to a scal­ar mul­tiple, is equal to \( \mu(e) \).) The Hecke op­er­at­ors \( T_\mathfrak{m} \), \( (\mathfrak{n}, \mathfrak{m})=1 \), are self-ad­joint with re­spect to this pair­ing. Hence, the usu­al con­clu­sions about Hecke op­er­at­ors be­ing di­ag­on­al­iz­able and their ei­gen­val­ues be­ing real are also val­id in this set­ting.

The Hecke op­er­at­ors may also be defined us­ing cor­res­pond­ences on \( X_0(\mathfrak{n}) \). Re­call the mod­uli in­ter­pret­a­tion of \( Y_0(\mathfrak{n}) \): it is the gen­er­ic fiber of the coarse mod­uli scheme for pairs \( (\phi,G_\mathfrak{n}) \) con­sist­ing of a Drin­feld \( A \)-mod­ule \( \phi \) of rank 2 over \( F \) with an \( F \)-ra­tion­al \( A \)-sub­mod­ule \( G_\mathfrak{n} \) of \( \phi[\mathfrak{n}] \) iso­morph­ic to \( A/\mathfrak{n} \) (\( F \)-ra­tion­al means that \( \sigma(G_\mathfrak{n})=G_\mathfrak{n} \) for all \( \sigma\in \operatorname{Gal}(F^{\mathrm{alg}}/F) \)). For \( \mathfrak{m} \lhd A \), the Hecke op­er­at­or \( T_\mathfrak{m} \) is defined as the cor­res­pond­ence on \( Y_0(\mathfrak{n}) \) giv­en by \[ T_\mathfrak{m} : (\phi,G_\mathfrak{n}) \mapsto \sum_{G_\mathfrak{m} \cap G_\mathfrak{n} = \{ 0 \} } (\phi/G_\mathfrak{m} , (G_\mathfrak{n} + G_\mathfrak{m})/G_\mathfrak{m}). \] It uniquely ex­tends to \( X_0(\mathfrak{n}) \). The res­ult­ing cor­res­pond­ence in­duces an en­do­morph­ism of the Jac­obi­an vari­ety \( J_0(\mathfrak{n}) \) of \( X_0(\mathfrak{n}) \), also de­noted \( T_\mathfrak{m} \). The \( \mathbb{Z} \)-sub­al­gebra of \( \operatorname{End}(J_0(\mathfrak{n})) \) gen­er­ated by \( (T_{\mathfrak{m}})_{\mathfrak{m} \lhd A} \) is ca­non­ic­ally iso­morph­ic to \( \mathbb{T}(\mathfrak{n}) \); this is a con­sequence of Drin­feld’s Re­cipro­city Law ([e10], The­or­em 2). Hav­ing this fact, one can use the clas­sic­al Shimura con­struc­tion to as­so­ci­ate to a \( \mathbb{T}(\mathfrak{n}) \)-ei­gen­form \( f\in \mathcal{H}_0(\mathfrak{n}, \mathbb{C}) \) an abeli­an vari­ety \( B_f \) whose num­ber of points over \( \mathbb{F}_\mathfrak{p} \), \( \mathfrak{p}\nmid \mathfrak{n} \), is com­puted us­ing the ei­gen­value of \( T_\mathfrak{p} \) act­ing on \( f \). In par­tic­u­lar, if \( f \) has ra­tion­al ei­gen­val­ues, then \( B_f \) is an el­lipt­ic curve over \( F \). Com­bin­ing this with some deep res­ults of Grothen­dieck and De­ligne, one can de­duce the fol­low­ing ana­logue of the Mod­u­lar­ity The­or­em (see ([e47], (8.3))).

The quo­tient \( \mathbb{T}(\mathfrak{n})/\mathfrak{E}(\mathfrak{n}) \) is a fi­nite ring. In­deed, oth­er­wise the per­fect­ness of the pair­ing \eqref{eqPairing} im­plies that there is a nonzero \( f\in \mathcal{H}_{0}(\mathfrak{n}, \mathbb{Q}) \) an­ni­hil­ated by \( \mathfrak{E}(\mathfrak{n}) \), which con­tra­dicts Weil’s bounds us­ing the Shimura con­struc­tion men­tioned earli­er.

Let \( \phi \) be a Drin­feld \( A \)-mod­ule of rank 2 over \( F \). Its tor­sion \( A \)-mod­ule \( (^\phi F)_\mathrm{tor}:= \bigcup_{a\in A} \phi[a](F) \) is fi­nite, as can be proved, for in­stance, by an ar­gu­ment in­volving the re­duc­tions of \( \phi \); see Re­mark 5.1 (2). Moreover, \( (^\phi F)_\mathrm{tor} \) can be gen­er­ated by two ele­ments \[ (^\phi F)_\mathrm{tor} \cong A/\mathfrak{m} \oplus A/\mathfrak{n}, \] where \( \mathfrak{m} \) and \( \mathfrak{n} \) are nonzero ideals of \( A \) and \( \mathfrak{m} \) di­vides \( \mathfrak{n} \). For rank-2 Drin­feld mod­ules we have a coun­ter­part of Con­jec­ture TEC.

Fol­low­ing Ogg’s philo­sophy on the ex­ist­ence of ra­tion­al points on mod­u­lar curves, we may also for­mu­late the fol­low­ing con­jec­ture which is an ana­logue of TEC\( ^+ \). It is sup­por­ted by the fact that the curve \( X_0(\mathfrak{p}) \) has nonzero genus if and only if \( \deg \mathfrak{p} \geq 3 \) (and this genus is then at least 2), and by the list of all ideals \( \mathfrak{n} \) of \( A \) with \( \deg \mathfrak{n}\geq 3 \) such that \( Y_0(\mathfrak{n}) \) has a \( F \)-ra­tion­al CM point; see [e58].

There is also an ana­logue of Con­jec­ture UBC for Drin­feld mod­ules of ar­bit­rary rank \( r \). If \( r\geq 2 \), it is re­min­is­cent of the uni­form bounded­ness con­jec­ture for the tor­sion of abeli­an vari­et­ies over a num­ber field.

In rank 2, Con­jec­ture UBC-DM is cur­rently open in gen­er­al but pro­gress has been made. Re­call that a first and im­port­ant step for the proof of Con­jec­ture UBC was Man­in’s res­ult [e5], namely a uni­form bound on the \( p \)-primary tor­sion of el­lipt­ic curves over a num­ber field \( L \), de­pend­ing on \( p \) and \( L \). Its ana­logue for Drin­feld \( A \)-mod­ules of rank 2 has been es­tab­lished by Poon­en [e51]. The fol­low­ing the­or­em of Sch­weizer im­proves this res­ult by mak­ing the con­stant de­pend only on the prime of \( A \) and the de­gree of \( L \) (see also [e84] for a gen­er­al­iz­a­tion to more gen­er­al func­tion fields than \( F \)).

As a con­sequence, Con­jec­ture UBC-DM in the rank-2 case is re­duced to the prob­lem of bound­ing uni­formly the num­ber of primes \( \mathfrak{p} \) in the primary de­com­pos­i­tion of the tor­sion, which is equi­val­ent to prov­ing that for any \( d\geq 1 \), there ex­ists a con­stant \( C > 0 \) such that if \( \mathfrak{p} \) is a prime of \( A \) with \( \deg\mathfrak{p} \geq C \) and \( L \) is an ex­ten­sion of \( F \) of de­gree \( \leq d \), then \( Y_1(\mathfrak{p}) \) has no \( L \)-ra­tion­al points.

Re­cently and for ar­bit­rary rank, Ishii proved a ver­sion of Man­in’s uni­form­ity res­ult on the \( \mathfrak{p} \)-primary tor­sion for one-di­men­sion­al fam­il­ies of Drin­feld mod­ules over a fi­nitely gen­er­ated ex­ten­sion of \( F \), which is the ana­logue of a res­ult of Cadoret and Tamagawa [e77] for 1-di­men­sion­al fam­il­ies of abeli­an vari­et­ies.

Be­sides this res­ult, not much is cur­rently known to­wards Con­jec­ture UBC-DM in ranks \( r\geq 3 \).

We make a few com­ments about the ana­logue of Con­jec­ture SUQ. In a series of pa­pers cul­min­at­ing in [e71], Richard Pink and his col­lab­or­at­ors proved the fol­low­ing ana­logue of Serre’s Open Im­age The­or­em.

The ana­logue of Con­jec­ture SUQ is the state­ment that the bound \( N(\phi, L) \) in The­or­em 5.5 can be made uni­form, i.e., in­de­pend­ent of \( \phi \). For \( r=1 \) this not hard to prove, but it re­mains a ma­jor open ques­tion for \( r\geq 2 \).

We now turn to known res­ults on Con­jec­tures TDM and TDM\( ^+ \), some of which have con­sequences for Con­jec­ture UBC-DM. In 2010, Pál made ma­jor pro­gress by prov­ing Con­jec­ture TDM\( ^+ \) for the case \( q=2 \), i.e., the field \( F = \mathbb{F}_2(T) \).

The pre­vi­ous the­or­em im­plies that Con­jec­ture TDM and Con­jec­ture UBC-DM hold for \( q = 2 \) and \( L = F = \mathbb{F}_2(T) \); see The­or­ems 1.4 and 1.6 in [e72]. In a dif­fer­ent dir­ec­tion and more re­cently, Ishii made more pro­gress to­wards Con­jec­ture TDM\( ^+ \) for ar­bit­rary \( q \) and levels of small de­gree.

For Con­jec­ture TDM, Sch­weizer proved that we al­ways have \( \deg \mathfrak{m} \leq 1 \) in \( (^\phi F)_\mathrm{tor}\cong A/\mathfrak{m}\oplus A/\mathfrak{n} \); see Pro­pos­i­tion 4.4 in [e58]. Moreover since any \( F \)-ra­tion­al point on \( Y_1(\mathfrak{p}) \) nat­ur­ally provides an \( F \)-ra­tion­al point on \( Y_0(\mathfrak{p}) \), The­or­ems 5.7 and 5.8 re­main val­id for the curve \( Y_1(\mathfrak{p}) \).

For fi­nite ex­ten­sions of \( F \), Ar­mana has ob­tained par­tial or con­di­tion­al res­ults to­wards Con­jec­tures TDM and UBC-DM. The first one fo­cuses on levels of small de­gree.

Giv­en a prime \( \mathfrak{p} \), the as­sump­tion in (iii) on the van­ish­ing or­der of \( L \)-series at the cent­ral value is not al­ways sat­is­fied. However, if one be­lieves in the philo­sophy that el­lipt­ic curves over \( \mathbb{Q} \) with rank great­er than 1 are ex­pec­ted to be rare and in its coun­ter­part for el­lipt­ic curves over \( F \), we should get many ex­amples of pairs \( (q,\mathfrak{p} \)) to which state­ment (iii) ap­plies. For in­stance it ap­plies to curves \( Y_1(\mathfrak{p}) \) for all prime levels \( \mathfrak{p} \) of de­gree 5 in \( \mathbb{F}_2[T] \) with \( L=F \), and for \( \mathfrak{p}=T^5-T^4-T^2-1 \in \mathbb{F}_3[T] \) with \( [L:F]\leq 3 \). There is also a sim­il­ar res­ult for \( F \)-ra­tion­al points on \( Y_0(\mathfrak{p}) \) when \( q\geq 5 \), un­der the same hy­po­thes­is as in (iii); see ([e78], The­or­em 7.8).

The second res­ult is con­di­tion­al but with a more gen­er­al con­clu­sion to­wards Con­jec­ture TDM. To state it we need some nota­tion. Sim­il­arly to clas­sic­al mod­u­lar forms, it is pos­sible to de­vel­op a the­ory of al­geb­ra­ic Drin­feld mod­u­lar forms of weight 2 us­ing sec­tions of the sheaf of re­l­at­ive dif­fer­en­tials on \( X_0(\mathfrak{p}) \). Let \( \mathcal{M}_{\mathfrak{p}} \) be the \( A[1/\mathfrak{p}] \)-mod­ule of doubly cuspid­al al­geb­ra­ic Drin­feld mod­u­lar forms of weight 2 and type 1 for \( \Gamma_0(\mathfrak{p}) \) and let \( \mathbb{T}_{\mathfrak{p}} \) be the Hecke al­gebra gen­er­ated by all Hecke op­er­at­ors act­ing on \( \mathcal{M}_{\mathfrak{p}} \). For a prime \( \mathfrak{l} \) of \( A \), let \( \mathbb{F}_\mathfrak{l} = A/\mathfrak{l} \) and \( \mathcal{M}_{\mathfrak{p}}(\mathbb{F}_\mathfrak{l}) = \mathcal{M}_\mathfrak{p}\otimes_{A[1/\mathfrak{p}]} \mathbb{F}_\mathfrak{l} \). By ex­tend­ing the res­ults of Sec­tion 4.1, any \( f \in \mathcal{M}_{\mathfrak{p}}(\mathbb{F}_\mathfrak{l}) \) has an ex­pan­sion at the cusp \( [\infty] \) of the form \( \sum_{i\geq 0} b_i(f) t^{1+i(q-1)} \) with coef­fi­cients \( b_i(f) \in \mathbb{F}_\mathfrak{l} \). In the Hecke al­gebra \( \mathbb{T}_{\mathfrak{p}} \), let \( I_e \) (resp. \( \widetilde{I_e} \)) be the the ideal which an­ni­hil­ates the ana­logue of the wind­ing ele­ment (resp. the wind­ing ele­ment mod­ulo the char­ac­ter­ist­ic of \( F \)); see [e78] for the defin­i­tions. Let \( \mathcal{M}_{\mathfrak{p}}(\mathbb{F}_\mathfrak{l})[\widetilde{I_e}] \) be the sub­space of \( \mathcal{M}_{\mathfrak{p}}(\mathbb{F}_\mathfrak{l}) \) an­ni­hil­ated by \( \widetilde{I_e} \).

The as­sump­tion in (i) on the ex­ist­ence of the ideal \( I \) is likely of a tech­nic­al nature. The as­sump­tion in (ii) is deep­er: it is re­lated to com­plic­a­tions that arise when ad­apt­ing Mazur’s form­al im­mer­sion ar­gu­ment to the Drin­feld set­ting. This will be dis­cussed in Sec­tion 5.2.

We re­view the main ideas be­hind the proofs of The­or­ems 5.3, 5.7, 5.8, 5.11 and 5.12, which are mostly in­spired by the work of Man­in and Mazur, with a fo­cus on the dif­fer­ences from the clas­sic­al set­ting and com­plic­a­tions that arise.

Sch­weizer’s uni­form bound for the \( \mathfrak{p} \)-primary tor­sion, The­or­em 5.3, is an ana­logue of a the­or­em of Kami­enny and Mazur [e36], it­self a stronger ver­sion of Man­in’s the­or­em [e5] for the \( p \)-primary part of the tor­sion of el­lipt­ic curves over \( \mathbb{Q} \). The proof of Sch­weizer, after Poon­en [e51], fol­lows the same ap­proach. The key in­gredi­ent es­sen­tially states that for any curve \( C \) over a glob­al field \( K \), the \( d \)-fold sym­met­ric power \( C^{(d)} \) has only fi­nitely many \( K \)-ra­tion­al points if \( C \) does not ad­mit a \( K \)-ra­tion­al cov­er­ing of \( \mathbb{P}^1_{K} \) of de­gree \( \leq 2d \). When \( K \) is a num­ber field, this is a the­or­em of Frey which de­rives from Falt­ings’ the­or­em on ra­tion­al points of sub­vari­et­ies of abeli­an vari­et­ies. When \( K \) is a func­tion field, Sch­weizer has ob­tained a sim­il­ar cri­terion from the Mor­dell–Lang con­jec­ture for abeli­an vari­et­ies over func­tion fields, proved by Hrushovski [e46]. He ap­plied it to the Drin­feld mod­u­lar curves \( X_0(\mathfrak{p}^e) \) for primes \( \mathfrak{p} \) to ob­tain The­or­em 5.3.

To dis­cuss the proofs of The­or­ems 5.7, 5.8, 5.11 and 5.12, we need to re­call Mazur’s ap­proach from [e14] for the study of \( \mathbb{Q} \)-ra­tion­al points on the clas­sic­al mod­u­lar curve \( X_0(p) \), \( p \) prime (see also [e106] in this volume and the over­view [e39] for ad­di­tion­al de­tails). Hand­ling points defined over a num­ber field \( K \) of de­gree \( d \) re­quires Kami­enny’s gen­er­al­ized ap­proach us­ing the \( d \)-fold sym­met­ric power of \( X_0(p) \). For sim­pli­city we fo­cus here only on \( d=1 \).

Let \( P=(E,C_p) \in X_0(p)(\mathbb{Q}) \) be a non­cuspid­al point for a prime num­ber \( p \). As­sume that the genus of \( X_0(p) \) is pos­it­ive. The ob­ject­ive of Mazur’s form­al im­mer­sion ar­gu­ment is to prove that the el­lipt­ic curve \( E \) has po­ten­tially good re­duc­tion at all primes \( \ell\neq \{2, p\} \); see Co­rol­lary 4.4 in [e14]. Con­sider the Abel–Jac­obi map \[ \begin{array}{rcl} X_0(p) & \longrightarrow & J_0(p) \\ Q & \longmapsto & (Q)-(\infty)\end{array} \] where \( J_0(p) \) is the Jac­obi­an vari­ety of \( X_0(p) \). The main in­gredi­ent is an op­tim­al quo­tient \( A \) of \( J_0(p) \) defined over \( \mathbb{Q} \) which sat­is­fies the fol­low­ing two prop­er­ties:

1. The Mor­dell–Weil group \( A(\mathbb{Q}) \) is fi­nite.
2. Let \( \varphi:X_0(p) \to J_0(p) \to A \), which ex­tends over \( \mathbb{Z} \) to \( \varphi:X_0(p)_{\mathrm{sm}} \to \mathcal{A} \), where \( X_0(p)_{\mathrm{sm}} \) is the largest open sub­set of \( X_0(p) \) smooth over \( \mathbb{Z} \) and \( \mathcal{A} \) is the Néron mod­el of \( A \) over \( \mathbb{Z} \). When look­ing at the fibers at any prime \( \ell \) such that \( \ell \nmid 2p \), the map \( \varphi_{\ell} : X_0(p)_{\mathbb{F}_\ell} \to \mathcal{A}_{\mathbb{F}_\ell} \) is a form­al im­mer­sion along the cuspid­al sec­tion \( [\infty] \).

Op­tim­al quo­tients sat­is­fy­ing (1) ex­ist: the idea is to con­struct them with the prop­erty that their Hasse–Weil \( L \)-func­tion does not van­ish at the cent­ral value and de­duce that their Mor­dell–Weil group is fi­nite, in the spir­it of the Birch and Swin­ner­ton-Dyer con­jec­ture. For \( A \), one may take Mazur’s Ei­s­en­stein quo­tient [e13] or Mer­el’s wind­ing quo­tient [e45] (the lat­ter one be­ing the largest quo­tient of \( J_0(p) \) sat­is­fy­ing this prop­erty).

Con­di­tion (2) is equi­val­ent to the sur­jectiv­ity of the in­duced map \( \varphi_{\ell}^*:\mathrm{Cot}_0(A_{\mathbb{F}_\mathfrak{l}}) \to \mathrm{Cot}_\infty(X_0(p)_{\mathbb{F}_\ell}) \) on the co­tan­gent spaces, which in this case is equi­val­ent to \( \varphi_{\ell}^* \) be­ing nonzero. It is in­struct­ive to re­for­mu­late this con­di­tion for the Abel–Jac­obi map \( X_0(p)_{\mathrm{sm}} \to \mathcal{J}_0(p) \) over \( \mathbb{Z} \) in terms of \( q \)-ex­pan­sions of cusps forms: on the co­tan­gent spaces at \( [\infty] \), it is simply the map \[ \begin{array}{ccc} S_2(\Gamma_0(p),\mathbb{Z}) & \longrightarrow & \mathbb{Z} \\ \sum_{n\geq 1} a_n q^n & \longmapsto & a_1 \end{array} \] where \( S_2(\Gamma_0(p),\mathbb{Z}) \) is the space of cusp forms of weight 2 with coef­fi­cients in \( \mathbb{Z} \). This map is nonzero, as a con­sequence of the clas­sic­al for­mula \begin{equation}\label{eq-a1Tnf} a_n(f)=a_1(f|T_n),\quad \text{for all } n\geq 1, \tag{5.1} \end{equation} for the ac­tion of the Hecke op­er­at­ors \( (T_n)_{n\geq 1} \) on cusp forms. For the map \( \varphi_{\ell}^* \), there is a sim­il­ar de­scrip­tion in­volving cusp forms an­ni­hil­ated by the Ei­s­en­stein ideal, resp. the wind­ing ideal of the Hecke al­gebra. One then uses a Hecke ei­gen­vector in \( \mathrm{Cot}_0(A_{\mathbb{F}_\mathfrak{l}}) \), which ne­ces­sar­ily has \( a_1\neq 0 \). This en­sures Con­di­tion (2) is sat­is­fied.

Us­ing (1) and (2), Mazur was able to prove that the el­lipt­ic curve \( E \) ne­ces­sar­ily has po­ten­tially good re­duc­tion at all primes \( \ell \neq \{2,p\} \) as fol­lows. If \( E \) has po­ten­tially mul­ti­plic­at­ive re­duc­tion at \( \ell \), the point \( P \) will spe­cial­ize on \( X_0(p) \) at \( \ell \) to one of the cusps. By ap­ply­ing the Atkin–Lehner in­vol­u­tion \( w_p \), we may as­sume that it spe­cial­izes to \( [\infty] \), there­fore the point \( \varphi(P) \) spe­cial­izes to 0 at \( \ell \). By the form­al im­mer­sion prop­erty (2) for \( \varphi \) at \( \ell \), we also have \( \varphi(P)\neq 0 \). By (1) we also know that \( \varphi(P) \) is a tor­sion point. This con­tra­dicts a spe­cial­iz­a­tion lemma for points of fi­nite or­der in group schemes.

The last part of Mazur’s proofs of Con­jec­tures TEC and TEC\( ^+ \) is to dis­card the cases where \( E \) does not have po­ten­tially good re­duc­tion at \( \ell \) when \( p \) is large enough. If the point \( P \) on \( X_0(p) \) comes from a tor­sion point of or­der \( p \) on \( E \), this fol­lows from known bounds for the or­der of the spe­cial­iz­a­tion of a tor­sion point at a prime \( \ell \) with \( p \neq \ell \), when \( E \) has good or ad­dit­ive re­duc­tion at \( \ell \). If \( P \) comes from a ra­tion­al cyc­lic sub­group \( C_p \) of or­der \( p \), Mazur stud­ies the iso­geny char­ac­ter, i.e., the rep­res­ent­a­tion \( \operatorname{Gal}(\mathbb{Q}^\mathrm{alg}/\mathbb{Q}) \to \mathrm{GL}_1(\mathbb{Z}/p\mathbb{Z}) \) com­ing from the nat­ur­al Galois ac­tion on \( C_p \). Con­straints com­ing from the ex­ist­ence of this iso­geny char­ac­ter with the Riemann hy­po­thes­is for the re­duc­tion of \( E \) mod­ulo \( \ell \notin\{ 2,p \} \) ul­ti­mately provide a fi­nite list of pos­sible val­ues for \( p \).

In this strategy, most of the steps ad­apt more or less eas­ily to the Drin­feld mod­u­lar curve \( X_0(\mathfrak{p}) \). As an op­tim­al quo­tient \( A \) of the Jac­obi­an \( J_0(\mathfrak{p}) \), one may take the Ei­s­en­stein quo­tient in­tro­duced by Tamagawa us­ing the Ei­s­en­stein ideal \( \mathfrak{E}(\mathfrak{p}) \), see Defin­i­tion 4.8. An­oth­er pos­sib­il­ity is the wind­ing quo­tient of [e72] or [e78]. In both cases their Mor­dell–Weil groups are fi­nite, by Schneider’s in­equal­ity in the Birch and Swin­ner­ton-Dyer con­jec­ture for abeli­an vari­et­ies over func­tion fields [e20], so there is an ana­logue of con­di­tion (1). The main obstacle, however, resides in the func­tion field coun­ter­part of the form­al im­mer­sion prop­erty in con­di­tion (2). In­deed, the ac­tion of Hecke op­er­at­ors on the ex­pan­sion of Drin­feld mod­u­lar forms is not well un­der­stood and a for­mula sim­il­ar to \eqref{eq-a1Tnf} is lack­ing (on this top­ic see also [e75]). This is the reas­on why The­or­ems 5.7, 5.8, 5.11 and 5.12 do not provide com­plete an­swers to the ana­logues of Ogg’s con­jec­tures. This also ex­plains as­sump­tion (ii) in The­or­em 5.12: if one re­moves the ideal \( \widetilde{I_e} \) and the prime place \( \mathfrak{l} \) from its for­mu­la­tion, the as­sump­tion be­comes equi­val­ent to the per­fect­ness of the \( \mathbb{C}_\infty \)-pair­ing between the space of Drin­feld mod­u­lar forms \( M_{2,1}^{0,0}(\mathfrak{n}) \) and its Hecke al­gebra, defined by \( (f,u) \mapsto b_1(f|u) \). If true, it would im­ply a “mul­ti­pli­city one” res­ult in \( M_{2,1}^{0,0}(\mathfrak{n}) \), which at the mo­ment is open in gen­er­al. (It is known to fail for oth­er con­gru­ence sub­groups; see Böckle ([e56], Ex­ample 15.4) for \( \Gamma_1(T) \) and Hat­tori ([e104], The­or­em 1) for \( \Gamma_1(T^n) \). These are strong in­dic­a­tions that mul­ti­pli­city one could fail for \( \Gamma_0(\mathfrak{n}) \).) Moreover Ar­mana has re­cently es­tab­lished that this pair­ing is not per­fect in a quite gen­er­al case, namely when \( \mathfrak{n} \) is prime of de­gree \( \geq 5 \); see [e111]. Al­though it is not enough to show that the ver­sion of this per­fect­ness stated in as­sump­tion (ii) is not sat­is­fied, it con­firms the sever­ity of the ob­struc­tion that arises when ad­apt­ing Mazur’s meth­od.

Be­cause of this ob­struc­tion, the un­con­di­tion­al state­ments that we cur­rently know, such as The­or­ems 5.7, 5.8 and 5.11, em­ploy work­arounds or vari­ants of the form­al im­mer­sion prop­erty at \( \ell \). They are es­sen­tial of two types:

• Fol­low­ing an idea of Mer­el and Par­ent for clas­sic­al mod­u­lar curves, Pál es­tab­lishes a vari­ant of the form­al im­mer­sion prop­erty at the place \( \infty \) of \( F \), in­stead of a fi­nite place \( \mathfrak{l} \) of \( A \); see Pro­pos­i­tion 7.14 in [e72]. To prove this vari­ant, he con­structs a reg­u­lar mod­el of the curve \( X_0(\mathfrak{p}) \) at \( \infty \), uses the in­cid­ence graph of the fiber of this mod­el and har­mon­ic co­chains on this graph. The study of iso­geny char­ac­ters is re­placed with an ad hoc ar­gu­ment. Pál’s strategy uses ex­tens­ively the hy­po­thes­is \( q=2 \) and ul­ti­mately provides The­or­em 5.7.
• If \( \mathfrak{p} \) has de­gree 3 or 4, or if the prime \( \mathfrak{p} \) is such that for any nor­mal­ized Hecke ei­gen­form \( f \in \mathcal{H}_{0}(\mathfrak{p},\mathbb{C}) \), we have \( \operatorname{ord}_{s=1}L(f,s) \leq 1 \), the situ­ation be­comes sim­pler. The wind­ing quo­tient is then \( J_0(\mathfrak{p}) \) or iso­gen­ous to \( J_0(\mathfrak{p})/(1+w_\mathfrak{p}) J_0(\mathfrak{p}) \) where \( w_\mathfrak{p} \) de­notes the Atkin–Lehner in­vol­u­tion. In this situ­ation, Ar­mana is able to avoid the form­al im­mer­sion prop­erty thanks to the fact that the cusp \( [\infty] \) is not a Wei­er­strass point on \( X_0(\mathfrak{p}) \) and to a lower bound of Sch­weizer on the gon­al­ity of \( X_0(\mathfrak{p}) \); see ([e78], Pro­pos­i­tion 7.6). These ar­gu­ments lead to The­or­em 5.11. Ishii’s work [e102], which is a gen­er­al study of iso­geny char­ac­ters com­ing from rank-2 Drin­feld mod­ules as in Mazur [e14], is also based on this work­around. Us­ing con­gru­ences com­ing from these char­ac­ters, he ob­tains sev­er­al fam­il­ies of con­di­tions which en­sure that \( Y_0(\mathfrak{p}) \) has no \( F \)-ra­tion­al points; see ([e102], The­or­ems 0.1, 0.2, 0.3). The con­di­tions he com­bines are of three dif­fer­ent types: on \( q \), on the prime \( \mathfrak{p} \) and on the set of fi­nite places where the Drin­feld mod­ule has po­ten­tially good re­duc­tion. Here again, it is not pos­sible to reach the full po­ten­tial of Mazur’s ap­proach be­cause the form­al im­mer­sion ar­gu­ment is miss­ing in gen­er­al. Ishii ob­tains The­or­em 5.8 for ra­tion­al points on \( Y_0(\mathfrak{p}) \) if \( \mathfrak{p} \) has de­gree 4 by com­bin­ing his work on iso­geny char­ac­ters and Ar­mana’s re­place­ment for the form­al im­mer­sion prop­erty in this case.

For a nonzero ideal \( \mathfrak{n}\lhd A \), let \( X_0(\mathfrak{n}) \) de­note the Drin­feld mod­u­lar curve in­tro­duced in Sec­tion 3. Its set of cusps is \( X_0(\mathfrak{n})(\mathbb{C}_\infty)-Y_0(\mathfrak{n})(\mathbb{C}_\infty) \). Let \( J_0(\mathfrak{n}) \) be the Jac­obi­an vari­ety of \( X_0(\mathfrak{n}) \). Sim­il­ar to \( \mathcal{C}_N \) ap­pear­ing in Con­jec­ture CJ-N, one defines the cuspid­al di­visor group \( \mathcal{C}_{\mathfrak{n}} \) as the sub­group of \( J_0(\mathfrak{n}) \) gen­er­ated by the di­visor classes \( [c]-[c^{\prime}] \) of dif­fer­ences of all cusps.

One in­ter­est­ing fea­ture of the the­ory of Drin­feld mod­ules is that Drin­feld mod­ules of rank \( r\geq 3 \) do not really have clas­sic­al ana­logues (Drin­feld mod­ules of rank 1 are sim­il­ar to the mul­ti­plic­at­ive group of a field and Drin­feld mod­ules of rank 2 are sim­il­ar to el­lipt­ic curves). Such Drin­feld mod­ules can be defined and stud­ied us­ing a single equa­tion \( \phi_T \) like el­lipt­ic curves, but the Galois rep­res­ent­a­tions arising from their tor­sion points have im­ages in \( \mathrm{GL}_r \) and their mod­u­lar vari­et­ies are \( (r-1) \)-di­men­sion­al. Hence, Drin­feld mod­ules of rank \( r\geq 3 \) are more com­plic­ated than el­lipt­ic curves but sim­pler than abeli­an vari­et­ies. This presents the in­triguing pos­sib­il­ity of prov­ing res­ults for these Drin­feld mod­ules which are known for el­lipt­ic curves but per­haps un­known or very hard for abeli­an vari­et­ies. One such res­ult is the fi­nite­ness of the cuspid­al di­visor group of their mod­u­lar vari­et­ies.

For \( r\geq 3 \), there are dif­fer­ent pos­sible com­pac­ti­fic­a­tions of \( Y_0^r(\mathfrak{n}) \) with de­sir­able prop­er­ties; see [e28] and [e96]. The Satake com­pac­ti­fic­a­tions of Drin­feld mod­u­lar vari­et­ies were con­struc­ted (at dif­fer­ent levels of gen­er­al­ity and de­tails of proof) by Gekel­er [e27], [e101], Kapran­ov [e28], Pink [e80], and Häberli [e100]. The con­struc­tions by Gekel­er, Häberli, and Kapran­ov are ri­gid-ana­lyt­ic, where­as Pink’s con­struc­tion is al­gebro-geo­met­ric. Häberli also proved that the ana­lyt­ic and al­geb­ra­ic Satake com­pac­ti­fic­a­tions give the same vari­ety. The Satake com­pac­ti­fic­a­tion can be de­scribed ana­lyt­ic­ally as fol­lows. For \[ \boldsymbol{z}=(z_1, \dots, z_r)\in \mathbb{P}^{r-1}(\mathbb{C}_\infty), \] define \[ d(\boldsymbol{z})=\dim_F(Fz_1+\cdots+Fz_r) \quad \text{and}\quad d_\infty(\boldsymbol{z})=\dim_{F_\infty}(F_\infty z_1+\cdots+F_\infty z_r). \] Then \[ 1\leq d_\infty(\boldsymbol{z})\leq d(\boldsymbol{z})\leq r\quad\text{ and }\quad \Omega^r=\{\boldsymbol{z}\mid d_\infty(\boldsymbol{z})=r\}. \] More gen­er­ally, for \( 1\leq i\leq r \), put \[ \Omega^{i, r} = \{\boldsymbol{z}\mid d_\infty(\boldsymbol{z})=d(\boldsymbol{z})=i\} \quad\text{ and }\quad \overline{\Omega^r}=\cup_{1\leq i\leq r} \Omega^{i, r}. \] The space \( \overline{\Omega^r} \) is in­vari­ant un­der the ac­tion of \( \Gamma_0(\mathfrak{n}) \) by lin­ear frac­tion­al trans­form­a­tions. The quo­tient \( X_0^r(\mathfrak{n})(\mathbb{C}_\infty)=\Gamma_0(\mathfrak{n})\setminus \overline{\Omega^r} \) is a pro­ject­ive con­nec­ted nor­mal vari­ety over \( \mathbb{C}_\infty \) of di­men­sion \( r-1 \) con­tain­ing \( Y_0^r(\mathfrak{n}) \) as an open sub­vari­ety, which has a ca­non­ic­al mod­el over \( F \). The cusps of \( X_0^r(\mathfrak{n}) \) are the (geo­met­ric­ally) ir­re­du­cible com­pon­ents of \( X_0^r(\mathfrak{n})(\mathbb{C}_\infty)-Y_0^r(\mathfrak{n})(\mathbb{C}_\infty) \) of di­men­sion \( r-2 \). The cuspid­al di­visor group \( \mathcal{C}^r_\mathfrak{n} \) of \( X^r_0(\mathfrak{n}) \) is the sub­group of the di­visor class group of \( X^r_0(\mathfrak{n}) \) gen­er­ated by the Weil di­visors \( [c]-[c^{\prime}] \), where \( c \) and \( c^{\prime} \) run over the cusps of \( X^r_0(\mathfrak{n}) \). The ana­logue of the res­ult of Man­in and Drin­feld in this set­ting is the fol­low­ing.

Proof. For \( r=2 \), this was proved by Gekel­er in [e23]. For \( r\geq 3 \) this is a res­ult of Kapran­ov [e28]; see also ([e112], The­or­em 7.8) and ([e117], The­or­em 10.7). The gen­er­al idea of the proof is to con­struct suf­fi­ciently many mod­u­lar units on \( X_0^r(\mathfrak{n}) \), i.e., func­tions whose di­visors are sup­por­ted on the cusps, so that the sub­group of de­gree-0 di­visors of these func­tions has fi­nite in­dex in the group gen­er­ated by all \( [c]-[c^{\prime}] \). A nat­ur­al meth­od for con­struct­ing mod­u­lar units uses the Drin­feld dis­crim­in­ant func­tions \( \Delta_{r}(z_1, \dots, z_r) \). Re­call that \( \Delta_{r} \) does not van­ish on \( \Omega^r \). One eas­ily shows that \( \Delta_{r}(z_1, \dots, z_r)/\Delta_{r}(\mathfrak{m} z_1, \dots, z_r) \) is in­vari­ant un­der \( \Gamma_0^r(\mathfrak{n}) \) for any \( \mathfrak{m}\mid \mathfrak{n} \), and hence that it defines a mod­u­lar unit on \( X_0^r(\mathfrak{n}) \) whose di­visor can be com­puted from the or­der of van­ish­ing of \( \Delta_{r}(\boldsymbol{z}) \) at the cusps. en­d­proof

Com­put­ing the group struc­ture of \( \mathcal{C}_{\mathfrak{n}}^r \) is much harder than prov­ing that this group is fi­nite. For clas­sic­al mod­u­lar curves an im­port­ant tool for solv­ing this prob­lem is a res­ult of Ligoz­at ([e12], Pro­pos­i­tion 3.2.1), which gives ne­ces­sary and suf­fi­cient con­di­tions for a func­tion of the form \( \prod_{m\mid N}\eta(m z)^{s_m} \), \( s_m\in \mathbb{Z} \), to be a mod­u­lar unit on \( X_0(N) \), where \( \eta(z) \) is the Dede­kind eta func­tion. (Re­call that \( \eta(z) \) is the 24th root of the clas­sic­al dis­crim­in­ant func­tion.) However, such a strong res­ult does not seem to hold over func­tion fields; for ex­ample, \( \Delta_r \) only has a \( (q-1) \)-th root in \( \mathcal{O}(\Omega^r)^\times \), but the ana­logue of 24 in this con­text is \( (q-1)(q^2-1) \).

In­stead, one fol­lows a dif­fer­ent strategy, spe­cif­ic to func­tion fields, which was ini­ti­ated by Gekel­er in [e49]. The key res­ult here is the ex­act se­quence (4.3). Giv­en a mod­u­lar unit \( u(\boldsymbol{z}) \) on \( X_0^r(\mathfrak{n}) \), to de­term­ine the max­im­al root that can be ex­trac­ted from \( u(\boldsymbol{z}) \) (and thus to de­term­ine the or­der of the un­der­ly­ing cuspid­al di­visor) one ap­plies \( \mathrm{dlog} \) to \( u(\boldsymbol{z}) \) to get a com­bin­at­or­i­al \( \mathbb{Z} \)-val­ued func­tion \( \mathrm{dlog}(u)\in \operatorname{Har}^1(\mathscr{B}^r, \mathbb{Z}) \). If one is able to com­pute the value of \( \mathrm{dlog}(u) \) on a few well-chosen edges of \( \mathscr{B}^r \), then \( u \) has an \( m \)-th root only if \( m \) di­vides the gcd of these val­ues. Gekel­er util­ized this strategy in [e49] to com­pute the or­der of the di­visor \( [0] - [\infty] \) on \( X_0^2(\mathfrak{n}) \), where \( \mathfrak{n} \) is either a prime or a square of a prime. Since \( [0] \) and \( [\infty] \) are the only cusps when \( \mathfrak{n}=\mathfrak{p} \) is prime, Gekel­er’s res­ult im­plies that \( \mathcal{C}^2_{\mathfrak{p}} \) is cyc­lic of or­der \begin{equation}\label{eqGekCp} \frac{\lvert\mathfrak{p}\rvert-1}{\gcd(q^2-1, \lvert\mathfrak{p}\rvert-1)}. \tag{6.1} \end{equation} Ho [e114] ex­ten­ded Gekel­er’s com­pu­ta­tion to ar­bit­rary prime power levels \( \mathfrak{p}^s \). We note that the for­mula for the or­der of \( [0] - [\infty] \) on \( X_0^2(\mathfrak{p}^s) \) when \( s \geq 3 \) does not spe­cial­ize to the cases \( s = 1 \) or \( s = 2 \).

In high­er ranks the only res­ult so far is the fol­low­ing gen­er­al­iz­a­tion of \eqref{eqGekCp} proved in [e112]: \( \mathcal{C}^r_{\mathfrak{p}} \) is cyc­lic of or­der \begin{equation}\label{eqPapWei} \frac{\lvert\mathfrak{p}\rvert^{r-1}-1}{\gcd(q^r-1, \lvert\mathfrak{p}\rvert-1)}. \tag{6.2} \end{equation}

In ana­logy with \( \mathcal{C}(N) \), define the ra­tion­al cuspid­al di­visor class group \( \mathcal{C}(\mathfrak{n}) := \mathcal{C}^2(\mathfrak{n}) \) of \( X_0(\mathfrak{n}) \) to be the sub­group of \( J_0(\mathfrak{n}) \) gen­er­ated by the lin­ear equi­val­ence classes of the de­gree 0 ra­tion­al cuspid­al di­visors on \( X_0(\mathfrak{n}) \). The group \( \mathcal{C}(\mathfrak{n}) \) is ex­pli­citly com­puted in the fol­low­ing cases:

• \( \deg(\mathfrak{n})=3 \); see [e87].
• \( \mathfrak{n}=\mathfrak{p}_1\mathfrak{p}_2 \) is a product of two dis­tinct primes; see ([e85], Sec­tion 6).
• \( \mathfrak{n}=\mathfrak{p}_1\cdots \mathfrak{p}_s \) is square-free, but \( \mathcal{C}(\mathfrak{n})_\ell \) is de­term­ined only for \( \ell\nmid q-1 \); see [e90]. In this case, \( \mathcal{C}(\mathfrak{n})_p=0 \), where \( p \) is the char­ac­ter­ist­ic of \( F \). Moreover, \( \mathcal{C}(\mathfrak{n})_\ell \) nat­ur­ally de­com­poses in­to a dir­ect sum of \( 2^{s-1} \) cyc­lic sub­groups each of which is an ei­gen­space with re­spect to the Atkin–Lehner in­vol­u­tions of \( J_0(\mathfrak{n}) \).
• \( \mathfrak{n}=\mathfrak{p}^s \) is a prime power; see [e114]. An in­ter­est­ing fact in this case is that \( \mathcal{C}(\mathfrak{n}) \) is “mostly” \( p \)-primary.

Let \( J_0(\mathfrak{n})(F)_\mathrm{tor} \) be the ra­tion­al tor­sion sub­group of the Jac­obi­an vari­ety \( J_0(\mathfrak{n}) \). By the Lang–Néron the­or­em, \( J_0(\mathfrak{n})(F)_\mathrm{tor} \) is a fi­nite abeli­an group. As earli­er, let \( \mathcal{C}_{\mathfrak{n}} \) be the cuspid­al sub­group of \( J_0(\mathfrak{n}) \), \( \mathcal{C}_{\mathfrak{n}}(F) \) be the sub­group of ra­tion­al points on \( \mathcal{C}_{\mathfrak{n}} \), and \( \mathcal{C}(\mathfrak{n}) \) be the ra­tion­al cuspid­al di­visor class group of \( X_0(\mathfrak{n}) \). We have \[ \mathcal{C}(\mathfrak{n}) \subseteq \mathcal{C}_{\mathfrak{n}}(F) \subseteq J_0(\mathfrak{n})(F)_\mathrm{tor}. \] The ana­logue of Con­jec­ture CJ-N in this set­ting is the fol­low­ing.

There is a nat­ur­al morph­ism \( X_1(\mathfrak{n})\to X_0(\mathfrak{n}) \), which, by Pi­card func­tori­al­ity, in­duces a morph­ism \( J_0(\mathfrak{n})\to J_1(\mathfrak{n}) \). The ker­nel of this morph­ism, de­noted \( \mathcal{S}(\mathfrak{n}) \), is fi­nite; see ([e85], Sec­tion 8). This is the Shimura sub­group in this con­text. Moreover, when \( \mathfrak{n} \) is square-free, \( \mathcal{S}(\mathfrak{n}) \) is a \( \mu \)-type étale group-scheme over \( F \), so its or­der is coprime to \( p \). For the proof of this claim, as well as the cal­cu­la­tion of the group struc­ture of \( \mathcal{S}(\mathfrak{n}) \), we refer to ([e85], Sec­tion 8). The ana­logue of Con­jec­ture SJ-p is the fol­low­ing.

Con­jec­ture CJD-\( \mathfrak{n} \) for prime \( \mathfrak{n}=\mathfrak{p} \), as well as Con­jec­ture SJD-\( \mathfrak{p} \), were proved by Pál in [e66]. Over­all, Pál’s ap­proach is modeled on Mazur’s, but with some in­ter­est­ing dif­fer­ences that we will ex­plain be­low. Com­bin­ing Pál’s ap­proach with some ideas from [e83] and [e110], one can prove par­tial res­ults to­wards Con­jec­ture CJD-\( \mathfrak{n} \):

• When \( \mathfrak{n} \) is square-free, we have \( \mathcal{C}(\mathfrak{n})_{\ell} = (J_0(\mathfrak{n})(F)_\mathrm{tor})_{\ell} \) for any prime \( \ell\nmid q(q-1) \); see [e90].
• When \( \mathfrak{n}=\mathfrak{p}^s \) is a prime power, we have \( \mathcal{C}(\mathfrak{p}^s)_{\ell} = (J_0(\mathfrak{p}^s)(F)_\mathrm{tor})_{\ell} \) for any prime \( \ell\nmid q(q-1) \); see [e115].