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Jacobians of Drinfeld modular curves. (English) Zbl 0848.11029
Similar to elliptic modular curves, Drinfeld modular curves are analytically constructed as quotients of the Drinfeld upper half-plane \(\Omega\) by arithmetic subgroups \(\Gamma\) of \(\text{GL}(2, A)\). Here \(A= {\mathcal O}_{\mathcal C}({\mathcal C}- \{\infty\})\) is the affine ring of a smooth complete geometrically connected curve \({\mathcal C}/ \mathbb F_q\) minus a closed point \(\infty\in {\mathcal C}\) and \(\Omega= C- K_\infty\), where \(K_\infty=\) completion of \(K= \text{Quot}(A)\) and \(C=\) completed algebraic closure of \(K_\infty\). There is a well-known formal analogy between the data \(\mathbb Z\), \(\mathbb Q\), \(\mathbb R\), \(\mathbb C\), \(H=\) complex upper half-plane, \(\text{SL}(2, \mathbb Z)\), elliptic curves \(E/\mathbb C\), elliptic modular curves…(the “classical side”) and \(A\), \(K\), \(K_\infty\), \(C\), \(\Omega\), \(\text{GL}(2, A)\), rank-two Drinfeld \(A\)-modules \(\phi/C\), Drinfeld modular curves…(the “Drinfeld side”). In particular, the \(C\)-points \(M_\Gamma(C)\) of a Drinfeld modular curve are in \(1-1\) correspondence with the isomorphism classes of \(\phi/C\) as above (with some level structure depending on \(\Gamma\)).
Far deeper than these formal similarities is Drinfeld’s reciprocity law [Theorem 2 in V. G. Drinfeld, Math. USSR, Sb. 23, 561–592 (1976); translation from Mat. Sb., New Ser. 94, 594–627 (1974; Zbl 0321.14014)]. It expresses the Galois representation associated with \(J_\Gamma\), the Jacobian of the compactification \(\overline M_\Gamma\) of \(M_\Gamma\), through automorphic data on \(\text{GL}(2, {\mathfrak A}_K)\), where \({\mathfrak A}_K\) is the adele ring of the ground field \(K\).
As follows from Drinfeld’s work (although it is nowhere explicitly stated), the analogue of the Shimura-Taniyama-Weil conjecture on the uniformization of elliptic curves through modular curves holds in our case:
STW/K: Each elliptic curve \(E/K\) with split multiplicative reduction at \(\infty\) is the quotient of a suitable Drinfeld modular curve \(\overline M_\Gamma\), or equivalently, appears up to isogeny in the Jacobian \(J_\Gamma\).
However, there are several important problems left open by Drinfeld’s work.
(A) The above results STW/K is a sheer existence statement. Of course, one would like to dispose of a construction that, given \(E/K\), produces a “Weil uniformization” \(p_E: \overline M_\Gamma\to E\). Equivalently, one would like to construct \(E\) (or some curve isogenous with \(E\)) out of the automorphic Hecke newform \(\varphi_E\), and to understand how properties of \(E\) are reflected in \(\varphi_E\) and vice versa.
(B) In the Drinfeld modular curve context, there are two different concepts that generalize classical modular forms, viz, automorphic forms, which are \(\mathbb C\)- or \(\mathbb Q_\ell\)- or \(\mathbb Q\)-valued functions on some adele groups, and Drinfeld modular forms, which are \(C\)-valued holomorphic functions on \(\Omega\). Both of these are needed for a full understanding of the curves \(\overline M_\Gamma\), so the question of their relationship arises.
These problems are closely related with the main result of the paper, the description (given in section 7) of \(J_\Gamma\) as a torus divided by some lattice. Satisfactory answers to both questions are given: See section 9, notably (9.6.1) for (A) (where an elliptic curve \(E\) is constructed from its newform \(\varphi_E\) by specifying the Tate period) and section 6, notably (6.5) for (B) (roughly speaking, Drinfeld modular forms of a certain type “are” the reductions mod \(p\) of \(\mathbb Z\)-valued automorphic forms).
The basic tool for the construction of \(J_\Gamma\) is the theory of theta functions for \(\Gamma\), i.e., of meromorphic functions on \(\Omega\) behaving nicely under \(\Gamma\) and at the “cusps” of \(\Gamma\backslash \Omega\). In the context of Schottky groups, these have been introduced by Manin-Drinfeld and studied by Gerritzen-van der Put and M. van der Put [Groupe Étude Anal. Ultramétrique 1981/82, Exp. No. 10 (1983; Zbl 0515.14027)].
For arithmetic groups \(\Gamma\subset \text{GL}(2, K)\) as above, several new problems arise, due to the non-compactness of \(\Gamma\backslash \Omega\) and the existence of torsion in \(\Gamma\). These are dealt with by a careful analysis of the relationship between modular data (on \(\Omega\) or \(\Gamma\backslash \Omega\)) and automorphic data (on \(\mathcal T\) or \(\Gamma\backslash {\mathcal T}\), where \(T\) is the Bruhat-Tits tree \(\mathcal T\) of \(\text{PGL}(2, K_\infty)\)). E.g. the absolute value of the theta pairing \((. , .)\) on \(\overline \Gamma= \Gamma^{ab}/\text{tor} (\Gamma^{ab})\) (which is a lattice in the \(\mathbb C\)-vector space \(\underline H_!({\mathcal T}, \mathbb C)^\Gamma\) of automorphic forms) turns out to agree with a conveniently normalized Petersson product (Theorem 5.7.1).
The result is as follows: For each \(\alpha\in \Gamma\), there exists a holomorphic theta function \(u_\alpha: \Omega\to C^*\) that satisfies \(u_\alpha(\beta z)= c_\alpha(\beta) u_\alpha(z)\). The pairing \(\Gamma\times \Gamma\to C^*\), \((\alpha, \beta)\mapsto c_\alpha(\beta)\) takes its values in \(K^*_\infty\) and induces via \(v_\infty: K^*_\infty\to \mathbb Z\) a symmetric pairing \((.,.): \overline\Gamma\times \overline\Gamma\to \mathbb Z\), which is positive definite by the above. Therefore \[ \begin{aligned} \overline c: \overline\Gamma\quad &\to \quad \operatorname{Hom}(\overline\Gamma, C^*):= T_\Gamma(C)\\ \text{class of } \alpha\quad &\mapsto c_\alpha\end{aligned} \] is injective, and the torus \(T_\Gamma\) divided by \(\overline c(\overline\Gamma)\) is an Abelian variety defined over \(K_\infty\), which happens to agree with \(J_\Gamma/K_\infty\) (Theorem 7.4.1).
As an application of the construction, it is shown how to obtain the strong Weil curve \(E_\varphi\) of a normalized rational Hecke eigenform \(\varphi\in \overline\Gamma \hookrightarrow \underline H_!({\mathcal T}, \mathbb C)^\Gamma\). In the commutative diagram \[ \begin{matrix} 1 & \rightarrow & \overline\Gamma & \rightarrow & T_\Gamma(C) & \rightarrow & J_\Gamma(C) & \rightarrow & 0\\ && \downarrow && \downarrow ev && \downarrow pr_\varphi\\ 1 & \rightarrow & \Lambda & \rightarrow & C^* & \rightarrow & C^*/\Lambda & \rightarrow & 0,\end{matrix} \] let the middle vertical arrow be defined by \(f\mapsto f(\varphi)\) and \(\Lambda:= ev(\overline\Gamma)\). Then \(\Lambda\subset K^*_\infty\) is a lattice, \(C^*/\Lambda= E_\varphi(C)\), and \(pr_\varphi\) is the strong Weil uniformization searched for.
As is shown in subsequent work of the first author [Analytical construction of Weil curves over function fields, J. Théor. Nombres Bordx 7, No. 1, 27–49 (1995; Zbl 0846.11037)], the degree of \(pr_\varphi\) and the valuation of the invariant \(j(E_\varphi)\) may be read off from the position of \(\varphi\) in the lattice \((\overline\Gamma, (. ,.))\), at least if the base ring \(A\) is a polynomial ring \(\mathbb F_q[T]\).

11G09 Drinfel’d modules; higher-dimensional motives, etc.
14G35 Modular and Shimura varieties
14H40 Jacobians, Prym varieties
11G05 Elliptic curves over global fields
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