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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 $$\mathbb F_q[T]$$.

MSC:
 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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