Analytical construction of Weil curves over function fields
The Shimura-Taniyama-Weil conjecture is valid for function fields \(K\) of one variable over a finite field. It is proved by combining deep results of Grothendieck, Jacquet-Langlands and Drinfeld [for details see the author and the reviewer, J. Reine Angew. Math. 476, 27--93 (1996; Zbl 0848.11029); cited [G-R] below]. Let \(\Gamma\) be an arithmetic subgroup of \(\text{GL}_2 (K)\). One knows, following [G-R], that \(\overline \Gamma = \Gamma^{ab}/ \Gamma_{tors}\) can be canonically interpreted as a \(\mathbb Z\)-module of automorphic forms for a convenient compact open subgroup of the adelization of \(\text{GL}_2 (K)\), with values in \(\mathbb Z\) and that transform like the special representation (and, up to tensorization by \(\mathbb Q\), all such automorphic forms can be recovered in that way). Moreover, there exists on \(\overline \Gamma \times \overline \Gamma\) a pairing, coming from the existence of theta functions (see [G-R]), that agrees with the (conveniently normalized) Petersson scalar product. Let \(\varphi \in \overline \Gamma\) be a normalized Hecke eigenform; we have a surjective morphism (the Weil uniformization) \(p_\varphi : J_\Gamma \to E_\varphi\) where \(J_\Gamma\) is the Jacobian of the modular curve associated to \(\Gamma\) and \(E_\varphi\) is the strong Weil curve with respect to \(\varphi\), this morphism being compatible with the uniformizations of \(J_\Gamma\) and \(E_\varphi\) at the place \(\infty\) of \(K\) ([G-R]). The author gives new main information about the Weil uniformization \(p_\varphi\), at least when the base field is \(K = \mathbb F_q (T)\). In particular, he proves that the degree of \(p_\varphi\) and the valuation of the modular invariant \(j (E_\varphi)\) can be calculated from the locus of \(\varphi\) in the lattice \(\overline \Gamma\) (equipped by its pairing). Enlightening examples are given in the last paragraph.
scientific article; zbMATH DE number 858013
scientific article