On the torsion of optimal elliptic curves over function fields (Q867366)

Let \(F = {\mathbb F}_q(t)\) be the rational function field in an indeterminate \(t\) over the finite field \({\mathbb F}_q\) with \(q\) elements, and let \(E/F\) be an elliptic curve with conductor \({\mathfrak p}\infty\), where \({\mathfrak p}\) (resp. \(\infty\)) is a finite (resp. the infinite) place of \(F\), and such that \(E\) has split multiplicative reduction at \(\infty\). It is known that such an elliptic curve \(E\) is isogenous to a factor of \(J = J_0(\mathfrak p)\), the Jacobian of the Drinfeld modular curve \(X_0(\mathfrak p)\) with conductor \({\mathfrak p}\). Such an \(E\) is optimal if it is even isomorphic to an abelian subvariety of \(J\), or equivalently, if there exists a morphism \(J \longrightarrow E\) with connected and smooth kernel. The main result (Theorem 1.1) of the paper states: (1) The induced homomorphism \(J(F)_{\text{ tor}} \longrightarrow E(F)_{\text{ tor}}\) on torsion subgroups is surjective (since \(J(F)_{\text{ tor}}\) is generated by the cusps, the group \(J(F)_{\text{ tor}}\) is well-known, cf. [4], [10]); (2) the specialization map from \(E(F)_{\text{ tor}}\) to \(\Phi_{E,{\mathfrak p}}\), the component group of the Néron model of \(E\) at \(p\), is an isomorphism; (3) \(E(F)_{\text{ tor}} = {\mathbb Z}/n{\mathbb Z}\) for some \(1 \leq n \leq 3\) (and subject to some further conditions). The proof uses arithmetical properties (established by Pal in [16]) of the Eisenstein ideal of the associated Hecke algebra and results of A. Schweizer's Ph.D. thesis (Saarbrücken 1996).