Let \(E\) be an elliptic curve over the rational function field \({\mathbb F}_q(t)\) with non-constant \(j\)-invariant \(j_E\). A theorem by J. Pesenti and L. Szpiro, proved in [Compositio Math. 120, No. 1, 83–117 (2000; Zbl 1021.11021)] for general function fields, says that in this special case the minimal discriminant \({\mathcal D}_E\) of \(E\) can be bounded by \[ \deg {\mathcal D}_E\leq 6 \deg_{ns}(j_E)\cdot (\deg {\mathfrak n}_E -2) \] where \({\mathfrak n}_E\) is the conductor of \(E\) and \(\deg_{ns}(j_E)\) is the inseparability degree of the map \(j_E\).
In general it is not possible to get rid of the annoying factor \(\deg_{ns}(j_E)\), as can be easily seen by applying the Frobenius isogeny to a given curve \(E\).
Now suppose that \(E\) has split multiplicative reduction at the place \(\infty=\frac{1}{t}\) of \({\mathbb F}_q(t)\). Then it is known that there exists a non-constant map from the Drinfeld modular curve \(X_0({\mathfrak n})\) to \(E\) where \({\mathfrak n}\in{\mathbb F}_q[t]\) with \({\mathfrak n}_E=(\infty)\cdot({\mathfrak n})\).
\(E\) is called a strong Weil curve (or an optimal elliptic curve in this paper) if this map does not factor through another elliptic curve.
The main results of the paper are about such optimal elliptic curves, namely: Theorem 1.2 which says that if \({\mathfrak n}\) is irreducible, then \(j_E\) is separable and hence \[ \deg {\mathcal D}_E\leq 6(\deg {\mathfrak n} -1); \] and Theorem 1.3 which for general \({\mathfrak n}\) gives an upper bound for \(\deg_{ns}(j_E)\) (and hence for \(\deg {\mathcal D}_E\)) only in terms of \(q\) and \(\deg {\mathfrak n}\).
The author shows that if \({\mathfrak n}\) is irreducible, then \(ord_{\mathfrak n}(j_E)\) is not divisible by the characteristic of \({\mathbb F}_q\). The proof of this easily stated fact (which immediately implies Theorem 1.2) is obtained from the rigid-analytic uniformization of the Jacobian of \(X_0({\mathfrak n})\) over the \({\mathfrak n}\)-adic completion of \({\mathbb F}_q(t)\) and requires high-powered machinery.
Theorem 1.3 is proved by relating \(ord_{\infty}(j_E)\) to the degree of the uniformization \(X_0({\mathfrak n})\to E\) and a special value of an \(L\)-function, which is estimated in an appendix.
We also mention that in a subsequent paper [Abelian subvarieties of Drinfeld Jacobians and congruences modulo the characteristic, preprint Saarbrücken (2005)] the author proves, among other results, that Theorem 1.2 remains true under the weaker assumption that \({\mathfrak n}\) is square-free, i.e. for all semistable optimal elliptic curves curves over \({\mathbb F}_q(t)\).
On the other hand, examples show that \(j_E\) need not be separable if \({\mathfrak n}\) has a multiple factor.