Strong Weil curves over Fq(T) with small conductor

Abstract

We continue work of Gekeler and others on elliptic curves over F q (T ) with conductor ∞ · n where n ∈ F q [T ] has degree 3. Because of the Frobenius isogeny there are infinitely many curves in each isogeny class, and we discuss in particular which of these curves is the strong Weil curve with respect to the uniformization by the Drinfeld modular curve X 0 (n). As a corollary we obtain that the strong Weil curve E/F q (T ) always gives a rational elliptic surface over F q .The first paper that systematically used Drinfeld modular curves and the Bruhat-Tits tree in order to classify elliptic curves over a function field was [Ge1].With an eye towards feasibility of practical calculations, what we are talking about here are elliptic curves over a rational function field F q (T ) and Drinfeld modular curves X 0 (n) for n ∈ F q [T ]. Every elliptic curve over F q (T ) that is modular, that is, covered by some X 0 (n), must have split multiplicative reduction at the place ∞ (= pole divisor of T ). Conversely, every elliptic curve over F q (T ) with conductor ∞ · n and split multiplicative reduction at ∞ is an isogeny factor of the Jacobian of X 0 (n).