Minimal number of points with bad reduction for elliptic curves over \(\mathbb{P}^1\) (Q2439541)

By classical results, there is no elliptic curve over \({\mathbb Q}\) and, more generally, no abelian variety over \({\mathbb Q}\) (cf. [\textit{J. M. Fontaine} Invent. Math. 81, No. 3, 515--538 (1985; Zbl 0612.14043)]) with good reduction everywhere. \textit{R. Schoof} [Compos. Math. 141, No. 4, 847--868 (2005; Zbl 1173.11333)] has answered the question about the existence of non-zero abelian varieties over \({\mathbb Q}\) with bad reduction at just one prime. In the case of function fields, there are examples of elliptic curves with good reduction everywhere coming from elliptic curves over the field of constants; however, with some additional hypothesis, one can obtain results similar to the previous ones. Let \(k\) be an algebraically closed field \(E\) be an elliptic curve over \(k(T)\). The authors proves the following two theorems: { Theorem 1}\ \ Let \(\text{char}(k)\neq 2,3\). (a) If \(E\) has a non-constant \(j\)-invariant, then there are at least three points of bad reduction; (b) if \(E\) is non-constant, then there are at least two points of bad recuction. { Theorem 2}\ \ Let \(\text{char}(k)\) be 2 or 3. (a) If \(E\) has a non-constant \(j\)-invariant, then there are at least two points of bad reduction; (b) if \(E\) is non-constant, then there is at least one point of bad reduction. \smallskip Both Theorems 1 and 2 are sharp. The main tool for the proof of Theorem 1 is \textit{Mason}'s \(abc\)-inequality [London Mathematical Society Lecture Note Series, 96, Cambridge (1984; Zbl 0533.10012)], while for the proof of Theorem 2 the author applies an algebraic lemma to the study of a globally minimal Weierstrass equation of \(E\).