The St\'ephanois theorem with only prime isogenies

Abstract

We present a strengthening of the proof of the St\'ephanois theorem. We follow the modular version by Waldschmidt, which is based in a suggestion by Daniel Bertrand, but it also applies to the original proof. The improvement is not in the result or the conditions, but in the need of weaker tools on the proof itself. More precisely, we only employ modular polynomials of prime degree, instead of polynomials of arbitrary level. Furthermore, one can restrict to primes in fixed arithmetic sequence. On the proof itself, the only crucial difference appears in Cinqui\`eme pas and on the final contradiction in Septi\`eme pas of Waldschmidt's proof, but for readability, we present a complete proof with this modification. This is part of a larger project to generalize the St\'ephanois theorem to the Igusa invariants of curves of genus two, as the Siegel modular polynomials in the literature are usually only considered for prime levels. The material is part of Chapter 7 of the author's PhD thesis.