On $p$-isogenies of elliptic curves with multiplicative reduction
George Ţurcaş
Babeş-Bolyai University, Cluj-Napoca, Romania
Abstract:
Given a fixed number field $K$, for which primes $p$ does there exist an elliptic curve an elliptic curve $E$, defined over $K$, admitting a $K$-rational $p$-isogeny and which has multiplicative reduction at the primes above some other prime $q$? We report on recent findings regarding the previous question, focusing on the case when K is quadratic. Time permitting, we will explain how these results, asserting non-existence of $p$-isogenies for certain elliptic curves, have applications to Diophantine equations.