Title:
Explicit isogenies of prime degree over quadratic fields
Abstract:
Let K be a quadratic field which is not an imaginary quadratic
field of class number one. We describe an algorithm to compute a
superset of the set of primes p for which there exists an elliptic
curve over K admitting a K-rational p-isogeny. Combining this algorithm
with recent work on the determination of quadratic points on low-genus
modular curves, we determine - conditional upon the Generalised Riemann
Hypothesis - the above set of isogeny primes for several quadratic
fields, providing the first such examples after Mazur's 1978
determination for K = Q. We will give a live demo of the Sage and
PARI/GP implementations of the algorithm.