Title: Amicable pairs and aliquot cycles for elliptic curves
Abstract: An amicable pair for an elliptic curve E/Q is a pair of primes (p,q) of good reduction for E satisfying #E(Fp) = q and #E(Fq) = p. Aliquot cycles are analogously defined longer cycles. Although rare for non-CM curves, amicable pairs are -- surprisingly -- relatively abundant in the CM case. We present heuristics and conjectures for the frequency of amicable pairs and aliquot cycles, and some results for the CM case (including the especially intricate j=0 case). We present some open problems and computational challenges arising from this work. This is joint work with Joseph H. Silverman.
Project
- smalljac
make smalljac code usable from Python (involves Cython); see this psage issue.
- use code:
- - replicate and extend data in Kate's talk - maybe try genus 2 analogue?
- cubic and sextic residue symbol
there's a ticket that has only partial implementation (cubic residue of rational prime and element of Q(sqrt(-3))) -- not at all a general implementation
- there are artin symbols etc. -- big machinery
m-th residue symbol implemented: number_field_ideal.py, number_field_element.pyx (old: code.sws)
some tests: residue_symbol_test.sws
- SAGE computes Kronecker symbol using GMP
- explicit calculation of Grossencharacters (aka Hecke characters).