| 概要 | Let E/Q be an elliptic curve. A pair of primes (p,q) is called an amicable pair for E if #E(F_p) = q and #E(F_q) = p. More generally, an elliptic aliquot cycle is a list of primes (p_1,...,p_n) satisfying an analogous condition. In this talk I will discuss properties of elliptic amicable pairs and aliquot cycles and explain how they appear in the study of elliptic divisibility sequences. In particular, I will explain why the CM and non-CM cases are strikingly different, and will discuss in detail the strange case of j=0, whose resolution required counting points over finite fields on certain curves of genus 4. This material is joint work with Katherine Stange. |