| A genus 2 curve whose Jacobian has complex multiplication may have (stable) bad reduction, though the Jacobian has potential good reduction. This phenomenon is the main subject of this talk; it has relevance to the construction of units in abelian extensions of quartic CM fields (motivated by Stark's conjecture) and to computation of class polynomials (motivated by applications to cryptography), for example, and bears interesting relation to recent work of Bruinier-Yang (generalizing work by Gross-Zagier). I shall described recent results, joint with Kristin Lauter (Microsoft Research), and some numerical evidence by Daniel Vallieres (McGill University), and explain the motivation mentioned above. Time allowing, I shall discuss the proofs. |